Name Period Divisibility Rules Algebra 9: Fill & Download for Free

To echo Tim Morgan's epic answer: all of them? There are lots and lots of ways to represent real numbers. I'll try to cover some of the more important ones, and survey a few of the exotic.Before we talk about representing real numbers, we should remind ourselves about natural numbers and their representations, rational numbers and their representations, and finally we will get to real numbers and their representations.Remember that natural numbers are also real numbers, so we're not wasting our time when we talk about natural numbers - it's just a special case. It's also a warm-up exercise.To learn a lot more about representing numbers and doing arithmetic using those representations, please do read Chapter 4 of one of the best books ever written, Knuth's The Art of Computer Programming.Natural NumbersThe natural numbers are simply the ordinary counting numbers 1, 2, 3 and so on. They already provide a good case study for overcoming the greatest psychological hurdle this answer tries to clear up:The same object can have many representations. The object and its representations are related but distinct entities.We're so used to the decimal notation of natural numbers that we tend to forget that the number 23 and the representation "23" are two different things.The number represented in decimal notation as "23" tells us how many circles there are here:oooooooooooooooooooooooThis number has lots of properties. It is odd, it is prime, in fact it is a regular prime, it is positive, it is greater than (the number represented by) 10, it is two less than a square and four less than a cube. All of these are properties of the number itself, in whichever form we choose to represent it.One thing this number doesn't have is any special relationship with the numbers 2 and 3. We use those numbers to represent it in decimal notation because this is our favorite notation that we've been using for centuries, but the number 23 and the numbers 2 and 3 have not much to do with each other. The reason the digits 2 and 3 are used is because[math]23 = 2 \times 10 + 3[/math]and this is how the decimal notation works: the number on the right counts 1's, the number next to it counts 10's, the next number counts 100's, and so on. But "10" doesn't have any special meaning in math or real life, so those 2 and 3 aren't especially important for 23.We use the decimal notation probably because we have ten fingers. It is also a fairly useful notation, because it only requires ten digits (as opposed to base 60, for instance, which requires sixty), and the representations are fairly compact (the year 2015 in binary is 11111011111 - a palindrome, but a long one), and the positional representation makes addition and multiplication fairly easy.What are other ways to represent natural numbers?We can use unary notation. Here we only use one digit, say 1, and we repeat it just as many times as the number we wish to represent:1: 12: 115: 1111123: 111111111111111111111112015: Er...no.One advantage of the unary system is that it makes it super easy to do addition: you just put one number next to the other.11111111 + 11111=1111111111111.Compare this to 8+5=13, which is rather tricky. Where did the "1" and the "3" come from?Different ways to represent numbers make certain tasks easier or harder.We can use binary notation, which is just like decimal notation only using powers of 2 instead of powers of 10. The various digits now represent ones, twos, fours, eights, sixteens, etc., instead of ones, tens, hundreds etc.23: 10111because if you take one 1, one 2, one 4 and one 16 (but no 8) and add them together you get the number which in decimal notation we call 23.We can similarly use ternary (base 3) notation, base 4, base 12, base 16, base 900 or any other base we want. The principle is always the same. In base 16, for example, twenty three is denoted "17". Yes! See how "2" and "3" are unimportant? The very same number now carries the digits "1" and "7".And in base 15 (fifteen), the same number is represented as 18. Wait, is it now an even number? No, of course not. Being even or odd is a property of the number. If you take 23 dots and try to arrange them in two rows, there'll be one left over. The number is odd, no matter what base you use.oooooooooooooooooooooooIt's just that in base 15, having an even digit on the right isn't the right way to tell if the number is even or odd. Just like in base 10 you can't tell if a number is divisible by 3 just by looking at the last digit.The ancient Babylonians used a system that was roughly positional (like our decimal system) but relied on base 60. This is why we still count seconds and minutes (for time) and seconds and minutes (in navigation, as angles) in sixties.The Romans used a system based on letters, as you probably know. 1 is I, 2 is II, 3 is III, 5 is V, 10 is X, 50 is L, 100 is C, 500 is D and 1,000 is M. To save space (sorta) they used IV for 4 instead of IIII (although IIII was also used early on), and similarly IX for 9, XC for 90 and so on. The year 1999 would be written MCMXCIX, which is about as confusing as it gets. For some reason, to this day movie credits often indicate the year in Roman numerals, as if this has more gravitas. The results are sometimes disastrous:As you can imagine, doing arithmetic with Roman numerals is torture, and it's not even clear how to represent large numbers. When the world gradually shifted to the modern base-10 positional system, many people had a hard time letting go of Roman numerals. Conway and Guy point out that there are records of European merchants writing things like M50iv for the year 1504.Are there other ways to represent natural numbers? Sure there are.We can use negative bases, such as (-10).[math]123_{-10} = 1 \times (-10)^2 + 2 \times (-10) + 3 = 83_{10}[/math](here I'm using the subscript 10 or -10 to indicate which base is being used). One cute feature of negative bases is that you can represent any whole number, either positive or negative, without a sign. In 1955, the inimitable Don Knuth wrote a paper on negative-radix systems (and complex-radix ones) in a science contest paper. One of his systems, the Quater-imaginary base, can easily be used to produce images like this Dragon fractal.In The Art of Computer Programming, Knuth discusses many other positional systems. One of his favorites is balanced ternary notation, which uses base 3 but with the digits [math]1, 0, \bar{1}[/math] instead of [math]0,1,2[/math], where [math]\bar{1}[/math] represents negative unity (-1). So for instance[math]1\bar{1}1_{\mbox{bt}} = [/math][math]9[/math][math] - 3 + 1 = 7[/math].We can also use systems where the value of each digit is not a power of some fixed base. For example, in this answer a system is discussed where the value of the [math]k[/math]th digit is [math]k![/math]. This means that the allowed values for each digit change along the number: the [math]k[/math]th digit can be anything between 0 and [math]k[/math]. For example,[math]221_! = 1 \times 1! + 2 \times 2! + 2 \times 3! = 17[/math].In 1972, Edouard Zeckendorf proved that every natural number can be represented uniquely as a sum of non-consecutive Fibonacci numbers. For example,[math]100 = 89 + 8 + 3[/math]which can be used to represent 100 as [math]1000010100_{Z}[/math], where the 1's indicate which Fibonacci numbers to use. This representation never uses two consecutive 1's.Some exotic representations of natural numbers have actual utility in mathematics. An important result in the combinatorics of convex polytopes (McMullen's conditions for [math]f[/math]-vectors of simplicial polytopes) is phrased and proved using a representation relying on binomial coefficients.Rational NumbersRational numbers are ratios of whole numbers, such as [math]\frac{1}{2}[/math] or [math]\frac{355}{113}[/math].By far the very best way to represent a rational number is just like this: use any way to represent the numerator and denominator as natural numbers (with a possible sign), and indicate that the rational number is their ratio.Why is this the best way? Because it captures the rational number precisely, as it is. When people use things like [math]33.3\% = 0.333[/math] to represent "one third", they are only using an approximation, which introduces errors,Many computers use a Floating point system to represent numbers, and this system typically only supports denominators which are powers of 10 (or 2, or some other base, but the damage is similar). This is quite terrible, since many numbers which could be represented completely accurately are only represented approximately. Whenever you actually need to work with rational numbers, you should use packages such as Python's fractions, which represent rational numbers as pairs of integers.One disadvantage of the ratio representations is that it's not unique. The same number, say [math]\frac{1}{3}[/math], can also be represented as [math]\frac{7}{21}[/math] or [math]\frac{100}{300}[/math]. In other words, two representations can be equal even if they look different. Beginning students often find this confusing: they are used to having two things equal only when they look exactly the same. But that's just not how things work.In school we learn to use "decimal notation" to represent numbers, like [math]\frac{1}{4} = 0.25[/math]. We tend to use this a lot, particularly in the form of percentages (a quarter is 25%). This is a fine thing to do, but let's remember a few things:First, the decimal representation is sometimes infinite, though for rational numbers the sequence of digits is always repeating. For instance,[math]\frac{1}{6} = 0.16666\ldots[/math]Second, some (not all) numbers have non-unique decimal representations:[math]1 = 0.9999\ldots[/math][math]0.25 = 0.249999\ldots[/math]Those are not major problems, but they do get many children (and adults) confused.Once again, we have an opportunity to distinguish properties of a number from properties of the representation of the number.A rational number may be an integer (or not), it may be smaller than 1 (or not), it may or may not be a square (of a rational number), and so on.A rational number cannot be "terminating" or "have period 7". Those are properties of various representations of the number.Real NumbersReal numbers are limits of certain sequences of rational numbers, called Cauchy sequences. This definition may seem unfamiliar, but it's important to get a sense of what it means. If you have a sequence of rational numbers, such as[math]1, \frac{14}{10}, \frac{141}{100}, \frac{1414}{1000}, \ldots[/math]you can check if those numbers keep getting closer to one another. This means that if someone challenges you make them close to 1/100, you can skip some of the first few numbers and get to a place where all of them are no more than 1/100 apart, and you can do this with a challenge of 1/1000 by skipping more initial numbers, or 1/1,000,000, or any such challenge. This is a Cauchy sequence.If you have such a sequence, there's a real number which is the limit of that sequence. It is the number - which may itself be rational, or not - which the sequence gets closer and closer to. This is one of the main ways real numbers are defined in modern math.And this is, in some ways, bad news.If I have a real number and I want to tell you what it is, I need to give you infinitely many rational numbers. Since communicating a rational number takes at least some minimal amount of time, this will require an infinitely long time. In most civilized societies it is considered very rude to take up an infinitely long amount of someone's time.You may have seen things like this written down:[math]\pi = 3.14159265...[/math]The number on the left, [math]\pi[/math], is a name we gave to some specific real number. But the number on the right is not fully specified. It gives some information about [math]\pi[/math], but not all of it - and in fact, for most real numbers, you can't get all of it.All this equation tells you is that [math]3.14159265 \leq \pi \leq 3.14159266[/math]. This is good information, and for many purposes it is sufficient: if you need to calculate the length of the circumference of some circular thing, you probably can get by with much less precise info. But just writing "three point one four one five nine dot dot dot" does not pinpoint a real number.In some cases, we can describe an infinite sequence of rational numbers using a finite description. For example, consider this sequence:[math]\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \ldots[/math]The rule here is very simple: in each fraction, the denominator is the numerator of the previous one, and the numerator is the sum of the previous numerator and denominator. So the next fraction is going to be [math]\frac{8+5}{8}=\frac{13}{8}[/math].This is a Cauchy sequence: the numbers get closer and closer. Its limit is a familiar number, the golden ratio [math]\frac{1+\sqrt{5}}{2}[/math]. So in this particular case we have a real number that does accommodate a finite description, in fact two of them: one as a limit of a finitely-describable sequence of rationals, and one as an expression involving arithmetical operations and square roots.Many familiar real numbers such as [math]\pi[/math], [math]e[/math], [math]\log_2(5)[/math] and [math]\sin(\pi/17)[/math] allow such finite descriptions, but they are outliers. Most real numbers cannot be finitely described in any way. If I have a secret real number [math]\zeta[/math], and I tell you that[math]\zeta = 4.669201609...[/math]I didn't really tell you what [math]\zeta[/math] is. You don't know if it's rational or irrational, you can't tell if it's a algebraic or not, all you know is its rough magnitude. If I ask you what is the nearest integer to [math]10^{100}\zeta[/math], you have no idea. You don't know what my number [math]\zeta[/math] is, and unless some very special circumstances are in place, there's no way for me to help you even if I wanted to.Therefore, a "representation" of a real number must be one of two things:Partial, providing some incomplete information about the number; orInfinitely long, and allowing a finite presentation only in some very special cases (i.e. for computable numbers.)So how do we represent real numbers, in either of these forms?Just like with natural numbers, we can use a positional number system in some base. When we write[math]\pi = 3.14\ldots[/math]We mean to say that[math]\pi = 3 \times 1 + 1 \times \frac{1}{10} + 4 \times \frac{1}{100} + [/math] a whole lot of things which we're not telling you, but they're altogether smaller than [math]\frac{1}{100}[/math].Of course, we can do the same things using other bases:[math]\pi = 11.001001000011_2\ldots[/math][math]\pi = 10.010211_3\ldots[/math][math]\pi = 3.243F6A88_{16}\ldots[/math](note that in base 16, or hexadecimal, we need 16 digits, so we typically use the numerals 0-9 and then the letters A-F). These examples serve as a good reminder that the "mystical" digits 3, 1, 4, 1, 5, etc. have nothing in particular to do with the essence of the number [math]\pi[/math].None of those representations will terminate, or repeat. Why? Because we know that [math]\pi[/math] is an irrational number - in fact, a transcendental one. Remember, this is a property of the number, having nothing to do with any particular representation. For that reason it makes no sense to ask if [math]\pi[/math] has a "rational representation" - it's not a rational number.Another very important representation of real numbers is the continued fraction representation.Pick up your calculator - or actually, your smartphone, and then launch the calculator app. If you're using an iPhone, turn it sideways so that the app becomes actually useful. Then hit the button labeled "[math]\pi[/math]".The value is 3 point something, so now let's subtract 3 to get a number less than 1.Now hit the "1/x" button to get a big number again.Now repeat! You have 7 point something something, so take away 7......and hit 1/x.If you retrace your steps, you'll see that so far you've shown that[math]\pi = 3 + \frac{1}{7 + \frac{1}{15 + \ldots}}[/math]and of course you can keep going:[math]\pi = 3 + \frac{1}{7 + \frac{1}{15 + \frac{1}{1 + \frac{1}{292 + \ldots}}}}[/math]This is the continued fraction representation of [math]\pi[/math], and to save clutter it is often written like this:[math]\pi = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, \ldots][/math]Since [math]\frac{1}{292}[/math] is a pretty small number, we know that we can get something pretty close to [math]\pi[/math] if we stop right there:[math]\pi \approx 3+\frac{1}{7+\frac{1}{15+\frac{1}{1}}} = \frac{355}{113}[/math].This is a really good rational approximation of [math]\pi[/math]. In general the continued fraction produces the "best" rational approximations of a number, in some precise sense.It's really fun to try this with some other numbers on your calculator. If you try square roots, you'll get continued fractions which are repeating, just like the base-[math]d[/math] representation of rational numbers.[math]\sqrt{2} = [1; 2, 2, 2, 2, \ldots][/math][math]\frac{1+\sqrt{5}}{2} = [1; 1, 1, 1, 1, \ldots][/math][math]\sqrt{23}=[4;1,3,1,8,1,3,1,8,\ldots][/math]Now try [math]e[/math].[math]e = [2;1,\, 2,1,1, \, 4,1,1,\, 6,1,1, \, 8,1,1, \ldots][/math]Lovely isn't it? The continued fraction isn't repeating (only quadratic irrationalities are repeating) but it does have a very simple pattern.No such pattern is known for [math]\pi[/math]. In fact we know very little about the continued fraction expansion of [math]\pi[/math], and that of most other real numbers. Remember though that we know precious little about the decimal expansion of [math]\pi[/math] as well. Nobody knows if it ever contains a run of 30 consecutive 7's in a row, for example.Most of the time we don't actually represent real numbers in decimal notation or continued fraction form. We often work with real numbers that have some concrete representation as a limit (or an infinite series, which is the same thing), or an algebraic combination of other numbers.[math]\pi = 4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\ldots)[/math][math]e = \frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots[/math][math]x = e^{\pi \sqrt{163}}[/math]Many people feel that those representations are insufficient in some sense; you find many questions in the classroom and online to the effect of "Yes but how do you compute that?" or "but this doesn't tell you what the number actually is". It's useful to understand that such representations are in no way inferior to the decimal representation we're so used to. In fact, [math]\pi = 3.1415926...[/math] is a great deal less informative than[math]\pi=2 \sum_{k=0}^\infty \frac{2^k k!^2}{(2k+1)!}[/math]Those are some of the ways real numbers can be represented.

I don’t know how to have a complete answer for this. But to a certain extent, the research I did on Vedic Mathematics impels me to say that they make sense. And here is what I came across in my research.Vedic MathematicsIn many ancient cultures and civilizations, the development of mathematics was necessitated on account of religious practices and observances. These required an accurate calculation of the times of certain festivals and of the times auspicious for the performances of certain sacrifices and rituals. They also required a correct knowledge of the times of rising and the setting of the sun and the moon, and of the occurrences of the solar and lunar eclipses. All these meant a good knowledge of arithmetic, plane and spherical geometry and trigonometry, and possibly also the know-how of the construction of simple astronomical instruments.[1]The impression that science started only in Europe was deeply embedded in the minds of educated people all over the world until recently. The alchemists of Arab countries were occasionally mentioned, but there was very little reference to India and China. Thanks to the work of the Indian National Science Academy and other learned bodies, the development of science in India during both the ancient and medieval periods has recently been studied. It is becoming clearer from these studies that India has consistently been a scientific country right from Vedic to modern times with the usual fluctuations that can been expected of any country. In fact, a research will not find another civilization except that of ancient Greece, which accorded an exalted place to knowledge and science as in India.[2]It has to be universally acknowledged that much of mathematical knowledge in the world originated in India and moved from East to West. The high degree of sophistication in the use of mathematical symbols and developments in arithmetic, algebra, trigonometry, especially the work attributed to Aryabhatta, is indeed remarkable and should be a source of inspiration to all of us in India. The articles which describe Indian contributions to science from the ancient times to the very modern period bring out quite clearly the continuity of scientific thought as part of our cultural heritage. It is however, unfortunate that the period of decline in India coincided with that of ascendancy of Europe. It is perhaps the contrast during this period that made Europeans believe that all modern science was European.[3]An impending danger which I recognized as part of this study is the fact that Vedic Mathematics is now understood and taught in some institutions merely as an alternate method in facilitating easy and quick mental calculations. So, my study is twofold in nature. The first goal is to have an integral understanding about the domain of Vedic mathematics and second is to have a better understanding about the misconceived notion about Vedic Mathematics, which is becoming prevalent as part of some political or religious agenda. This is a very subjective and personal inquiry towards the disciplines dealt with in Vedic Mathematics which had substantial impact in the Mathematical pursuits of probably all the Indian Mathematicians and Schools of Mathematics.History and Development of Vedic MathematicsThe mathematical tradition in India goes back at least to the Vedas. For compositions with a broad scope covering all aspects of live, spiritual as well as secular, the Vedas show a great fascination for large numbers. As the transmission of the knowledge was oral the numbers were not written, but they were expressed as combinations of powers of 10, and it would be reasonable to believe that when the decimal place value system for written numbers came into being it owed a great deal to the way numbers were discussed in the older compositions.[4] (See the second section of this chapter)It is well known that Geometry was pursued in India in the context of construction of vedis[5] for the yaj̀nas of the Vedic period. The Śulvasūtras[6]contain elaborate descriptions of constructions of vedis and also enunciate various geometric principles. These were composed in the first millennium BC, the earliest Baudhāyana Śulvasūtra dating back to about 800 BC. The Śulvasūtra geometry did not go very far in comparison to the Euclidean geometry developed by the Greeks, who appeared on the scene a little later, in the seventh century BC. It was however an important stage of development in India too. The Śulvasūtra geometers were aware, among other things, of what is now called the Pythagorean theorem, over two hundred years before Pythagoras (all the four major Śulvasūtras contain explicit statement of the theorem), addressed (within the framework of their geometry) issues such as finding a circle with the same area as a square and vice versa and worked out a very good approximation to the square root of 2, in the course of their studies.[7]Though it is generally not recognized, the Śulvasūtras geometry was itself evolving. This is seen in particular from the differences in the contents of the four major extant Śulvasūtras. Certain revisions are especially striking. For instance, in the early Śulvasūtras period the ratio of the circumference to the diameter was, like in other ancient cultures, thought to be 3, as seen in the Sūtra of Baudhāyana, but in the Manava Śulvasūtra a new value was proposed, as 3 and one-fifth; interestingly the Sūtra describing it ends with an exultation “not a hair breadth remains”, and though we see that it is still substantially off the mark, it is a gratifying instance of an advance made. In the Manava Śulvasūtra one also finds an improvement over the method described by Baudhāyana for finding the circle with the same are as that of a given square.[8]Vedic Hindus evinced special interest in two particular branches of Mathematics, namely geometry (Śulva) and astronomy (Jyotiṣa). Sacrifice (Yaj̀na) was their prime religious avocation. Each sacrifice has to be performed on an altar of prescribed size and shape. They were very strict regarding this and thought that even a slight irregularity in the form and size of the altar would nullify the object of the whole ritual and might even lead to adverse effect. So, the greatest care was taken to have the right shape and size of the sacrificial altar. Thus the problems of geometry and consequently the science of geometry originated.[9]As it is evident, available sources of Vedic Mathematics are very poor. Almost all works on the subject have perished. At present we find only a very short treatise on Vedic astronomy in three rescensions, namely, in Ārca Jyotiṣa, Yājuṣa Jyotiṣa and Atharva Jyotiṣa. There are six small treatises on Vedic Geometry belonging to the six schools of the Vedas. Thus for an insight into Vedic Mathematics, we have to now depend more on secondary sources such as the literary works.[10]The study of astronomy began and developed chiefly out of the necessity for fixing the proper time for the sacrifice. This origin of the sciences as an aid to religion is not at all unnatural, for it’s generally found that the interest of a people in a particular branch of knowledge, in all climes and times, has been aroused and guided by specific reasons. In the case of the Vedic Hindu that specific reason was religious. In the course of time, however, those sciences outgrew their original purposes and came to be cultivated for their own sake.[11]The Chāndogya Upaniśad (VII.1.2, 4) mentions among other sciences the science of numbers (rāśi). In the Muņdaka Upaniṣad {I.2. 4-5} knowledge is classified as superior (parā) and inferior (aparā). In the second category is included the study of astronomy (jyotiṣa). In the Mahābhārata (XII.201) we come across a reference to the science of stellar motion (nakṣatragati). The term gaņita, meaning the science of calculation, also occurs copiously in Vedic literature. The Vedāńga jyotiṣa gives it the highest place of honour amongst all the sciences which form the Vedāńga. Thus it was said: ‘As are the crests on the heads of peacocks, as are the gems on the hoods of snakes, so is the gaņita at the top of the sciences known as the Vedāńga’. At that remote period gaņita included astronomy, arithmetic, and algebra, but not geometry. Geometry then belonged to a different group of sciences known as kalpa.[12]Vedic Astronomy – JyotiṣaVedāńga Jyotiṣa is one of the six ancillary branches of knowledge called the Ṣad- Vedāńgas, essentially dealing with astronomy. It must be remarked that although the word Jyotiṣa, in the modern common parlance, is used to mean predictive astrology, in the traditional literature, the word always meant the science of astronomy. Of course, mathematics was considered as a part of this subject. Vedāńga Jyotiṣa is the earliest Indian astronomical work available.[13]The purpose of Vedāńga Jyotiṣa was primarily to fix suitable times for performing the different sacrifices. It is found in two rescensions: the Ŗgveda Jyotiṣa and the Yajurveda Jyotiṣa. Though the contents of both the rescensions are the same, they differ in the number of verses contained in them. While the Ŗgvedic version contains only 36 verses, the Yajurvedic contains 44 verses. This difference in the number of verses is perhaps due to the addition of explanatory verses by the adhvaryu priests with whom it was in use.[14]Authorship and DateIn one of the verses of the Vedāńga Jyotiṣa, it is said, “I shall write on the lore of Time, as enunciated by sage Lagadha”. Therefore, the Vedāńga Jyotiṣa is attributed to Lagadha.[15]According to the text, at the time of Lagadha, the winter solstice was at the beginning of the constellation Śrāviṣṭhā (Delphini) and the summer solistice was at the midpoint of Āśleṣā. Since Vārāhamihira (505 – 587AD) stated that in his own time the summer solstice was at the end of the first quarter of Punarvasu and the winter solstice at the end of the first quarter of Uttarāṣāḍhā, There had been a precession of one and three quarters of nakṣatra or 230 20`. Since the rate of precession is about a degree in 72 years, the time interval for a precession of 230 20` is 72 * 231/3 = 1680 years prior to Vārāhamihira’s time i.e. around 1150 BC. According to the famous astronomer Prof. T. S. Kuppanna Sastri, if instead of the segment of nakṣatra, the group itself it smeant, which is about 30 within it, Lagadha’s time would be around 1370 BC. Therefore, the composition of Lagadha’s Vedāńga Jyotiṣa can be assigned to the period of 12th to 14th century BC.[16]The Vedāńga Jyotiṣa belongs to the last part of the Vedic age. The text proper can be considered as the records of the essentials of astronomical knowledge needed for the day to day life of the people of those times. The Vedāńga Jyotiṣa is the culmination of the knowledge developed and accumulated over thousands of years of the Vedic period prior to 1400 BC.[17]Furthermore, there is considerable material on astronomy in the Vedic Samhitas. But everything is shrouded in such mystic expressions and allegorical legends that it has now become extremely difficult to discern their proper significance. Hence it is not strange that modern scholars differ widely in evaluating the astronomical achievements of the early Vedic Hindus. Much progress seems, however, to have been made in the Brāhmana period when astronomy came to be regarded as a separate science called nakṣatra-vidyā (the science of stars). An astronomer was called a nakṣatra-darśa (star observer) or ganaka (calculator).[18]Numerous Discoveries AnticipatedNumerous amounts of mental calculations have been done in the Vedic era. The distance of the heaven from the earth have been calculated differently in various works. All of them are figurative expressions indicating that the extent of the universe is infinite. There is speculation in the Ŗg-Veda (V.85.5, VIII.42.1) about the extent of the earth. It appears from passages therein that the earth was considered to be spherical in shape (I.33.8) and suspended freely in the air. (IV.53.3) The Śatapatha Brāhmaņa describes it clearly as parimaņdala (globe or sphere). There is evidence in the Ŗg-Veda of the knowledge of the axial rotation and annual revolution of the earth. It was known that these motions are caused by the sun.[19] In fact, all the postulations and conclusions had been arrived at couple of millenniums prior to the discovery of the same by the westerners.There are also evident mentions about the zodiacal belt, the inclinations of the ecliptic with the equator and the axis of the earth. We see that the apparent annual course of the sun is divided into two halves. We also see that the ecliptic is divided into twelve parts or sings of the zodiac corresponding to the twelve months of the year, the sun moving through the consecutive signs during the successive months. The sun is called by different names at various parts of the zodiac, and thus has originated the idea of twelve ādityas or suns.[20]The Ŗg-Veda (IX.71.9 etc.) says that the moon shines by the borrowed light of the sun. The phases of the moon and their relations to the sun were fully understood. Five planets seem to have been known. The planets Sukra or Vena (Venus) and Manthin are mentioned by name.[21]Knowledge Through ObservationsIt appears from a passage in the Taittrīya Brāhmaņa (I.5.2.1) that Vedic astronomers ascertained the motion of the sun by observing with the naked eye the nearest visible stars rising and setting with the sun from day to day. This passage is considered very important ‘as it describes the method of making celestial observations in old times’. Observations of several solar eclipses are mentioned in the Ŗg-Veda, a passage of which states that the priests of the Atri family observed a total eclipse of the sun caused by its being covered by Svarbhānu, the darkening demon (V.40.5-9).[22] The Atri priests could calculate the occurrence, duration, beginning, and the end of the eclipse. Their descendants were particularly conversant with the calculation of eclipses. At the time of the Ŗg-Veda, the cause of the solar eclipse was understood as the occultation of the sun by the moon. There are also mentions of lunar eclipses.[23]Concept of Time and SeasonsThe Vedic people had considerable knowledge about the seasons of for sowing, reaping etc. Apart from that, they had acquired knowledge required for their religious activities, like the times and periodicity of the full and new moons, the last disappearance of the moon and its first appearance etc. This type of information was necessary for their monthly rites like the darśapūrņamāsa and seasonal rites cāturmāsya.[24]Vedic Hindus counted the beginning of a season on the sun’s entering a particular asterism. After a long interval of time, it was observed that the same season began with the sun entering a different asterism. Thus they discovered the falling back of the seasons with the position of the sun among the asterisms. There are also clear references to the vernal equinox in the asterism Punarvasu. There is also evidence to show that the vernal equinox was once in the asterism Mŗgaśirā from whence, in course of time, it receded to Kŗttikā. Thus there is clear evidence in the Samhitās and Brāhmaņas of the knowledge of the precession of the equinox. Some scholars maintain that Vedic Hindus also knew of the equation of time.[25] The practical way of measuring time is mentioned as the time taken by a specific quantity of water to flow through the orifice of a specified clepsydra (water-clock), as one nāḍikā i.e., 1/60 part of a day. The day is divided into 124 bhāgas (or parts) so that the ending moments of parvas and tithis can be given in whole units. The day is also divided into 603 units called kalās so that the duration of the lunar nakṣatra is given in whole units as 610 kalās. The nakṣatra is divided into 124 amśas so that the nakṣatraspassed at the ends of the parvas may be expressed in whole amśas. [26]During the Yajurveda period, it was known that the solar year had 365 days and a fraction more. In the Kŗṣna Yajurveda (Taittrīya Samhita VII. 2.6) it is mentioned that the extra 11 days over 12 lunar months, Caitra, Vaiśākha etc. (totaling to 354 days), complete the ŗtus by the performance of the ekādaśarātra or eleven day sacrifice. Again, the Taittrīya Samhita says that 5 days more were required over the sāvana year of 360 days to complete the season adding specifically that 4 days are too short and 6 days too long.[27]In the Yajurveda period, they had recognized the six ŗtus (seasons) in a period of 12 tropical months of the year and named them as follows:SeasonsMonthsi.Vasanta ŗtuMadhu and Mādhavaii.Grīṣma ŗtuŚukra and Śuciiii.Varṣa ŗtuNalcha and Nabhasyaiv.Śarad ŗtuIṣa and Ūrjav.Hemanta ŗtuSaha and Sahasyavi.Śiśira ŗtuTapa and TapasyaThe sacrificial year commenced with Vasanta ŗtu. Thy had alos noted that the shortest day was at the winter solstice when the seasonal year Śiśira began with Uttarāyana and rose to a maximum at the summer solstice.[28]Vedic ArithmeticIndia’s recognized contribution to mathematics was chiefly in the fields of arithmetic and algebra. In fact, Indian arithmetic is what is now used world over. The topics discussed in the Hindu Mathematics of early renaissance included the following:-[29]a.Parikarma (The four fundamental operations)b.Vyavahāra (determination)c.Rajju (meaning rope referring to geometry)d.Trairāśi (the rule of three)e.Yāvat tāvat (simple equations)f.Kalasavarņa (operations with fractions)g.Varga and Vargamūla (square and square root)h.Ghana and ghana-mūla (cube and cube root)i.Prastāra and Vikalpa (Permutations and Combinations)Sources of information on Vedic arithmetic being very meagre, it is difficult to define the topics for discussion and their scope of treatment. One problem that appears to have attracted the attention and interest of Vedic Hindus was to divide 1000 into 3 equal parts. It is unknown how the problem could be solved, for 1000 is not divisible by 3. So, an attempt has been made to explain the whole thing as a metaphorical statement. But a passage in the Śatapatha Brāhmaņa (III.3.1.13) seems clearly to belie all such speculations, saying: ‘When Indra and Viṣṇu divided a thousand into three parts, one remained in excess, and that they caused to be reproduced into three parts. Hence even now if any one attempts to divide a thousand by three, one remains over.’ In any case, it was a mathematical exercise.[30]Vedic Hindus developed the terminology of numeration to a high degree of perfection. The highest terminology that ancient Greeks knew was ‘myriad’ which denoted 104 and which came into use only about the fourth century BC. The Romans had to remain content with a ‘mille’ (103). But centuries before them, the Hindus had numerated up to parārdha (1014) which they could easily express without ambiguity or cumbrousness. The whole system is highly scientific and is very remarkable for its precision.[31]Scales of NumerationFrom the time of the Vedas, the Hindus adopted the decimal scale of numeration. They coined separate names for the notational places corresponding to 1, 10, 102, 103, 104, 105, etc. and any number, however big, used to be expressed in terms of them. But in expressing a number greater than 103 (sahasra) it was more usual to follow a centesimal scale. Thus, 50.103 was a more common form than 5.104. In Taittrīya Upaniṣad (II.8) the centesimal scale has been adopted in describing the different orders of bliss. Brahmānanda, or the bliss of Brahman, has been estimated as 10010 times the measure of one unit of human bliss.[32]In cases of actual measurements, the Hindus often followed other scales. For instance, we have in the Śatapatha Brāhmaņa (XII.3.2.5 et seq.) the minute subdivision of time on the scale of 15. The smallest unit prāṇa is given by 1/155 of a day. In the Vedāńga jyotiṣa (verse 31), a certain number is indicated as eka-dvi-saptika. If it really means ‘two-sevenths and one’ as it seems to do, then it will have to be admitted that there was once a septismal scale.[33]Representation of NumbersThe whole vocabulary of the number-names of the Vedic Hindus consisted mainly of thirty fundamental terms which can be divided into three groups, the ones, tens and the powers of ten. Furthermore, we can understand that Vedic Hindus had a unique and powerful method of their own in representing numbers. Because from the seals and inscriptions of Mohenjo-daro we can see that in the third millennium before the Christian era, numbers were represented in the Indus valley by means of vertical strokes arranged side by side or one group upon another. There were probably other signs for bigger numbers. Those rudimentary and cumbrous devices of rod-numerals were, however, quite useless for the representation of large numbers mentioned in the Vedas. In making calculations with such large numbers, as large as 1012, Vedic Hindus must have found the need for some shorter and more rapid method of representing numbers. This and other considerations give sufficient grounds for concluding that the Vedic Hindus had developed a much better system of numerical symbols.[34]From a reference in the Aṣṭādhyāyī of Pāṇini, we come to know that the letters of the alphabet were used to denote numbers. Another favourite device of Vedic Hindus to indicate a particular number was to employ the names of things permanently connected with that number by tradition or other associations, and sometimes vice versa. Applications of this are found in the earliest Saṁhitās. This practice of recording numbers with the help of letters and words became very popular in later times, especially amongst astronomers and mathematicians.[35]Holiness Attributed to NumbersIt appears that Vedic Hindus used to look upon some numbers as particularly holy. One such number is 3. In the Ṙg-Veda, the gods are grouped in three (I.105.5) and the mystical ‘three dawns’ are mentioned (VIII.41.3, X.67.4). Cases of magic where 3 is employed in a mysterious occult manner occur in the Ṙg-Veda and the Atharva-Veda. Even the number 180 is mentioned in the Ṙg-Veda as three sixties (VIII.96.8) and 210 as three seventies (VIII.19.37). The number regarded as most sacred seems to have been 7. Thus in the Ṙg-Veda, we get ‘seven seas’ (VIII.40.5), ‘seven rays of the sun’ (I.105.9), and ‘seven sages’ (IV.42.8, IX.92.2, etc.); and the number 49 is stated as seven sevens. Instances of combinations of these two numbers also occur. Thus 21 is stated as three sevens in the Ṙg-Veda (I.133.6, 191.12) and the Atharva-Veda (I.1.1), and 1,470 as three seven seventies in the Ṙg-Veda (VIII.46.26).[36]Classification of NumbersNumbers were divided into even (yugma, literally pair) and odd (ayugma, literally non pair), but there is no further subdivisions of numbers. There is an apparent reference to Zero and recognition of the negative number in the Atharvaveda. Zero is called kṣudra (XIX.22.6) meaning trifling’; the negative number is indicated by the epithet anṛca (XIX.23.22), meaning ‘without a hymn’; and the positive number by ṛca (XIX.23.1), meaning ‘a sacred verse’. These designations were replaced in later times by ṛṇa (debt) and dhana (asset).[37] The fractions like half, quarter, one-eighth and one-sixteenth are referred to for the first time, in the history of mathematics, in the Ŗgveda. These fractions are respectively called ardha, pāda, śapha and kalā.[38]Number SeriesVedic Hindus became interested in numbers forming series or progressions. The TaittrīyaSaṁhitā (VII.2.12-17) mentions the following arithmetical series:(i)1, 3, 5,...19,.......99;(ii)2, 4, 6, ...............100;(iii)4, 8, 12, .............100;(iv)5, 10, 15, ............100; and(v)10, 20, 30, ..........100.The arithmetical series are classified into ayugma and yugma. The Vajasaneyi Saṁhitā (XVIII.24.25) has given the following two instances:(i)1, 3, 5, .............31 and(ii)4, 8,12,...............48.The first series occurs also in the Taittrīya Saṁhitā (IV.3.10). The Paňcavimśa Brāhmana (XVIII.3) describes a list of sacrificial gifts forming a geometrical series of some interest and particular nature.12, 24, 48, 96, 192, ..............49152, 98304, 196608, 393216.This series reappears in the Śrauta-sūtras. Some method for the summation of series was also known. Thrice the sum of an arithmetical progression whose first term is 24, the common difference 4 and number of terms 7 is stated correctly in the ŚatapathaBrāhmaņa as 756.[39] . Moreover, from the texts available to us, it is obvious that Vedic Hindus knew how to perform fundamental arithmetical operations even with elementary fractions.[40]Regarding solutions of quadratic equations, we see that problems for solving quadratic equations of the form ax2 = c and ax2 + bx = c are stated and proved. We also find some indeterminate equations of the first degree (wrongly called as Diophantine equations by modern mathematicians). Furthermore, an important feature, peculiar to the Vedic mathematics, was to use geometrical diagrams and methods to solve algebraic problems and identities. Such geometrical techniques are beautifully adopted in the Śulvasūtras to solve algebraic equations.[41]The ŚulvasūtrasThe most ancient mathematical texts of the Vedic lore are the Śulvasūtras which form a part of the Śrauta section of the Kalpa Vedāńga. In the Śulvasūtras are seen very remarkable and rich principles of mathematics, particularly geometry. The word Śulva (or Śulba) is derived from the verb-root, Śulv or Śulb which means to measure. Since for measure length and breadth rope (rajju) was used, the word Śulva, in course of time came to mean a rope. In fact, Geometry (now referred to as Rekhāgaņita) was called Śulva Sāstra or Rajju Sāstra in ancient http://times.It is believed that these Śulvasūtras were composed around eight or nine centuries before Christ. [42]In the Vedic religion, every household man (barring the Sanyasis who would concentrate on meditation for years uninterruptedly) had to do certain acts of worship every day. It would be sinful if he neglected them. For purposes of worship, he would constantly maintain in his house three types of Agnis or fires sheltering them in certain altars of special designs. The required altars had to be constructed with great care so as to conform to certain specific shapes and areas.[43] While the three Agnis were to be used for the daily or routine Pujas or acts of worship, there were more elaborate sacrifices or Pujas for attaining cherished objects or wants. They were called Kamyagnis. The sacrificial altars for these Kamyagnis required more complicated constructions involving combinations of rectangles, triangles and trapeziums. It is clear that these processes require a clear knowledge of the properties of triangles, rectangles and squares, properties of similar figures, and a solution of the problem of ‘squaring the circle’ and its converse, ‘circling the square’ (i.e., to construct a square equal in area to a given circle, and vice versa).[44]Source and OriginOnly seven of the Śulvasūtras are known at present. They are known by the names Baudhāyana, Āpasthamba, Kātyāyana, Manava, Maitrayana, Varaha and Vadhūla after the names of the Rishis or sages who wrote them. The Kātyāyana Sūtra belongs to the section of the Vedas called Sukla Yajurveda while all the rest belong to Krishna Yajurveda. The Bodhayana, Āpasthamba and Kātyāyana Śulvas are of importance from the mathematical point of view.[45]The dates of these Śulvasūtras have been estimated to be between 800 BC and 500 BC. There is no knowledge about the existence of any Śulvas prior to these seven Sūtras. It must be emphasized that the writers of the Śulvasūtras only wrote down and codified the rules for the constructions of the altars, which were in vogue from ancient times. They were not the persons who specified and directed the rules for the constructions of the altars.[46]Simple Theorems in ŚulvasūtrasThe Śulvas explain a large number of simple geometrical constructions- constructions of squares, rectangles, parallelograms and trapeziums. The following geometrical theorems are either explicitly mentioned or clearly implied in the constructions of the altars of the prescribed shapes and sizes.[47]a.The diagonal of a rectangle divides it into two equal parts.b.The diagonal of a rectangle bisect each other and the opposite areas are equal.c.The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into equal halves.d.A rectangle and a parallelogram on the same base and between the same parallels are equal in area.e.The diagonals of a rhombus bisect each other at right angles.f.The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. (The famous theorem known after the name of Pythagoras)g.Properties of similar rectilinear figures.h.If the sum of the squares on two sides of a triangle be equal to the square on the third side, then the triangle is right-angled. (This is the converse of the Pythagoras theorem, nay the Śulva theorem!)How the Mathematical Theorems were derived?These above mentioned theorems of Śulvasūtras cover roughly the first two books and the sixth book of Euclid’s ‘Elements’. How these theorems were actually obtained is a matter for which no definite answer is available. We all know that Euclid’s geometry is based upon certain axioms and postulates as I pointed out in the first chapter[48] and the proofs involve strict logical application of these. The logical methods of Greek Geometry are certainly not discernible in Hindu geometry. No book on Hindu mathematics explains the system of axioms and postulates assumed, and this itself should go some way in refuting the concocted claim that Hindu mathematics is borrowed from the Greeks. At the same time, it may not be correct to conclude that the above theorems were asserted as a matter of experience and measurement. The people who could make out and solve complicated problems of arithmetic, algebra and spherical trigonometry should be credited with some amount of logic in their work. The Śulvas are not formal mathematical treatises. They are only adjuncts to certain religious works. The question has to end with these remarks.[49] And probably from those very remarks does spring up, valid support for the idea, hypothesis and ultimate destiny of my attempt in this research paper.To make the view clearer, it could be stated that what Pythagoras’ theorem states was known, proved and applied in cases by the Vedic Hindus even before Pythagoras was born in the 6th century BC. Of course, the Vedic scholars did not prove the theorem though they stated and used it. But then there is no evidence that Pythagoras proved it either! The well known proof given by Euclid in this regard may be his own. But the theorem itself was known and widely used from very early times as we mentioned some 5 or 6 centuries prior to the birth of Euclid. It is interesting to note that A. Burk even argues that the much travelled Pythagoras borrowed the result from India![50]Geometrical Constructions contained in ŚulvasThe Vedis discussed in the Śulvasūtras are of various forms. Their constructions require a good knowledge of the properties of the square, the rectangle, the rhombus, the trapezium, the triangle, and of course, the circle. The following are among the important geometrical constructions used in the Śulvasūtras.[51](i)To divide a line-segment into any number of equal parts(ii)To divide a circle into any number of equal areas by drawing diameters (Baudhāyana, II. 73-74, Āpasthamba VII. 13-14)(iii)To divide a triangle into a number of equal and similar areas (Baudhāyana, III. 256)(iv)To draw a straight line at right-angles to a given line (Kātyāyana I. 3)(v)To draw a straight line at right-angles to a given line from a given point on it. (Kātyāyana I. 3)(vi)To construct a square on a given side(vii)To construct a rectangular of given length and breadth. (Baudhāyana I. 36-40)(viii)To construct an isosceles trapezium of given altitude, face and base. (Baudhāyana I. 41, Āpasthamba V. 2-5)(ix)To construct a parallelogram having the given sides at a given inclination. (Āpasthamba XIX. 5)(x)To construct a square equal to the sum of two different squares. (Baudhāyana I. 51-52, Āpasthamba II. 4-6, Kātyāyana II. 22)(xi)To construct a square equivalent to two given triangles.(xii)To construct a square equivalent to two given pentagons. (Baudhāyana III. 68, 288, Kātyāyana IV. 8)(xiii)To construct a square equal to a given rectangle in area. (Baudhāyana I. 58, Āpasthamba II. 7, Kātyāyana III. 2)(xiv)To construct a rectangle having a given side and equivalent to a given square.(xv)To construct an isosceles trapezium having a given face and equivalent to a given square or rectangle. (Baudhāyana I. 55)(xvi)To construct a triangle equivalent to a given square. (Baudhāyana I. 56)(xvii)To construct a square equivalent to a given isosceles triangle. (Kātyāyana IV. 5)(xviii)To construct a rhombus equivalent to a given square or rectangle. (Baudhāyana I. 57, Āpasthamba XII. 9)(xix)To construct a square equivalent to a given rhombus. (Kātyāyana IV. 6)Squaring a CircleOne of the greatest problems that had remained unsolved for centuries in the history of mathematics, till recently, was what is popularly known as “squaring the circle”, i.e., to construct – using only ruler and compasses – a square whose area is equal to that of a given circle. IT is really remarkable that this very problem and vice versa, was tackled by the authors of the Śulvasūtras. They gave practical methods for constructing a square whose area is equal to that of a given circle and vice versa. Of course, their constructions involved approximating the value of the well-known constant number Pi to 3.088 (Pi is the ratio of the circumference of any circle to its diameter). The approximation made by the Vedic Ŗṣis is quite justifiable and admirable in the light of crude mathematical tools and methods available to them thousands of years ago. IT is now, however, in modern mathematics established that an exact construction to “Square a circle” or “to circle a square” (in the sense of the areas) is impossible. Incidentally, it must be pointed out that the approximate value of Pi used in the Śulvasūtras is certainly far better than the Biblical value, 3 (see Kings VII. 23 and Chronicles IV. 2) given many centuries later! In fact, it is only as late as in 1761 AD that Lambert proved that Pi is “irrational” and only in 1882 AD that Lindemann established further that Pi is transcendental.[52]Vedic Knowledge of Irrational NumbersThe essentially arithmetical background of the Śulva mathematics must be contrasted with the essentially geometrical background characteristic of Greek mathematics. Simple fractions and operations on them are available in the Śulvas. Moreover, we find the use of Irrational numbers in them. Surds (Irrational numbers) of the form √2, √3 etc. are called Karanis, thus √2 is dwi-karani, √3 is tri-karani, √1/3 is triteeya karani, √1/7 is saptama karani, so on.[53]The shape of Ashwamedhiki Vedika is an isosceles trapezium whose head, foot and altitude are respectively 24√2, 30√2 , 36√2 prakramas. Its area is stated to be 1944 prakramas (square is to be understood).Area = 24√2 * 1/2(30√2 * 36√2) = 1944.This indicates knowledge of the method of finding the area of a trapezium and simple operations on surds.[54]A remarkable approximation of √2 occurs in each of the three Śulvas Bodhayana, Āpasthamba and Kātyāyana.√2 = 1 + 1/3 + 1/ (3.4) – 1/(3.4 34)....This gives √2 = 1.4142156........., whereas the true value is 1.414213............ The approximation is thus correct to five decimal places, and is expressed by means of simple unit fractions. The problem evidently arises in the construction of a square double a given square in area. The Śulvas contain no clue at all as to the manner in which this remarkable approximation was arrived at. Many theories or plausible explanations have been proposed.[55]This point to the fact that the India, thanks to Vedic mathematics, was the first nation to use irrational numbers. It has to be believed that the Vedic Hindus knew its irrationality. The Greeks also used irrational numbers. If AB is a given segment, Pythagoras and others described the methods of constructing the segments of length√2AB, √3AB, √5AB, etc. But no rational approximation to √2, √3 etc. are found in Greek mathematics, nor are there any problems of the arithmetical operations on irrational numbers. This is easily explained, because the requisite knowledge of arithmetic was not available to the Greeks. It will also be borne in mind that according to unprejudiced estimates, the Śulvasūtras are about two or three centuries prior to Pythagoras.[56][1] Iyengar, The History of Ancient Indian Mathematics, 6.[2]Raja, “The Cultural Heritage of India”,1.[3] Raja, “The Cultural Heritage of India”, 2.[4] Dani, “Ancient Indian Mathematics - A conspectus”, 236.[5] The Vedic Culture had the unique feature of performances of five rituals, known as yajnas. These involved well laid out altars, the vedis, and fire platforms, known as citi or agni, elaborately constructed in the form of birds, tortoise, wheel, etc.[6] The SulvaSūtras are compositions, in the form of manuals for construction of vedis and citis, but they also discuss the geometric principles involved.[7] Dani, “Ancient Indian Mathematics - A conspectus”, 238.[8] Dani, “Ancient Indian Mathematics - A conspectus”, 239.[9] Datta, “Vedic Mathematics”, 18.[10] Datta, “Vedic Mathematics”, 19.[11] Datta, “Vedic Mathematics”, 18.[12] Datta, “The Scope and Development of Hindu Ganita”, 480.[13] Rao, Indian Mathematics and Astronomy, 23.[14] Rao, Indian Mathematics and Astronomy, 23.[15] Rao, Indian Mathematics and Astronomy, 23.[16] Rao, Indian Mathematics and Astronomy, 24.[17] Rao, Indian Mathematics and Astronomy, 24.[18] Datta, “Vedic Mathematics”, 19.[19] Ghosh, “Studies on Rig-Vedic Deities”, 11.[20] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 25.[21] Datta, “Vedic Mathematics”, 20.[22] Datta, “Vedic Mathematics”, 21.[23] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 26.[24] Rao, Indian Mathematics and Astronomy, 25.[25] Mookerjee, “Notes on Indian Astronomy”, 137.[26] Rao, Indian Mathematics and Astronomy, 25.[27] Rao, Indian Mathematics and Astronomy, 25.[28] Rao, Indian Mathematics and Astronomy, 26.[29] Rao, Indian Mathematics and Astronomy, 3.[30] Datta, “Vedic Mathematics”, 30.[31] Rao, Indian Mathematics and Astronomy, 3..[32] Datta, “Vedic Mathematics”, 31.[33] Datta, “Vedic Mathematics”, 31.[34] Datta, “Vedic Mathematics”, 33.[35] Datta, “Vedic Mathematics”, 34.[36] Hopkins, “Numerical Formulae in the Veda and their Bearing on Vedic Criticism”, 279.[37] Shukla, “The Deceptive Title of Swamiji’s Book”, 36.[38] Rao, Indian Mathematics and Astronomy, 14.[39] Shukla, “The Deceptive Title of Swamiji’s Book”, 37.[40] Mehta, Positive Sciences in the Vedas, 114.[41] Rao, Indian Mathematics and Astronomy, 12.[42] Rao, Indian Mathematics and Astronomy, 12.[43] Mehta, Positive Sciences in the Vedas, 117.[44] Iyengar, The History of Ancient Indian Mathematics, 6.[45] Iyengar, The History of Ancient Indian Mathematics, 7.[46] Iyengar, The History of Ancient Indian Mathematics, 8.[47] Rao, Indian Mathematics and Astronomy, 14.[48] Chapter I (1.2.1) Geometry, 16-21.[49] Iyengar, The History of Ancient Indian Mathematics, 9.[50] Rao, Indian Mathematics and Astronomy, 15.[51] Rao, Indian Mathematics and Astronomy, 15-17.[52] Rao, Indian Mathematics and Astronomy, 18.[53] Iyengar, The History of Ancient Indian Mathematics, 13.[54] Iyengar, The History of Ancient Indian Mathematics, 13.[55] Iyengar, The History of Ancient Indian Mathematics, 14.[56] Iyengar, The History of Ancient Indian Mathematics, 15.

Mathematics is an important discipline in ancient Chinese science. It has a long history and brilliant achievements. Chinese mathematics originated from the Yangshao culture. It has a history of more than 5,000 years. In the Zhougong period, count was one of the six arts. In the Spring and Autumn Period, the calculation of the decimal system has become widespread. The famous Japanese mathematician Samio Saifu pointed out that the development of Chinese mathematics has a history of two to three thousand years. Such a long history of development has never been seen in all countries of the world. Greece had a history of only one thousand years from the 6th century BC to the 4th century AD; Arabic mathematics was limited to the 8th century to the 13th century.Budding stageIn the late period of the sprouting primitive communes of ancient Chinese mathematics, after the emergence of the private ownership system and the exchange of goods, the concept of number and shape had further development. The pottery unearthed during the Yangshao culture period was marked with the symbol 1234. By the end of the primitive communes, they had begun to replace knot notes with text symbols.The pottery unearthed in Banpo, Xi’an, has an equilateral triangle consisting of 1 to 8 dots and a pattern of 100 small squares in the sub-squares. The premises of the Banpo site are both round and square. In order to draw a circle and determine straightness, people also created rules, moments, standards, ropes and other mapping and measurement tools. According to the records of "Historical Records of Xiabenji", these tools were used during the summer flood control.originAccording to "easy. "Correspondence" records: "The ancient knot rope and rule, the future saints easy to use the book". There are many counted words in Oracle inscriptions. From one to ten, and one hundred, one thousand, and ten thousand are special count texts. There are a total of 13 independent symbols. The counts are written in co-scripts, among which there are decimal system notation. The largest number is 30,000. Computation is an ancient Chinese calculation tool, and this calculation method is called calculation. The era of calculations has become untestable, but what is certain is that the calculations have been common in the Spring and Autumn Period.With counting calculations, there are vertical and horizontal methods:When a multi-digit number is represented, the decimal value system is adopted. The number of values ​​is arranged from left to right, and the vertical and horizontal phases (the rule is: one vertical and ten horizontal, one hundred horizontal and one thousand stiff, thousands and ten, and one hundred thousand and one hundred equivalent). , and zero is represented by a space. Calculations establish good conditions for addition, subtraction, multiplication, and division.In the field of geometry, "Historical Records. In the book of Xia Benji, he used the rules, moments, standards, ropes, and other tools for drawing and measuring. He has already discovered the special case of the “three-string four-string five” Pythagorean Theorem (Western Pythagorean Theorem). . During the Warring States Period, the "Kao Gong Ji" written by Qi Guoren summarized the norms of handicraft technology at that time, contained some measurement contents, and involved some geometric knowledge, such as the concept of angles.Shang Dynasty TimesIn the middle of the Shang Dynasty, a set of decimal numbers and notation was produced in Oracle. The largest number was 30,000. At the same time, the Yin people used ten heavenly and twelve earthly branches to form Jiazi, Yiwu, and Bingzi. Ding Yi and other 60 names recorded the date of 60 days. In the Zhou Dynasty, the eight kinds of symbols formed by the yin and yang symbols were used to represent eight kinds of things and sixty-four characters, representing 64 kinds of things.In the first century BC, the Zhouyi calculations mentioned the method of measuring the height, depth, breadth, and distance by moment in the early Western Zhou Dynasty, and listed examples of the hook-shaped hooks, strands, strings, strings, and ring moments. . The book of the Book of Rites mentions that the children of the nobles in the Western Zhou Dynasty must learn the numbers and counting methods from the age of nine. They must be trained in rituals, music, shooting, chanting, books and numbers as one of the “six arts”. Has begun to become a specialized course.Spring and Autumn and Warring StatesOn the occasion of the Spring and Autumn and Warring States period, the calculation has been widely used. The calculation method has used the decimal value system. This kind of counting method has epoch-making significance for the development of the world mathematics. Measurement mathematics in this period has been widely used in production, and there has been a corresponding increase in mathematics.The contending in the Warring States Period also promoted the development of mathematics, and in particular, the debate over formal names and some propositions was directly related to mathematics. The masters think that after the abstract concept of nouns is different from their original entities, they propose that “there is no squareness and rules cannot be a circle”, and the “big one” (infinity) is defined as “big, no, and small” ( Infinity is defined as "small to no inner". It also put forward the proposition that "one foot of a rut, one day will take half of it, and the last will never be exhausted."MohistThe Mohist thinks that the name comes from the object and the name can reflect things from different aspects and different depths. Mohist gives some mathematical definitions. For example, circle, square, flat, straight, minor (tangential), end (point), and so on.The Mohist disagreed with the proposition that "one foot of a mile" and proposed a "non-half" proposition to refute: When a line segment is divided into half and half indefinitely, there will be a "non-half" that cannot be further divided. "Not half" is the point.The famous proposition discusses that the finite length can be divided into an infinite sequence, and the Mohist proposition indicates the change and result of this infinite segmentation. The mathematical definitions of famous artists and Mohism and the discussion of mathematical propositions are of great significance to the development of ancient Chinese mathematical theories.System formationThe Qin and Han dynasties were the rising period of the feudal society, and both the economy and culture developed rapidly. The ancient Chinese mathematics system was formed in this period. Its main symbol is that mathematics has become a specialized discipline, and the emergence of mathematical works represented by the "Nine Chapters of Arithmetic".The "Nine Chapters of Arithmetic" is a summary of the development of the mathematics of the Warring States, Qin and Han feudal societies that established and consolidated, and in terms of its mathematical achievements, it can be regarded as a masterpiece of mathematics in the world. For example, fractional arithmetic, current technique (Western called tri-rate method), open square and open cube (including quadratic equation numerical solution method), insufficiency technique (Western called double strategy), various area and volume formulas, linear equations The levels of solution, addition and subtraction of positive and negative calculations, and the Pythagorean method (especially the Pythagorean theorem and the method for obtaining the number of shares) are all very high. Among them, the solution of equations and the addition and subtraction of positive and negative numbers are far ahead in the development of mathematics in the world. As far as its characteristics are concerned, it has formed an independent system that is centered on calculations and completely different from ancient Greek mathematics.The "Nine Chapters of Arithmetic" has several notable features: the use of a set of mathematics questions in a class-by-category form; the formulas are all developed from the method of computational notation; they are dominated by arithmetic and algebra; they rarely involve the nature of graphs; Application, lack of theoretical elaboration, etc.The emphasis on the application of mathematics in the Qin and Han Dynasties. Cheng Shuji's "Nine Chapters of Arithmetic" in the early years of the Eastern Han Dynasty ruled out the discussion of the definition and logic of nouns and the Mohists that appeared in the commemorative period of the Warring States period. It emphasized the mathematical problems and their solutions that were closely integrated with the production and life at that time.The "Nine Chapters of Arithmetic" was transmitted to North Korea and Japan during the Sui and Tang dynasties and became the mathematics textbooks of those countries. Some of its achievements, such as the decimal system, current technique, and insufficiency technique, also spread to India and Arabia, and spread to India through India and Arabia, which promoted the development of mathematics in the world.Mathematics DevelopmentWei and Jin DynastiesThe metaphysics that appeared during the Wei and Jin periods was not confined to the Confucian school of Confucianism and thoughts were active; it solicited to win, but also used logical thinking and analysis of reason, all of which are conducive to the theoretical improvement of mathematics. Wu Guo and Zhao Shuang's Note on Zhou Yi's Classics, Xu Yue's essays on "The Nine Chapters of Arithmetic" in the late Han Dynasty and early Wei Dynasty. Note that the "Nine Chapters in Arithmetic" note and the "Nine Chapters in Difference Figures" appeared in Liu Wei's and Liu Wei's early Jin Dynasty. In this period. The work of Zhao Shuang and Liu Hui laid the theoretical foundation for the ancient Chinese mathematics system.Zhao Shuang was one of the earliest mathematicians who proved and deduced mathematical theorems and formulas in ancient China. The “Pythagorean Circle Chart and Note” and “Hikari Chart and Note” supplemented by him in the book “Zhou Yi Jing Jing” are very important mathematical documents. In the “Pythagorean Circle Chart and Note” he proposed the use of string graphs to prove the Pythagorean Theorem and the five formulas for solving the hook-shaped fractals; in the “Day-High-Graph and Note”, he used the graph area to prove the common application of the Han Dynasty The formula, Zhao Shuang's work is groundbreaking and plays an important role in the development of ancient Chinese mathematics.At the same time as Liu Huihui and Zhao Shuang, he inherited and developed the ideas of the famous masters and Mohist during the Warring States period, and he advocated a strict definition of some mathematic terms, especially important mathematical concepts. He believed that mathematical knowledge must be “analyzed” before Making mathematics concise and concise, to the reader. His "Nine-Chapter Arithmetic" note is not only a general explanation and derivation of the methods, formulas and theorems of the "Nine Chapters of Arithmetic", but also has great development in the process of the discussion. Liu Hui created a cutting circle and used the idea of ​​limit to prove the area formula of the circle, and for the first time used the theoretical method to calculate the ratio between 157/50 and 3927/1250.Liu Hui proved that the volume ratio of the right-angled square pyramid to the right-angled tetrahedron was always 2:1 by using the infinite segmentation method, which solved the key problem of the general cubic volume. In the demonstration of the volume of square cones, cylinders, cones, and cones, Liu Hui proposed the correct way to completely solve the volume of the ball.After the Eastern Jin DynastyAfter the Eastern Jin Dynasty, China had long been in a state of war and North-South split. The work of Zu Chongzhi's father and son was the representative work of the development of mathematics in the south after the economic and cultural shift to the south. They took the traditional mathematics forward a step further on the basis of Liu Hui's “Nine Chapters of Arithmetic”. Their mathematics work mainly includes calculating the pi between 3.1415926 and 3.1415927, proposing the ancestral principle and proposing the solution of the second and third equations.It has been speculated that Zu Chongzhi calculated the area of ​​the 6144-flange and the 12288-flange inscribed on the basis of Liu Hui’s cutting technique, and obtained this result. He also used the new method to obtain two fractional values ​​for the pi, namely the rate of 22/7 and the density of 355/113. Zu Chongzhi's work has enabled China to lead the calculation of pi for more than a thousand years than the West.The son of Zu Chongzhi sums up Liu Hui’s work and puts forward that “power and potential are not only different”, that is, two dimensions of equal height. If the horizontal cross section of any height is equal, the two volumes are equal. Is the famous ancestral justice. The ancestors used this axiom to solve Liu Hui's unresolved ball volume formula.During the Tang and Tang DynastiesThe emperor's large-scale construction project objectively promoted the development of mathematics. At the beginning of the Tang Dynasty, Wang Xiaotong's “The Ancient History of Confucian Classics” mainly discussed the computational aspects of earthwork, engineering division, acceptance, and warehouse and cellar calculations in civil engineering, reflecting the situation of mathematics in this period. Wang Xiaotong developed a digital cubic equation without using mathematical symbols, which not only solved the needs of society at that time, but also laid the foundation for the establishment of Tianyuan's technique. In addition, Wang Xiaotong's solution to the traditional Pythagorean method is also solved using a digital cubic equation.At the beginning of the Tang Dynasty, the rulers inherited the control system. In 656, the Imperial College set up a computer science museum with 30 doctoral students and teaching assistants. The Notes of the Book of the Compilation were edited by Tai Shiling Li Shufeng et al. As a textbook for the students of the computer science school, the calculations for these subjects are also based on these calculations. The "Book of Ten Classics" edited by Li Yufeng and others is of great significance for the preservation of classical works of mathematics and the provision of documentation for mathematical research. Their comments to the Zhouyi Sutra, the Nine Chapters, and the Island Opera can be helpful to readers. During the Sui and Tang Dynasties, due to the needs of the calendar, the days mathematicians created the interpolation method of quadratic function, which enriched the content of ancient Chinese mathematics.Calculating is the main calculation tool in ancient China. It has the advantages of simplicity, image, and concreteness. However, it also has the disadvantages of large deployment area, easy to mess with when speeding up operations and causing errors, etc. Therefore, the reform was started very early. Among them, Taiyi, Biyi, Sancai, and abacus are all beaded abacus and are technically important reforms. Especially the "Abacus," it inherited the advantages of five-litre-decade-advancement and place-based system, and it also overcomes the shortcomings of calculating and calculating inconveniently, and the advantages are obvious. However, since the multiplication and division algorithm at that time still cannot be performed in one row. The beads have not yet been worn and are not convenient to carry, so they are still not widely used.After the mid-Tang period, the commercial prosperity, the increase in digital computing, and the urgent need for reforms in calculation methods. From the "New Book of Tang" and other documents left by the calculation of bibliography, we can see that this algorithm reform is mainly to simplify the multiplication and removal algorithm, Tang Dynasty's The algorithmic reform allows the multiplication and division to be performed in a single row. It applies both to calculations and to abacus.Education establishmentIn the Sui Dynasty, civil engineering was greatly promoted, which objectively promoted the development of mathematics. At the beginning of the Tang Dynasty, Wang Xiaotong wrote “The Ancient History of Confucian Classics” mainly through the practical problems of earthwork, engineering division and acceptance, and warehouse and cellar calculations. He discussed how to establish cubic polynomial equations geometrically and developed the “Nine Chapters of Arithmetic”. "In the less broad, the "complications chapter" in the prescription theory.The period of the Sui and Tang Dynasties was the period when the feudal bureaucratic system was established in China. With the establishment of the imperial examination system and the Guozijian system, mathematics education has made considerable progress. In 656, Guozijian set up a mathematics school with a doctor of mathematics and teaching assistants, and was compiled by the Tai Shiling Li Shufeng and others to compile notes on the “calculus of the ten books” (including the “Zhoukejingkejing”, “the Nine Chapters of Arithmetic”, “the Island Opera”, “ The Sun Zi Bi Jing Jing, Zhang Qiu Jian Su Jing Su, Xia Hou Yang Su Jing, Gu Fu Gu Su Jing, Wu Cao Su Jing, Wu Jing Ma Ji, and Fu Ji Gong were used as students in the computer science classroom. Textbooks. It has played an important role in the preservation of ancient mathematical classics.Since some major astronomical discoveries in the period of the Southern and Northern Dynasties began to be implemented into the calculation of calendars at the turn of the Sui and Tang Dynasties, some important mathematical achievements emerged in the calendar of the Tang Dynasty. In the year 600 AD, during the formulation of the "Emperor's Calendar," the Sui Dynasty, Liu Huan, put forward the first-ever interpolating quadratic interpolation formula in the world. This is an outstanding creation in the history of mathematics. It was developed as an unequal interval quadratic interpolation formula.In the late Tang Dynasty, computing technology has been further improved and popularized. There are many kinds of practical arithmetic books, and they are simple and straightforward for multiplication and elimination algorithms.Math ProsperityNorthern Song DynastyIn 960, the establishment of the Northern Song Dynasty ended the separatist situation of the five generations and ten nations. The unprecedented prosperity of agriculture, handicrafts, and commerce in the Northern Song Dynasty, and the rapid advancements in science and technology, the three inventions of gunpowder, compass, and printing were widely used in this economic upsurge. In 1084, the secretary of the province printed and published the "Ten Books of the Compilation" for the first time. In 1213, Bao Ting's book was reprinted. All these have created good conditions for the development of mathematics.11th century to 14th centuryFrom the 11th to the 14th century, about 300 years ago, there appeared a group of famous mathematicians and mathematics works, such as Jia Xian's "The Yellow Emperor's Nine Chapters Algorithm", Liu Yi's "Proposal on Ancient Roots," and Qin Jiulu's "Several Number Nine". Zhang Li, Li Ye's "Measurement of Circular Oceanscope" and "Yigu Performing Stage", Yang Hui's "Detailed Explanation of Nine Chapter Algorithms", "Daily Use Algorithm" and "Yang Hui Algorithm", Zhu Shijie's "Academic Enlightenment" and "Siyuan Jade Kam" "And so on, many fields have reached the peak of ancient mathematics, and some of these achievements were also the peaks of mathematics in the world at that time."From opening squares, opening cubes to four or more squares, it is a leap in understanding, and Jia Xian is the one who has made this leap forward. Yang Hui included Jia Xian’s “Method of Enhancing the Kaiping Kaiping” and “Method of Enhancing Passengers by Opening and Closing” in the “Nine Chapter Algorithms”. In the “Detailed Explanation of the Algorithm of the Nine Chapters”, Jia Xian’s “origin of predicative method” was included. The example of "increase method for hunting grass" and the method of increasing the power to open the fourth power. Based on these records, it can be ascertained that Jia Xian has discovered a bi-coefficient table and created a multiplier-opening method. These two achievements have a major impact on the mathematics of the entire Song and Yuan dynasties. Among them, the Jia Xian triangle was proposed 600 years earlier than the Pascal triangle of the West.It is Liu Yi that generalizes the augmentation-opening method to the digital higher-order equation (including the case where the coefficient is negative). In the “Ya Hui algorithm”, the “Multiplication and Subtraction of Field Ratios” method, the original 22-quadratic equations and a quartic equation are introduced. The latter is the earliest example of solving higher-order equations with more than three times using the augmentation-opening method. .Qin Jiulu is a master of higher-order equation solutions. In order to adapt to the calculation procedure of the augmentation and opening method, Pan Jiuju defines the constant term as a negative number and classifies the high-order equation solution into various types. When the root of the equation is non-integer, Qin Jiulu takes the fraction that continues to be rooted, or uses the sum of the coefficients of each power of the root reduction transformation equation as the denominator, and the constant is the numerator to represent the non-integer part of the root. This is the "Chapter 9 Arithmetic". 》and Liu Huibin deal with the development of irrational number methods. In seeking the second digit of the root, Qin Jiulu also proposed a test division method in which the second term of the root of the term coefficient is divided by one term coefficient, which is more than 500 years earlier than the earliest method of Horner in the West.The astronomers Wang Yu and Guo Shoujing of the Yuan Dynasty solved the problem of the interpolated value of the cubic function in the “Time History”. formula.Using Tianyuan (equivalent to x) as an unknown symbol to formulate a high-order equation, which was called Tianyuanshu in ancient times. This is the first introduction of symbols in the history of Chinese mathematics and the use of symbolic operations to solve the problem of establishing high-order equations. The earliest known Tianyuan technique is Li Ye’s “Measurement of Circular Oceanscope”.Zhu Shijie’s quadruple high-order simultaneous equations representation was developed on the basis of Tianyuan’s technique. He placed the constants in the center, and the powers of the four elements were placed in the directions of up, down, left, and right. All other items are placed in the four quadrants. Zhu Shijie’s greatest contribution is to propose a four-element elimination method. The method is to choose one yuan as an unknown number. The polynomials composed of other elements are used as the coefficients of this unknown number. They are listed as a number of one-dimensional higher-order equations, and then are gradually eliminated by the mutual multiplication and destructive method. This unknown number. Repeat this step to eliminate other unknowns, and finally solve it with the augmentation open method. This is a significant development of the linear method set solution, more than 400 years earlier than the Western equivalent method.Song and Yuan periodThe climax of China's ancient computing technology reforms also appeared during the Song and Yuan dynasties. Song Yuanming's historical documents contain a large number of practical arithmetic bibliographies of this period, and the number is far more than that of the Tang Dynasty. The main content of the reform is still multiplication and division. At the same time as algorithmic reforms, beaded abacus may have appeared in the Northern Song Dynasty.The mathematicians of the Song and Yuan Dynasties all opposed to the mysticism of the Confucianists in varying degrees. Although Qin Jiulu once advocated the same source of mathematics and Taoism, he later realized that there is no mathematics for "going through the gods," but only the mathematics of "all things in the affairs of the world"; Mo Ruo is in the preface to "Siyuan Yujian" The "imaginary truth" and "virtual truth" represent the thoughts of a highly abstract thinking; Yang Hui's research on the aspect structure reveals the nature of Luo Shu and effectively criticizes the iconic mysticism. All these are undoubtedly important factors in promoting the development of mathematics.The climax of China's ancient computing technology reforms also appeared during the Song and Yuan dynasties. Song Yuanming’s historical documents contain a large number of practical arithmetic bibliographies of this period, which are far more numerous than the Tang Dynasty. The main content of the reform is still multiplication and division. "Leaving head and riding" was first seen in Zhu Shijie's "Academic Enlightenment". The "Nine Return" first appeared in Shen Kuo's "Meng Xi Bi Tan". Yang Hui further improved it in "Multiplication and Transformation of the General Theory" (1274) and Zhu Shijie in the "Enlightenment of the Study of Mathematics". The "reduction" was first seen in "Academic Enlightenment," and "Crashing" and "Starting One" were first proposed by Zhu Shijie, Ding Ju (author of the Ding Ju algorithm, 1355) and He Pingyu (author of "Detailed Algorithm". (1,373) and Jia Heng (author of "Algorithm Algebra") concretized it. The appearance of "reserving the head and multiplying" and "removing" enables the multiplication and division to be carried out in a single row without any modification, just like the current method of abacus. At the same time as the algorithm reform, beaded abacus may have appeared in the Northern Song Dynasty. However, if you think of modern abacus as a beaded abacus and a complete set of algorithms and arguments, then it should be said that it was finally completed in the Yuan Dynasty.The prosperity of mathematics in the Song and Yuan Dynasties is the inevitable result of social and economic development and the development of science and technology, and is the inevitable result of the development of traditional mathematics. In addition, mathematicians’ scientific thinking and mathematical thinking are also very important. The mathematicians of the Song and Yuan Dynasties all opposed to the mysticism of the Confucianists in varying degrees. Li Ye had criticized Zhu Xi’s writings and said that there are many places where it does not work. He pointed out that it is possible to say that mathematics is difficult to understand, but it is not correct to say that mathematics cannot be understood. He believes that mathematics comes from the natural world, and that “whether it can promote the theory of nature” can “clear the number of nature”. Although Qin Jiulu once advocated the same source of mathematics and Taoism, he later realized that there is no mathematics for "going through the gods," and there is only mathematics for "all things in the world."Chinese and Western fusionMing and Qing DynastiesChina entered the late period of the feudal society from the Ming Dynasty. After the end of the 16th century, Western elementary mathematics was introduced to China one after another. This made the Chinese mathematics research appear to be a fusion of Chinese and Western cultures. After the Opium War, modern mathematics began to spread to China. Into a period dominated by learning Western mathematics; by the late 19th century and early 20th century, modern mathematics research really began.From the early Ming Dynasty to the middle of the Ming Dynasty, commodity economy developed, and it is the popularization of abacus that is compatible with this kind of commercial development. The appearance of Kuiben's "Four Sentences" and "Lu Ban Mu Jing" at the beginning of the Ming Dynasty shows that abacus is very popular. The former is a textbook for children to read and read, and the latter includes the abacus as a household essential item in the general wood furniture manual.Decline in developmentThis period refers to the 1582 period when the Ming dynasty was established in the middle of the fourteenth century until the end of the Ming Dynasty. In mathematics, besides abacus, there has been an overall debilitating situation, which involves the limitations of medium calculation, the deletion of mathematics content in the 13th century examination system, and the complex issues of Ming Dynasty Daxing eight-segment examination system. Many Chinese and foreign mathematicians are still discussing The reasons involved.The greatest achievement of the Ming Dynasty was the popularization of abacus. Many abacus textbooks appeared, and Zhi Chengzhi’s “Accomplishment of Zhizhi Algorithm” (1592) came out. The theory of abacus has become a system, marking the completion of the transition from planning to abacus. However, due to the prevalence of abacus, the calculations are almost extinct. Ancient mathematics based on calculations has also been gradually lost, and mathematics has been stagnant for a long time.AbacusThe abacus algorithm and word of mouth have also gradually improved. For example, Wang Wensu and Cheng Dawei increased and improved the crash, and began to swear; Xu Xinlu and Cheng Dawei added and reduced the threshold and widely used the division in the division, thereby realizing all the oral operations of the four operations of the abacus; Zhu Zaihe Cheng Dawei applied the method of calculating open squares and opening cubes to abacus, and Cheng Dawei used abacus to solve the numbers quadratic and cubic equations. Cheng Dawei’s works spread widely at home and abroad and have a great impact.In 1582, Italian missionary Matteo Ricci went to China. After 1607, he successively translated the first six volumes of Geometry Original with the volume of "Measurement of Righteousness" with Xu Guangqi. Counting fingers. In 1629, Xu Guangqi was appointed by the Ministry of Justice to supervise the calendar, under which he compiled the 137 volumes of the Chongzhen almanac. The "Chong's almanac" mainly introduces the geocentric theory of the European astronomer Dicho. As the mathematics foundation of this doctrine, Greek geometry, some trigonometry in Yushan, Europe, and calculation tools such as Napier's calculation and Galileo's scale were also introduced.In the incoming mathematics, the most influential is the "geometry original." "Geometry" is China's first mathematical translation work. Most of the mathematical terms are original, and many of them are still in use today. Xu Guangqi believes that "it is not necessary to doubt" and "do not need to change" to it. "There is no one who does not learn from the world." After the Qing Dynasty invaded the Central Plains, science was once again thrown into the "cold palace." Not only did the late part of the book fail to be translated, even the first half that Xu Guangqi had already translated was no longer issued. The scientific writings brought by Western missionaries became the exclusive hobby of Emperor Kangxi, Yongzheng or Emperor Qianlong.The second most widely used is trigonometry. The books on Western trigonometry include "Measurements", "Cutting the Circular Eight Lines" and "Measurement Completeness." "Measurement" mainly describes the nature of the triangular eight-line (sine, cosine, tangent, cotangent, secant, cosecant, positive vector, and yoke), table creation methods, and table methods. In addition to the addition of some plane triangles missing in the "Measurement", "Measurement Completeness" is more important than product and difference formulas and spherical triangles. All of these were used with translations at the time of the calendar work.In 1646, the Polish missionary Mooney Court came to China and followed him to study Western science including Xue Fengqi and Fang Zhongtong. After the death of the Mooney House, Xue Fengqi was composed as a “learning pass” according to what he learned and wanted to link the Chinese, French, and Western law together. The mathematics content in the "Learning Society" mainly includes the proportional logarithm table "Proportional Four-line New Table" and "Triangle Algorithm". The first two books describe the logarithms of the British mathematicians Napier and Briggs inventing amendments. In addition to the spherical triangle introduced in the "Chongzhen almanac", the latter book still has half-width formulas, semi-arc formulas, Derby's ratios, and Nessler's ratios. Fang Zhongtong's "Several Degrees" explains the logarithmic theory. The introduction of the logarithm is very important and it is applied immediately in the calculation of the calendar.Scholars at the beginning and end of the Qing dynasty studied Chinese and Western mathematics, and many books were handed down. The major influences were Wang Xi’s “graphics”, Mei Wending’s “Mei Shi Congshu Series” (13 of them are mathematics books, 40 volumes in total), and Nian Xi.尧 "Inspection" and so on. Mei Wending is a mastermind of Western mathematics. He collated and studied the linear equations solution, the Pythagorean form solution, and the high-powered positive root method in traditional mathematics. Nian Xie's "Inspection" was the first Chinese book to introduce western perspectives.Qing Kangxi valued Western science, but only as his own hobby. In 1712, Kangxi was ordered to serve as a compilation official for Ren Mengyi, and together with Chen Houyao, He Guozong, Ming Antu, and Yang Daosheng, they compiled astronomical algorithms. In 1721, he completed 100 volumes of the “Legal Origins” and published it in 1723 in the name of “The Imperial Ding” of Kangxi. Among them, "Machinery Theory" is mainly responsible for Mei Xuancheng, and is divided into two editions. The edition includes "Geometry Original" and "Algorithm", which are all translated from French works. The following editions include Arithmetic, Algebra, Plane Geometry Triangle, and Stereo. Elementary mathematics, such as geometry, with a prime, logarithm, and trigonometric function table. Because it is a more comprehensive encyclopedia of elementary mathematics and has the name of "Yingding" of Kangxi, it has a certain influence on the mathematics research at that time.The mathematicians in the Qing Dynasty did a lot of communication work on Western mathematics and achieved many original achievements. These achievements, such as comparisons with traditional mathematics, have progressed, but they have lagged behind those of contemporary Western countries.After the reign of Emperor Yongzheng was retired, Western science stopped importing into China and imposed a high-pressure policy internally. As a result, the average scholar could neither touch Western mathematics nor dare to question the teachings of the world. Therefore, he buried himself in the study of ancient books. During the Qianjia period, a school of Qiangjia, which was mainly based on textual research, was gradually formed.With the collection and annotation of the "Ten Books" and the Song and Yuan mathematics books, there has been a climax to the study of traditional mathematics. Among them, Jiao Xun, Wang Lai, Li Rui, Li Shanlan and others who can break through the old frame and have inventions and creations. Their work has evolved in comparison with algebra in the Song and Yuan Dynasties; compared with Western algebra, it has been late in time, but these achievements were independently obtained without the influence of Western modern mathematics.At the same time as the climax of traditional mathematics research, Qian Yuan and Li Rui wrote a biography of astronomy mathematician - "Chou Renzhuan," collecting more than 270 deceased astronomers and mathematicians from the Yellow Emperor to Jiaqing. There are less than 50 mathematical works handed down, and 41 missionaries who introduced western astronomy mathematics since the late Ming Dynasty.After the Opium War of 1840, Western modern mathematics began to spread to China. The first is the British establishment of the Mohai Library in Shanghai to introduce Western mathematics. After the Second Opium War, government bureaucrats such as Zeng Guofan and Li Hongzhang started the "Westernization Movement." They also advocated introducing and learning western mathematics, and organized and translated a number of modern mathematics books.Among the more important ones are Li Xiaolan and Wylie Yali’s translation of “Algebra Math” and “Graduation of Micro Generations”; Hua Weifang and the English novelist Fu Lanya have translated the “Algebraic Technique”, “Micro Accumulation Traceability”, and “Determination Mathematics”; Zou Liwen And Di Kaowen compiled the "Shape Book", "Algebra Book", "Computational Mathematics"; Xie Hongxuan and Pan Shenwen's translation of the "Generation Covenant" "Eight-line preparation" and so on.The “Generation of Micro Generations” is China’s first translation of calculus; “Algebra” is a symbolic almanac of the English mathematician De Morgan; “Detecting Mathematics” is the first translation of probability theory. In these translations, many mathematical terms and terms have been created and are still in use today, but the mathematical symbols used have generally been eliminated. After the reform of the Reform Movement in 1898, new law schools were set up in different parts of the country, and some of the above-mentioned works became major textbooks.While translating Western mathematic works, Chinese scholars also conducted some researches and wrote some books. The most important ones are Li Shanlan's "Acupuncture Converting Method" and "Examination Root Method"; Xia Benxiang's "Diagram of Cave Art" "is caused by Qufu, Songs and Illustrations, etc., are the results of the research of Chinese and Western academic ideas.As the modern mathematics of input requires a process of digestion and absorption, and the corruption of the rulers in the late Qing Dynasty, under the impact of the Taiping Heavenly Kingdom Movement, under the plunder of the great powers, it was in jeopardy and there was no time for mathematics studies. It was not until the May 4th Movement in 1919 that the study of modern Chinese mathematics really began.