1-6 Write A Positive Or A Negative Number For Each Example: Fill & Download for Free

Because 1 + 1 = 2!TL;DRI’m not going to answer this short, however if you’re willing to follow, you’re going to watch probably the most beautiful and simplest idea of math. This gonna be a simple ride, but it would teach us the whole idea of math in just less than one hour. So, buckle up Dorothy!Counting stuffs, 1 + 1 = 2, yes, counting stuffs, by only and only this one simple rule, we built almost the entire foundation of the so called mathematics. Watch. We're not going to invent any new rule, NONE at all, to derive this result that - x - = + (minus times minus equals plus).Now, for the first step, let’s forget that you ever understood a number (1, 2, 3, etc), an operator (+, -, x, :, etc), or any thing else that you ever knew from mathematics. We’re going back to the time 5000 years ago when a number was barely invented (in Mesopotamia). Then, take a look at the stuffs at your desk. How are we going to count it? This is how we’re going to do it. For every one stuff at the desk, we put a mark (*) on a piece of wood, a clothes, a paper, or whatever, e.g.:* * * * *Those are 5 marks for probably: a laptop, a glass, a phone, a wallet, and a TV remote control.But soon, we're going into trouble. For example, how if we are going to also count all the stuffs in the room? Then we would end up with something like this:* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *Oo sh#t, no way! So, we invent a symbol that we could agree together to write the result of that counting more practically. NOTE: there's no new rule here, no at all, we just agree on how to write the result of the counting. Here are the symbols that we invent:1 for *2 for * *3 for * * *4 for * * * *(and so on)Nice! We could stop at an arbitrary number of symbols, but we choose to stop at the symbol "9", and agree that for every * * * * * * * * * * (ten marks) we would write it to the left. Very nice! For example:* * * * * * * * * * * *Would be written as: 12. How if we have counted pass one hundred? We apply the same agreement, that is for every ten of ten marks, we write it to the left. E.g.: 112, and so on. With this simple agreement, we have had a powerful tool to write our counting.When we're in the middle of counting our stuffs, supposed there's a call from the nature, so we stop, and go to the bathroom. We're back and start to count again. So, on the paper that we write our marks, would be something like this:* * * * * * * * (nature calls, stop, and then continuing) * * * *But that is, of course, the same as writing it as:* * * * * * * * * * * *The fact that the nature calls, doesn't change the result of the counting. We just stop, and then continue to write our marks. However it does give us the idea of addition operation. It means to continue the counting, and that's just it! First, we invent a new symbol "=" to denote "the result is", and write the addition process as following:* * * * * * * * (nature calls, stop, and then continuing) * * * * = * * * * * * * * * * * *You would notice immediately that writing "nature calls, stop, and then continuing" is too cumbersome. So we invent again a new symbol "+". NOTE: again, there's no new rule, this is still just about counting. We just stop, then count again. It's still exactly the same process: counting stuffs! So we write it as following:* * * * * * * * + * * * * = * * * * * * * * * * * *As we have invented some symbols for those marks, we could choose to write it as following:8 + 4 = 12It's the same thing all over again.Are you still with me? Good. Then the scenario changes a little bit. Your Mom tells you something like this:"There must be 12 stuffs at your room, I've counted it up to 8, would you continue it for me?"Sure, no big deal. But then, we would think something like this. Well, if it's supposed to be 12 stuffs, then we must expect that we should count just another 4 stuffs. Note there's nothing new here, but this our thinking process is the same as writing it as following:(Mom has counted until) 8 + (the remaining stuffs to be counted) = 12Try to replace the "the remaining stuffs to be counted", following our counting rule, then it starts to look like a game. Indeed it's a game. Mathematics is no more than a game, and this game surprisingly has only one rule: COUNTING STUFFS. Of course, we could not pretend that we don't know the answer (it's 4), but pretend that we haven't used to the "new" symbols we just invented. So we do something like this:* * * * * * * * + (some marks) = * * * * * * * * * * * *Sure, we could count it one by one, but it would be easier and save time if we arrange it like this:* * * * * * * * * * * ** * * * * * * *We don't have to count it all! We just need to count the missing marks which is:* * * *And when we look at the table of our new symbols above, we would find that it could be written as number "4". Again, there's no new rule here. (Sorry to remind you again and again about this -- but this is the most important idea that I would like to convey.) It's just playing the same game: counting. But it gives us a new idea in the process. That's the so called subtraction operation. Imagine, our Mom mentioned 12 stuffs, then has counted it up to 8 stuffs, then our counting should give the result 4. If we write it, it would be something like this:12 (Mom have counted it until) 8 (so our counting should give the result) 4Remember that we have invented the symbol for "the result is" which is "=". But that "Mom have counted it to" is too cumbersome, because we haven't invented a symbol for it, so why not we invent another symbol for this process? Right, the symbol we choose is "-". So we write it again as:12 - 8 = 4It has the same meaning as we arrange our marks so 12 marks are put above and 8 marks are put below and then we count the missing marks which is 4. And it has the same meaning as:8 + 4 = 12Taking away? How if we take away 12 marks from 12 marks, how many marks left? Nothing. No new rule, because it's just the same game we play above:12 + (some count) = 12Or write it using the new symbol "-" as:12 - 12 = (nothing)But it's the same thing if we do it to all kind of numbers:5 - 5 = (nothing)16 - 16 = (nothing)31 - 31 = (nothing)(and so on)Well, it's too cumbersome. Of course, we invent another symbol: "0" (called zero). No new rule, it's just another agreement on a symbol. It's just about how to write it, instead to write 12 - 12, 5 - 5, 16 - 16, 31 - 31, or whatever, it's simpler to just write it with this new symbol "0":12 - 12 = 05 - 5 = 016 - 16 = 031 - 31 = 0(and so on)We have had some of our symbols, now we're going to do something ABSOLUTELY amusing, like creative human being does it all the time. How if we encounter something like this:8 - 12 = (what should we put here?)No! No, no, no: we could NOT just combine the symbol "-" and the symbol "4", and make "-4". There is a "hole" in this kind of thinking. You may not feel it because you use to the concept of negative number. But we haven't arrived at that concept yet! Be noted that we haven't had something like "-4" -- that is a bare minus sign without a number at its left.That's not how mathematics developed! This is a very important lesson in mathematics (and also in life!), nothing could be concluded from out of nowhere, that's called gambling, and we would never win! That's wrong, wrong, and wrong.( Edit : thanks to Gary Williams who pinpointed the previous logic I used is kinda of broken. )So, here is how we attack this problem properly. Notice that we could use the symbol “-” as following:12 - 3 = 99 - 5 = 4And shortcut the symbols by writing them as following:12 - 3 - 5 = 9 - 5 = 4No we are going to use this finding to solve the problem 8 - 12 = (what should we put here?). There's no jumping in logic because there's no new symbol (and no new rule). Alas, we have invented the symbol “0”, so let's also use it here:8 - 12 = 8 - 8 - 4 = 0 - 4It’s the same with what we are doing before: take 8, and take another 4, so it's the same as take 12 in one take.I know what you are thinking now, but stop it first!So, we have this: "0 - 4". What is it? (No, it's not a negative number -- we haven't invented it yet.) When we're facing this kind of question, and feel a little bit confused, always go back to what these symbols mean. It means the counting of our marks. So it's simply means that we have 8 marks, but we want to take away 12 marks, thus AFTER we take away 8 marks, we're short for 4 marks.* * * * * * * *(try to take away)* * * * * * * * * * * *(thus we're in short for) * * * *That should be easy to understand.From now on, I'm going to make an excuse from not always mentioning something antique, such our marks, but we are going to start to play this game differently. It's called the abstract thinking. It's how a mathematician runs his day. Don't worry, it's just a rather scary call, but what I mean, we would just be more sticking to the pattern of the game, WITHOUT looking back to where it comes from too often. You would find it very useful. Here it goes.Every time we subtract something bigger from something smaller, we could always write it... yes, just write it, like these:1 - 5 = 1 - 1 - 4 = 0 - 43 - 9 = 3 - 3 - 6 = 0 - 616 - 31 = 16 - 16 - 15 = 0 - 15Then, as you've guessed, we are going to invent another new symbol. Rather than writing "0 - ..." each time, we would just write "-...". We remove the symbol "0", because there's no use to write it every time. We still have to leave the "-" sign though, to differentiate it from the plain number. Then we call it as negative number. You could breath, now we have had the so called negative number! You may find it too artificial, but that's the correct way to do it; that's to ensure there's NO JUMPING in the process. Mathematicians call it rigorous.(Well… NOT really rigorous yet, but it’s always good to start somewhere with something more intuitive!)So now we could write it using the new symbol of negative number "-..." as:1 - 5 = 1 - 1 - 4 = -43 - 9 = 3 - 3 - 6 = -616 - 31 = 16 - 16 - 15 = -15(It means the same thing, the symbol "-" means "short for", so "-4" just means we're short for "4".)Then we notice some pattern! We could just run the subtraction backward and adding the symbol "-" to the result. That is if we get "1 - 5", we could just write it as "5 - 1" and then add the symbol "-" in front of it. No new rule, we just watch some pattern, and we SHORTCUT the thinking process to get the final process. That's just it. And that is my friend, what I mean by an abstract thinking in mathematics, and that's very useful (because it could shortcut our thinking process). So if we also cut out the writing, mimicking the way we think it now (in abstract), we would have something clearer:1 - 5 = -43 - 9 = -616 - 31 = -15(Subtract the smaller one from the bigger one, then add the symbol "-" to its left.)Wow, it has been quite long, I hope you got the idea already! Now I'm going to speed up this rather boring explanation, by thinking in more and more abstract. Yes, speed it up, Johnny!We're going to do something like this:2 + 2 + 2 = 6It's too cumbersome, so we invent a multiplication symbol "x". That's just counting the number of the same numbers then write it as following:3 x 2 = 6It means the same thing. Then we could have something like this:2 x 3 = 3 + 3 = 6Hmm... let's try with other numbers:3 x 4 = 4 x 3 = 125 x 6 = 6 x 5 = 30(and so on)It seems obvious that switching the numbers in a multiplication doesn't change the result. But we could not just go on and calculate it for all numbers! So we need another way to be really really sure that it is the way it is. Let's get back to what it means in the first place: counting number! For example if we have 3 x 2, it means:* * * * * *But we could write it as:* ** ** *If we turn our head to look at it from its left (or its right), or simply just to rotate it, of course the count of the marks won't change! But it would look like this:* * ** * *See, it's 2 x 3. Thus we could repeat the same procedure to any kind of number (draw the marks, then see it from its left / right), the result would be the same. Well, we have our first equation! So rather than writing it for all numbers, we invent a new kind of symbols. No new rule -- it's just about how we write it. This new kind of symbol works as following:For the same numbers, we denote it with the same letters.For different numbers, we denote it with different letters.This game is called algebra. With this we could write "2 x 3 = 3 x 2" as:a x b = b x aBut that symbol "x" looks confusing with the letter symbols, so we agree to throw it away, and write it more elegantly as following:ab = baNow how about "a x -b"? Because we may haven't used to our new symbols, let's put back our numbers, for example: "3 x -2"? What does it mean? According to our agreement, it's the same as:-2 + -2 + -2If you still feel a little bit confused, let's write it as:0 - 2 + 0 - 2 + 0 - 2Or in our old day marks notation, we're short for 2:* * ** * * * *Then short for another 2:* * ** * * * *Then, another 2:* * ** * * * *So if we combine them all:* * * * * * * * ** * * * * * * * * * * * * * *We're short for 6. Do the same procedure for all kind of number we would have result that "3 x -2" is the same as "3 x 2" with the symbol "-" at its left, or:3 x -2 = -3 x 2 = -6In our new algebra game, we could write it as:a x -b = - a x b = -abHow if we add "b" to "c" then multiply it by "a", that is:b + cThen multiply it by aTo avoid confusion, we invent two symbols "(" and ")" to mean that whatever inside those two symbols would be counted first. So for a such operation we could write it elegantly as:(b + c) a = (by switching the numbers, then we would get)a (b + c)Let's put some number back, e.g.:2 x (3 + 4)Which is of course: (no new rule -- still counting number)2 x (3 + 4) = 2 x 7 = 14But we notice something beautiful if we use back our marks notation, that is 2 x 7 is written as:* * * * * * ** * * * * * *If we arrange it a little bit differently, it would not change the result, but there's something interesting pattern there:* * * | * * * ** * * | * * * *Bingo! It's the same as:(2 x 3) + (2 x 4)So:2 x (3 + 4) = (2 x 3) + (2 x 4)We repeat the same procedure for all kind of numbers, surely we would get the same result, thus we write it as following:a (b + c) = ab + acThat, we got our second equation!This game is so fascinating that people play it for thousands of years, but we are going to stop it just here. We have had every thing we need to answer:WHY "-a x -b = ab"??(Or WHY "- x - = +"?)Here it goes... BECAUSE:-a -b =-a -b + 0 = (adding 0 would not change the counting)-a -b + ab + -ab = (subtracting something from the same thing equals 0)-a -b + -ab + ab = (shift the "-ab" term to the left)(Notice that (-a x -b) + (-a x b) has the same pattern with previous result (a x b) + (a x c) = a x (b + c), that we could write it as -a x (-b + b), so...)-a -b + -ab + ab =-a (-b + b) + ab =-a 0 + ab = (note that -a x 0 = -(a x 0) = -(0 + 0 + 0 + ...) = -0 = 0)abOr if you prefer to put our numbers back, e.g. the number "2", then we would have:-2 x -2 =-2 x -2 + 0 = (adding 0 would not change the counting)(-2 x -2) + (2 x 2) + (-2 x 2) = (subtracting something from the same thing equals 0)(-2 x -2) + (-2 x 2) + (2 x 2) = (shift the (-2 x 2) term to the left)-2 x (-2 + 2) + (2 x 2) =-2 x 0 + (2 x 2) = (note that -2 x 0 = -(2 x 0) = -(0 + 0) = -0 = 0)0 + 4 =4You may wonder why don't I just cut all the craps, and bring you directly to algebra! That's because algebra is not the most fundamental building block of math, and I think you would get much more of the fun by watching how, for 5000 years, we as human beings collectively build this wonderful tool called algebra, and thus math. (I hope!)Anyway, that's how mathematics developed, and would stay that way until unforeseeable future. So every time you see a mathematician, you could say to him / her confidently:"Whatever the result you get, I could tell you that's ONLY BECAUSE 1 + 1 = 2."Thank you.

The question is, “If imaginary numbers aren't mathematically possible, why do they exist and why does it matter? What purpose do they serve that is essential to human life?”Imaginary numbers are not part of the system of real numbers. For example, if you want to know the square root of 4, it is 2 (i.e. 2x2=4). But if you want to know the square root of -4, there is no such real number (if R is a real number, RxR can never equal -4). In that sense, imaginary numbers are not mathematically possible.They exist because there are reasons to need something that does equal the square root of -4 (or any other negative number). The following discussion answers the question, in context with other similar questions about numbers.Consider the domain of numbers. It is built entirely of simple operations on simple objects. Yet we shall see that even very simple statements in this simple domain may be either true or false, depending on what universe of numbers we use. We will consider this carefully and at length, as an example of how people can hold conflicting views and express disagreement, based on their underlying (but normally unspoken) assumptions.As children, we learned first that the numbers (known as the positive integers) are 1, 2, 3, etc. The universe of mathematics, at that stage of our development, consisted in the ability to recite the numbers, and later, to count objects. The digit zero was a convenience, needed for writing the number 10 and its multiples. We could learn about addition and multiplication, but subtraction and division left us with some big questions, such as, “Why can’t you subtract 5 from 4?”, or “What’s the exact answer to dividing 7 by 3?” We could confidently assert that there are no numbers between 5 and 6. If our big sister had told us that 5 minus 8 is -3, or that 11 divided by 2 is 5.5 and is a number between 5 and 6, we would have said, “Those aren’t numbers!” And we would have been correct, considering the universe of numbers available to us. There are a number of things that are false for the positive integers, but true for a larger universe of numbers. Two of the true statements in the universe of positive integers are, “Any two numbers can be added to get another number” and “Any two numbers can be multiplied to get another number.” Three false statements are, “Any two numbers can be subtracted to get another number,” “Any two numbers can be divided to get another number,” and “Between any two numbers there is another number.”Later in our education, we learned about fractions (or ratios, formally known as “rational numbers”) and negative numbers. With these added to our universe, the three formerly false statements became true (excluding division by zero, which is undefined). The statements did not change, but their meaning changed, because of implicit reference to a different universe of available numbers. Thus 4 minus 5 is -1, 7 divided by 3 is 2-1/3, and some numbers between 5 and 6 are 5-1/2, 5-3/4, and 5-11/16. Note that there is some ambiguity about notation: without context, it is not possible to know whether 5-1/2 means “five-and-one-half” or “five minus one-half.” This ambiguity is not unusual: as a system becomes more complex and expressive, opportunities for ambiguity increase. In this case, we could avoid ambiguity by writing “5+1/2” or “5&1/2”, but the typographic convention “5-1/2” already exists, and we cannot make it disappear just by our own choice of usage. Note also that to write “5-1/2” as a fraction (that is, to explicitly display it as a rational number) we need to combine the whole number with the fractional part and write “11/2” (eleven over two).Now suppose that we want to indicate the lengths of the sides of a triangle. Certainly, these should all be numbers. However, if we limit ourselves to just the rational numbers (those that can be expressed as fractions) we can express the sides of some triangles, but not of others. It can be shown by purely geometric proof that for a right triangle (one having one 90-degree angle), the square of the length of the hypotenuse (diagonal side) is equal to the total of the squares of the other two sides (legs). For example, if the legs have lengths 3 and 4, the square of the length of the hypotenuse is 9+16, or 25. In this case, we can see that the length of the hypotenuse is another integer, 5. However, for the simple and obvious case of legs of length 1, the square of the hypotenuse is 2. This means that the length of the hypotenuse is √ 2 (the square root of 2). It is not difficult to prove that √ 2 is not a rational number—that is, it cannot be expressed as a fraction. It can be shown also that the circumference of a circle of radius 1 is not a rational number. In the universe of rational numbers, then, it is false to say “If the lengths of two sides of a triangle are numbers, so is the length of the third,” or, “If the radius of a circle is a number, so is its circumference.” This is obviously unsatisfactory, and so the universe of numbers is expanded to what is known as the “real” numbers. By this we mean all numbers that can be written in decimal notation as “a...bc.defg...”, where the letters indicate decimal digits, the three dots between a and b indicate any finite number of intervening digits, the dot between c and d is the decimal point, and the dots after g indicate any number, finite or infinite, of additional digits after the decimal point. We will see shortly that the term “real” reveals a certain prejudice.Before discussing this prejudice, though, let us see what statements are true and false in the universe of real numbers. In this universe, we can say, “If the lengths of two sides of a triangle are numbers, so is the length of the third side”, and, “If the radius of a circle is a number, so is its circumference.” However, we cannot say, “Every number has a square root.” While it is true to say that every positive real number has a square root, negative real numbers do not have real square roots. Just as the universe of the positive integers does not allow subtraction of a larger number from a smaller one, likewise the universe of real numbers does not allow the square root of a negative number. Why not? Because the only way a product of two real numbers can be negative is for one to be positive and the other negative. Thus the square root of a negative number would have to be either zero (clearly wrong) or simultaneously positive and negative (not possible). So in the universe of real numbers, the statement, “Every number has a square root,” is false. Now, this does not pose a problem if all we want to do is measure the sizes of objects, and other similar computations. However, it turns out that there are reasons for wanting the square root of a negative number.This brings us to the prejudice referred to earlier: the square root of a negative number is known as an “imaginary” number. This makes linguistic sense, in contradistinction to the “real” numbers; but the “imaginary” numbers are no more (or less) imaginary than, say, negative numbers or the square root of 2. Perhaps it would be better to call them “surreal” numbers; be that as it may, the name “imaginary” is conventional. The canonical imaginary number is the square root of minus one, denoted “i”. In the universe of “complex” numbers, consisting of the real numbers, the imaginary numbers, and the sums and products of real and imaginary numbers, the statement, “Every number has a square root,” is true.There are further extensions of the number system used for various purposes, but the ones described thus far are sufficient to illustrate the problems of communication resulting from different assumptions about the set of numbers. We have described the positive integers, all integers, the rational numbers, the real numbers, and the complex numbers. We now recapitulate, in tabular form, the truth and falsity of some statements relative to each number set.—Truth: A Path for the Skeptic, Mathematical Oddities.Other answers to this question have fully explained the usefulness and the necessity of imaginary numbers; this writer will refrain from repeating what they have said.

I'm taking this from the beginning of Tristan Needham's Visual Complex Analysis.Imaginary numbers were an idea that floated around the minds and scribblings of people doing algebra, mostly in Italy in the mid-16th century. However, most people who thought about them said they were dumb and left it at that.People wanted to solve quadratic equations, but something like [math]x^2 + 1 = 0[/math] simply has no solutions - not in the real numbers anyway. This is obvious because anything squared is zero or positive, so adding one to it can never get you to zero. You could introduce the "imaginary unit" [math]i[/math] and say that [math]i^2 = -1[/math], but if all you want to do is solve quadratic equations, this is extremely artificial. After all, the plot looks like this:It clearly doesn't have any solutions. Why make up new numbers for the sole purpose of being solutions to quadratic equations? That was the state of things for some time.Quadratics are fairly easy to solve, though. The best way to prove you were a badass mathematician in the 16th century was to step it up and solve a cubic equation. It wasn't considered useful. It was just an obvious problem out there that nobody knew how to do.The most general cubic in the variable [math]u[/math] is[math]a[/math][math] u^3 + b u^2 + c u + d = 0[/math]If you divide everything by [math]a[/math][math][/math], it gets a little simpler, and if you also make a substitution [math]x = u-\frac{b}{3a}[/math], you can get rid of the squared term. You're left with[math]x^3 - 3p x - 2q = 0[/math]where [math]p[/math] and [math]q[/math] are functions of [math]a,b,c,d[/math] that you can work out if you're interested.Well, various guys were working on this with the usual drama. They bickered over priority, called each other mean names, bit their thumbs at each other, etc. They even had public cubic-equation-solving exhibition matches. (This isn't in Needham, but I remember reading it somewhere.) Eventually, some dude named Cardano proved he had the biggest math dick by publishing the solution[math]x = \sqrt[3]{q + \sqrt{q^2 - p^3}} + \sqrt[3]{q - \sqrt{q^2 - p^3}}[/math]It looks strange, but if you try it out on some examples, you can see it works. Needham gives the example [math]x^3 = 6x + 6[/math]. (Needham credits Cardano with publishing the formula in his book, but it appears that others figured it out first. See the clarification by Dan in the comments.)Unfortunately when [math]p^3 > q^2[/math] this gives us complex numbers.You could just dismiss these complex numbers as useless like before, but that's less valid here since every cubic has at least one real solution. That's because cubics look like thisCubics shoot to negative infinity on one side and positive infinity on the other, so they must cross zero somewhere.Thirty years later, a guy named Bombelli looked at the cubic [math] x^3 - 15x - 4= 0[/math]. By trial and error, you can see that this has a root at [math]x=4[/math]. If you use Cardano's formula, though, you get imaginary numbers. The result is[math]x = \sqrt[3]{2 + 11\sqrt{- 1}} + \sqrt[3]{2 - 11\sqrt{-1}}[/math]This is suspicious. We know the answer is 4. The answer from the formula kind of looks like it has a 4 hiding in there, since it's 2 plus something and 2 minus something, but with cube roots and square roots of negative numbers thrown in.So now Bombelli had a reason to investigate complex numbers more deeply. He had an equation with a known solution, and a formula with negative square roots that looked like it might produce that solution. Setting [math]i = \sqrt{-1}[/math] and using the normal rules of algebra, Bombelli showed that[math](2 \pm i)^3 = 2 \pm 11i[/math]this lets us go back to the result of Cardano's formula and write it as[math]x = (2 + i) + (2 - i) = 4[/math]At that point, Bombelli realized these things were useful. You could take problems that were already understood and produce solutions that were understood, but only if you went through complex numbers first. That was the turning point in their history. They had a long infancy, though. The geometric interpretation of complex numbers as points on the complex plane did not come along until the early 19th century.Of course today they are ridiculously useful across all the major fields of mathematics, to the point where even as a non-mathematician I was nonetheless coerced into spending many sleepless nights not understanding them in college.Needham references a book by John Stillwell called Mathematics and Its History, so that would be a place to look for more information.