Subtracting Positive Negative Numbers 83 Practice 1: Fill & Download for Free

“Do calculators use PEMDAS?”It depends on the calculator.If the calculator is not described as a scientific calculator, as what might be handed out as a giveaway item or sold for around a dollar and only adds, subtracts, multiplies and divides (and perhaps extracts square roots), then no, it will operate strictly in order your typing in the operations, so:3 + 21 × 5 = 120 [not matching PEMDAS 108].In fact, if you watch the display as you type the keys:After pressing 3 : display shows 3.After pressing 3 + : display shows 3.After pressing 3 + 2 : display shows 2.After pressing 3 + 21 : display shows 21.After pressing 3 + 21 × : display shows 24 (having seen the ×, it knows the number 21 is complete so completes the previous operation of 3 + 21 = 24.After pressing 3 + 21 × 5 : display shows 5.After pressing 3 + 21 × 5 = : display shows 120.An operation is always carried out immediately upon completion of entry of the second operand (detected by the pressing of another operation key or the = key)—it does not wait to see that the next operation might have higher priority, since to such calculators all operations have equal priority.If you intended the multiplication to exercise its priority over addition, then you must enter the multiplication before the addition, such as 21 × 5 + 3.Such calculators do not provide parentheses to override the left to right order—the calculator provides only enough memory for itself to hold 2 numeric values and one pending operation. Usually there is one extra memory location for the user to store intermediate results, so to calculate 2 × 3 + 4 × 5 with the normal precedence of operation to get 26 instead of 50, you will need to:Make sure the user memory slot is clear (contains 0), usually be pressing the MC key. Then key in 2 × 3 = to see 6 in the display. Then press the M+ key, which adds this 6 to the 0 currently in the user memory and puts the result in the user memory. Then key in 4 × 5 + MR = (when the + is encountered the 4 × 5 is evaluated as 20, and the MR key recalls the 6 stored in user memory to be the second operand for the +. If you have three products to add, the second product should be processed using the M+ key instead, and then use the + MR = on the final product. It requires keeping on your toes and typically having paper and pencil handy (as often the user needs more than one memory location) if you find the so-called “algebraic operation” second nature, as it is very easy to get confused with what the calculator is doing versus what you want it to do.Many calculators in the $10 and up price range are referred to as “scientific calculators” or as “using algebraic operation”, as described in algebra class, which means they regard different operations to have different precedence levels, such as multiplication before addition and subtraction regardless of how the expression is keyed in, with parentheses provided to override the natural order. There have been two main variants of the precedence rules. Some of these calculators require that multiplication always be indicated by pressing the × key; others allow the multiplication to be implied, such as 2(3) instead of 2 × 3, and in a few cases 2π where the calculator retains the π symbol in the display, instead of showing 3.141592654, and regarding it as the product of 2 and π.The TI-82 and older TI calculators, as well as older Casio calculators (but I do not remember model numbers), would follow standard practice of professional mathematicians and physicists and performed implicit multiplications before all divisions and all other multiplications (with divisions and the latter multiplications being treated as co-equal and performed left to right). Thus:6 ÷ 2 ( 1 + 2 ) is evaluated as 6 ÷ 2(1 + 2) = 6 ÷ 2(3) = 6 ÷ 6 = 1, whereas6 ÷ 2 × ( 1 + 2 ) is evaluated as 6 ÷ 2 × (1 + 2) = 6 ÷ 2 × (3) = 3 × 3 = 9.We see that when the multiplication is written implicitly, it is done before the division because it has higher precedence than the division, whereas when the multiplication is written explicitly, it is done after the division because it is equal in precedence to the division and is to the right of the division. And, yes, such a distinction between implicit multiplication and explicit multiplication has been done traditionally by most professional mathematicians and physicists, a practice that largely continues to this day, though in recent years a significant number of professional mathematicians and some professional physicists have changed their position from the above evaluations to regarding both expressions (implicit and explicit multiplication) to be meaningless with consequent undefined value—neither 1 nor 9 nor any other value. In order for the expressions to have semantic meaning, they must be rewritten with explicit grouping symbols (either an added pair of parentheses or writing the division vertically with a vinculum [horizontal bar] rather than slash with the components being put above the vinculum, below the vinculum, or to the right). To see the TI-82 order of operations, look at https://math.oregonstate.edu/files/math/home/programs/undergrad/TI_Manuals/ti82Guidebook.pdf (https://math.oregonstate.edu/files/math/home/programs/undergrad/TI_Manuals/ti82Guidebook.pdf), page 1–20. You will see that implicit multiplication in the form I discussed is at level 3 below powers and roots, while multiplication and division in general are together way on down at level 7, with addition and subtraction together at level 8.It seems appropriate in my mind that calculators would interpret and evaluate mathematical expressions the same way as professional mathematicians and physicists do. However, TI and Casio received a substantial and rapidly growing number of complaints from secondary school mathematics teachers that the calculators were not evaluating the expression 6 ÷ 2(1 + 2) according to the rules of PEMDAS that are being taught in school (which would insist on an answer of 9, regardless of the multiplication being implicit or explicit), and such a situation is totally unacceptable (in the minds of the teachers)—so unacceptable that the school districts would be seeking a calculator vendor that does it “correctly” so that the calculators used in the schools will give the same results as the textbooks. (A similar situation happened about the same time with 0⁰, which the calculators, in agreement with many professional mathematicians and physicists, produced a result of 1, whereas the secondary school textbooks inappropriately said it is undefined, so this “issue” got addressed mostly at the same time.) Of course, TI and Casio got on the problem right away and put out new models of their calculators (the TI-83 in the case of TI) that performed the calculations “correctly”. And how was “correctly” determined? Based on standard mathematical usage by professional mathematicians and physicists? Of course not! The best way to determine the correct resolution is to determine the impact on profit—everybody should know that. Money speaks and if the ones with more money say something is so, then it is so, even though the professionals indicate, with rationale, why that is the wrong answer technically even if not financially. To TI and Casio, calculators are instruments for producing revenue, not instruments for handling mathematical calculations—though, of course, to enhance revenue by attracting buyers of the calculators, the calculators need to be advertised in such a way as to appear useful to the buyer by the way they handle mathematical calculations. You see, professional mathematicians are not big purchasers of calculators because most do not do basic arithmetic that need numeric expressions evaluated in their research but work with abstract concepts or formulas involving interrelationships among certain variable entities; the few that are involved with extensive numeric calculations for statistical data analysis (such as actuarial data) processing so much data that it is handled on computers, not handheld calculators. Professional physicists tend to be much more involved with numeric calculations, but, like the mathematicians, they involve huge amounts of data and calculations that necessitate use of computers, not handheld calculators. On the other hand, many school districts buy large numbers of calculators so that each student in a classroom can have one to work with during class for demonstrations, so we are talking a lot more calculators going to classrooms than professional mathematicians and physicists, which means a lot more revenue from the schools, so the school teachers exercise a lot more clout than professional mathematicians and physicists in how calculators behave. So, yes, many of the more recent calculators to adhere to PEMDAS plus extensions for unary operations that PEMDAS does not address. To see the TI-83 order of operations, look at https://math.oregonstate.edu/home/programs/undergrad/TI_Manuals/ti83Guidebook.pdf (https://math.oregonstate.edu/home/programs/undergrad/TI_Manuals/ti83Guidebook.pdf), page 1–22. You will see that several layers of the TI-82 rules on order of operations have been compressed, so that now [explicit] multiplication, implied multiplication, and division are all together at level 5, and PEMDAS is adhered to while adding some new categories that PEMDAS does not address.For a discussion as to why professional mathematicians and physicists generally regard the TI-82 behavior as superior to the TI-83 behavior in handling the 6 ÷ 2(1 + 2), read Howard Ludwig's answer to What is a good way to introduce PEMDAS to young students? (NOTE: The discussion in that answer is not focused on the one expression but on how PEMDAS violates conventional professional practice, which is an issue of outright false information, and on how, even if it had been correct information, how problematic PEMDAS is as an educational tool due to multiple opportunities for serious confusion.)If the reader is astute, the two previous paragraphs might have triggered some surprise or confusion. Did I just say that professional mathematicians and physicists interpret mathematical expressions in a manner contrary to PEMDAS, and, if, so, did I really mean to say that? Yes and Yes. You might have been told that PEMDAS is the only set of rules you will ever need to know because it will handle all mathematical expressions and all properly educated people use it to interpret all mathematical expressions—and, assumedly, professional mathematicians and physicists are properly educated in how to properly write and interpret mathematical expressions, because that is what they spend a lot of time doing. Sorry, but PEMDAS is incomplete and will not provide direction on how to fully interpret all mathematical expressions, and when the gaps are filled in for PEMDAS, it is necessary to break some of the core tenets of PEMDAS, meaning that it cannot be thought of as working correctly in the big scheme of things even for expressions it was explicitly intended to handle.Now, there is yet one other category of calculators and it applies, in so far as I know, to only one brand, HP. These calculators do not use strict left-to-right evaluation nor do they use “algebraic operation”; instead they use something called RPN, which is a quite different way of writing expressions—one which in its simplest form does not need parentheses so the traditional HP calculators have no key for parentheses. RPN is an acronym for Reverse Polish Notation (and the “Polish” refers to the nationality of Poland, not making something shiny). The reason it was called this by HP, is that a Polish mathematician and logician by the name of Jan Łukasiewicz—a name that most Americans cannot remember how to spell or pronounce and involves a letter that does not exist in the English alphabet, so Americans remedy that problem by simply referring to his nationality rather than his name—came up with a notation to handle expressions of symbolic logic that would never require parentheses. In traditional expression of binary operations, the operator is placed between the two operands (sometimes called infix) as 2 + 3, but with multiplication taking precedence over addition, you need to write parentheses around the addition to indicate it is overriding the standard precedence of multiplication over addition if you need the addition done first. Łukasiewicz would, instead, place both operands after a binary operator (sometimes called prefix), as + 2 3 to add 2 and 3. If you wanted 3 + 4 × 5, you would write 3 4 5 × +, but if you wanted the addition done first instead, you would write 3 4 + 5 ×, with no parentheses needed in either case. HP with RPN reversed the order so that the operator came after both operands as 2 3 +. With this notation, there is no precedence of operations—they are all equal. Order of operations is handled, quite literally, in the same order the operations are written on a line, with operands preceding the operators. I used only the earlier HP calculators that had only RPN available. I have heard that later HP calculators added the ability to use algebraic order in case you were not comfortable with RPN, but the ability to use RPN had a lot of hard-core fans and was also still supported.Thus, you see that the order of operations depends rather heavily on which calculator you are using. The best thing is to read the manual where it deals with that topic. Since you know enough to ask about PEMDAS, you are probably not going to be using a strict left-to-right input-order calculator (it would drive you nuts if you had to use it for any significant amount of calculation), but there are still several variations of the scientific calculators. This also applies to calculator programs that are included or can be readily added to computers or smart phones. The best thing is to try out a few things (and remember the outcomes) so you can make proper use of your calculator)—some are minor nits while others are critical:3 + 4 × 5: Does the calculator give you the correct 23 or the incorrect 35?−3²: Does the calculator give you the correct −9 or the incorrect 9?0⁰: Does the calculator give the correct 1 or incorrectly indicate an error?6 ÷ 2(1 + 2): Does the calculator allow you to input the expression this way or does it require you to put an explicit × between the 2 and the (1 + 2)? If it allows skipping the ×, does the calculator give you the more appropriate 1 or the PEMDAS 9?0.5!: Does the calculator support calculating factorials? If so, does it accept integer + 1/2 values as operands (0.5! should result in an error indication or give the same result as one-half the square root of π (0.886 226 925…). A few calculators will allow even more general real numbers.4³: To calculate arbitrary powers, does the calculator require putting the base first and then the exponent, or the exponent first and then the base? If the calculator has a button for the nth root of b, which is to be input first—n or b?For unary operations (operations with one operand, like square root, cos, ln, !) does the calculator require you to input the operation of the operand first?1000000 × (√2 − 1.414 213): The answer should be 0.5623731. How many of the decimal places (not counting the 0) did it give you? The number of decimal places it gave you, plus 6, is the number of significant digits it works with. For handheld calculators, this number will typically be 10, 12, 13, or 14, and typically only 10 will be displayed. The Microsoft Windows calculator on PCs goes out past 30 correct digits; Excel is usually 15 correct digits. (You will need more digits than what I gave you to show what Excel or the Microsoft calculator can do.)40 ÷ 6: Does the calculator show all 6’s after the 1, or is there a 7 at the very end? Some calculators will round off the internal value for the display; others merely lop off the digits not presented on the display. You might need to adjust the display value if that is what you want to use or present but the display did not round the value.Programming languages for computers also vary. Most of them do not have many symbolic operators (except for APL, which has lots of rather obscure symbolic operators), and usually all operators need to be explicit (typically no implied multiplication supported); these operators typically cover basic arithmetic (possibly with exponentiation or remainder evaluation), value comparisons (equality, inequality, less than, greater than), and basic logic operations (negation, conjunction, inclusive and exclusive disjunction). Most languages have a hierarchy of operator precedence to indicate the order of operations; where such hierarchies exist and overlap, they mostly agree—the most common disagreements the relationship between unary arithmetic negation with other arithmetic operations (although the disagreements usually do not have any consequence), and the relationship among conjunction and the two disjunctions.Smalltalk evaluates strictly left to right unless overridden with parentheses.APL evaluates strictly right-to-left unless overridden with parentheses.Lisp and Scheme have no hierarchy but are structured as prefix or Polish Notation; however, instead of getting rid off all parentheses, every operation is regarded as a function call, and all function calls require being set off with parentheses, in part because, some operations take an arbitrary number of operations, so you need parentheses to bound which operands to which operator.For some languages the rules get so complicated, that extra doses of parentheses are recommended. It can be hard enough to keep it straight for yourself as you write code, because one language may have as many as 17 levels of precedence, and the relative positions of particular operators in the precedence hierarchy varies some from language to language. If you frequently deal with multiple languages, it is a common source of error to mix up the rules for different languages. That in and of itself is motivation to use extra doses of parentheses, especially when many people come back to deal with some code they wrote 6 months before and have no idea how it worked any more. Even if you are sure you know the rules and will not mess up, how about giving the reader a break. There are a few things that should be obvious (like multiplication over addition and subtraction) so the use of parentheses adds potentially confusing clutter and should not be used, but there are cases where things are confusing and the parentheses should be added, even if they are technically not needed since you are not overriding any defined hierarchy.Some languages allow the compiler to rearrange mathematical expressions as long as they do not change the meaning mathematically. For example, A + B + C might be processed to compute (A + B) + C, A + (B + C), or even (A + C) + B; A * B + A * C might be processed as A * (B + C). If there is any reason you really need things done in a certain order, you may need to use more parentheses than normal or break the evaluation of the expression into multiple pieces handled in separate statements, and you might need to turn off compiler optimization of critical parts of code.As you can probably tell, I have had a wide variety of experience in various fields of study: physics research (including experimental research that led to PhD); mathematics from numerous perspectives (automated theorem proving research, solving some interesting equations for applied problems, finding efficient [both general use and optimized for a specific processor] algorithms for computing functions on data, competitive mental calculations for speed and accuracy, etc.); radar engineering and development of radar systems; software development (conventional mainframe software, soft-real-time embedded, hard-real-time embedded—single processor and multi-processor) in a wide variety of programming languages (as well as study of the theory and application of programming language design, compiler optimization, analysis of algorithms, etc.). I have tried to make use of learning from one area to help me to a better job in another area, which has sometimes meant knowing to try out unconventional approaches for a new area. Much of my experience has fortunately been proactive in order to avoid mistakes by having good knowledge and planning, but, of course, inevitably my share of lessons have been based on making my own mistakes.

A keto or ketogenic diet is a low-carb, moderate protein, higher-fat diet that can help you burn fat more effectively. It has many benefits for weight loss, health, and performance, as shown in over 50 studies.1 That’s why it’s recommended by a growing number of doctors and healthcare practitioners.2Here, you’ll learn how to eat a keto diet based on real foods. Get started with our visual guides, recipes, meal plans, and simple 2-week Get Started program. It’s everything you need to succeed on keto.1. What is a keto diet?The keto diet is a very low-carb, higher-fat diet. It’s similar in many ways to other low-carb diets.While you eat far fewer carbohydrates on a keto diet, you maintain moderate protein consumption and may increase your intake of fat. The reduction in carb intake puts your body in a metabolic state called ketosis, where fat, from your diet and from your body, is burned for energy.You can quickly learn more about the basic ideas behind the keto diet in this video course: What “keto” meansA “keto” or “ketogenic” diet is so named because it causes your body to produce small fuel molecules called “ketones.”5 This is an alternative fuel source for your body that can be used when blood sugar (glucose) is in short supply.When you eat very few carbs or very few calories, your liver produces ketones from fat. These ketones then serve as a fuel source throughout the body, especially for the brain.6The brain is a hungry organ that consumes lots of energy every day, and it can’t run on fat directly. It can only run on glucose – or ketones.7On a ketogenic diet, your entire body switches its fuel supply to run mostly on fat, burning fat all day long. When insulin levels drop very low, fat burning can increase dramatically. It becomes easier to access your fat stores to burn them off.8This is great if you’re trying to lose weight, but there can also be other benefits, such as less hunger and a steady supply of energy — without the sugar peaks and valleys that often occur when eating high-carb meals. This may help keep you alert and focused.9When the body produces ketones, it enters a metabolic state called ketosis. The fastest way to get there is by fasting – not eating anything – but nobody can consistently fast forever.10A keto diet, on the other hand, also results in ketosis and can be eaten indefinitely. It has many of the benefits of fasting – including weight loss – without having to fast long term.11For more details about pros and cons in different situations, check out our full guide: Is a keto diet right for you?Are you a doctor or do you need your doctor to help you with medications on a keto diet? Have a look at our low carb for doctors guide.Disclaimer: While the ketogenic diet has many proven benefits, it’s still controversial. The main potential danger regards medications, e.g. for diabetes, where doses may need to be adapted (see above). Discuss any changes in medication and relevant lifestyle changes with your doctor. Full disclaimerThis guide is written for adults with health issues, including obesity, that could benefit from a ketogenic diet.Controversial topics related to a keto diet, and our take on them, include saturated fats, cholesterol, whole grains, red meat, whether the brain needs carbohydrates and restricting calories for weight loss. Learn more2. What to eat on a keto dietHere are typical foods to enjoy on a ketogenic diet. The numbers are net carbs per 100 grams (3.5 ounces) of food.13 To remain in ketosis, lower is generally better: What’s the most important thing to do to reach ketosis? Avoid eating too many carbs. You’ll likely need to keep carb intake under 50 grams of net carbs per day, ideally below 20 grams.14The fewer the carbs, the more effective the diet appears to be for reaching ketosis, losing weight or improving type 2 diabetes.15Counting carbs can be helpful at first. But if you stick to our recommended foods and recipes you can stay keto even without counting.Here’s what you should avoid on a keto diet – foods containing a lot of carbs, both the sugary and the starchy kind. This includes starchy foods like bread, pasta, rice and potatoes. These foods are very high in carbs.You should also avoid low-fat diet products. A keto diet should be moderately high in protein and will probably be higher in fat, since fat provides the energy you’re no longer getting from carbohydrate. Low-fat products usually provide too many carbs and not enough protein and fat.17What can you drink on a ketogenic diet? Water is the perfect drink, and coffee or tea are fine too. Ideally, use no sweeteners, especially sugar.18A splash of milk or cream in your coffee or tea is OK, but beware that the carbs can add up if you drink multiple cups in a day (and definitely avoid caffe lattes!). The occasional glass of wine is fine too.Check out our full guides to keto drinks and keto alcohol.For more on specific topics – like what fruits or nuts to eat on a ketogenic diet – check out our popular visual guides: How low carb is a keto diet?A keto diet is a very strict low-carb diet, containing less than 20 grams of net carbs per day.Here are three examples of low-carb meals, based on how many carbs you aim to eat in a day.19 Note that only the left plate is reliably ketogenic for everyone: Ketogenic 0-20We recommend starting out by following the dietary advice as strictly as you can. When you’re happy with your weight and health, you could carefully try eating a few more carbs (if you want to). Learn moreDo you have another translation or a significant improvement of one of the earlier ones? E-mail us.3. Keto benefitsThe benefits of a ketogenic diet are similar to those of other low-carb and higher-fat diets, but it appears to be more powerful than liberal low-carb diets.20 Think of keto as a super-charged, low-carb diet, maximizing the benefits. However, it may also increase the risk of side effects a bit.Turning your body into a fat-burning machine can be beneficial for weight loss. Fat burning is significantly increased, while insulin – the fat-storing hormone – levels drop greatly.21 This appears to make it far easier for body fat loss to occur, without hunger.22More than 30 high-quality scientific studies show that, compared to other diets, low-carb and keto diets result in more effective weight loss.23Our new 10-week program helps you lose weight in a healthy and sustainable way.On a keto diet you’re likely to gain better control of your appetite. It’s a very common experience for feelings of hunger to decrease dramatically, and studies prove it.24This usually makes it easy to eat less and lose excess weight – just wait until you’re hungry before you eat.25 It also makes intermittent fasting easier, something that can enhance efforts to reverse type 2 diabetes and speed up weight loss, beyond the effects of keto only.26Plus, you could save time and money by not having to snack all the time. Many people only feel the need to eat twice a day on a keto diet (often skipping breakfast), and some just eat once a day, termed OMAD.27Not having to fight feelings of hunger could also potentially help with problems like sugar or food addiction.28At last, feeling satisfied can be part of the solution. Food can stop being an enemy and become your friend, or simply fuel — whatever you prefer.Studies prove that a ketogenic diet is excellent for managing type 2 diabetes, sometimes even leading to complete reversal of the disease.29 It makes perfect sense, since keto lowers blood-sugar levels, reduces the need for medications and reduces the potentially negative impact of high insulin levels.30Since a keto diet may reverse existing type 2 diabetes, it’s likely to be effective at preventing it, as well as reversing pre-diabetes.31Note that the term “reversal” in this context simply means that the disease gets better, normalizing glucose control while minimizing the need for medications. In the best case, blood glucose returns to normal without the need for medications.However, lifestyle changes only work when you do them. If a person returns to the lifestyle he or she had when type 2 diabetes appeared and progressed, over time it is likely to return and progress once again.Many studies show that low-carb diets improve several important risk factors for heart disease, including the cholesterol profile, which includes high-density lipoprotein (HDL) cholesterol and triglycerides. Total and low-density lipoprotein (LDL) cholesterol levels are usually impacted modestly.32Some people use ketogenic diets specifically for increased mental performance. Also, it’s common for people to experience an increase in energy when in ketosis.36On keto, the brain doesn’t need dietary carbs. It’s fueled 24-7 by ketones along with a smaller amount of glucose synthesized by your liver. There is no need for dietary carbohydrates.37Therefore, ketosis results in a steady flow of fuel (ketones) to the brain, thus avoiding problems experienced with big blood sugar swings.38 This may sometimes result in improved focus and concentration, and resolution of brain fog, with improved mental clarity.39A keto diet can result in a calmer stomach, less gas, fewer cramps and less pain, often resulting in improvements in IBS symptoms.40For some people this is the top benefit, and it often only takes a day or two to experience it.41Learn moreKetogenic diets can in theory increase your physical endurance by improving your access to the vast amounts of energy in your fat stores.42The body’s supply of stored carbohydrates (glycogen) only lasts for a couple of hours of intense exercise, or less. But your fat stores carry enough energy to potentially last for weeks.43Beyond this effect, another potential benefit is the reduction in body fat percentage that can be achieved on a keto diet (see weight loss, above). This reduction in body fat weight is potentially valuable in a number of competitive sports, including endurance sports.The ketogenic diet is a proven and often effective medical therapy for epilepsy that has been used since the 1920s. Traditionally it was used primarily for children, but in recent years adults have benefited from it as well.44Using a ketogenic diet for epilepsy can allow some people to take fewer anti-epileptic drugs or none at all, while potentially still remaining seizure-free. This may reduce drug side effects and thus increase mental performance.45We love receiving stories from people who are using a keto diet to dramatically improve their health. We’ve been sent thousands of such stories, and we publish some of the most amazing ones – over 300 so far – for you to read.494. How to get into ketosis on a keto dietHere are the seven most important things to increase your level of ketosis, ranked from most to least important: Restrict carbohydrates to 20 digestible grams per day or less – a strict low-carb or keto diet. Fiber does not have to be restricted, it might even be beneficial for ketosis.50How much is 20 grams of carbs? Use our visual guide to find out. Or simply use our keto recipes and meal plans, which are designed to keep you under 20 grams. No calorie counting required.Often, just restricting carbs to very low levels results in ketosis. So this may be all you need to do. But the rest of the list below will help make sure that you’re successful.Eat enough fat to feel satisfied. A keto low-carb diet is normally a higher-fat diet, because fat supplies the energy that you are no longer getting from carbs.51 This is the big difference between a keto diet and starvation, which also results in ketosis. A keto diet is sustainable while starvation is not.52If you feel as if you’re starving, you’re likely to feel tired and want to give up your diet. But a ketogenic diet should help you avoid getting too hungry, making it sustainable and possibly making you feel great.53So eat enough protein foods and low-carb veggies, with enough added fat to feel satisfied. If you’re hungry all the time, check that you are getting adequate amounts of protein at most meals and, if so, add more fat to your meals (like more butter, more olive oil, or some delicious sauces).Our keto recipes have plenty of fat included, but you can adjust up or down, according to your own needs.Maintain a moderate protein intake. A keto diet is not meant to be a very-high-protein diet. We recommend 1.2 to 2 grams of protein per kg of reference body weight per day.54 This means about 100 grams of protein per day if your lean body mass weight is around 70 kilos (155 pounds). Check out our target protein ranges to find out how much protein you should be aiming for each day.Despite concerns that people on keto diets eat “too much” protein, this does not seem to be the case for most people. Because it is very filling, most people find it difficult to overeat protein.55Although amino acids from protein foods can be converted to glucose, under experimental conditions, only a small percentage actually are.56 This may be related to individual factors, such as degree of insulin resistance.57 However, even people with type 2 diabetes usually do well with the adequate levels of protein Diet Doctor recommends, if their diets are also low carb.58At the same time, inadequate protein intake over extended periods of time is a serious concern. It can result in loss of muscle and bone, especially as you age.This is why our keto recipes are designed with the right amount of protein. See our protein guide.Avoid snacking when not hungry. Eating more often than you need, just eating for fun, or eating because there’s food around, reduces ketosis and slows down weight loss.59 Though using keto snacks may minimize the damage when you’re hungry between meals, try to adjust your meals so that snacks become unnecessary.If necessary, add intermittent fasting. For example, skip breakfast and only eat during 8 hours of the day, fasting for 16 hours (i.e. 16:8 fasting). This is effective at boosting ketone levels, as well as accelerating weight loss and improving insulin resistance.60 It’s also usually easy to do on keto.Add exercise. Adding any kind of physical activity while on low carb can increase ketone levels moderately.61 It can also help speed up weight loss and improve type 2 diabetes.62 Exercise is not necessary to get into ketosis, but it may be helpful.Sleep enough and minimize stress. Most people benefit from a minimum of seven hours of sleep per night, on average. And try to keep stress under control. Sleep deprivation and stress hormones raise blood sugar levels, slowing ketosis and weight loss.63 Plus they might make it harder to stick to a keto diet and resist temptations.64 So while handling sleep and stress will not get you into ketosis on their own, they are still worth thinking about.Keto supplements are not required. Note what’s not on the list above: you do not need expensive supplements, like exogenous ketones or MCT oil (medium-chain triglycerides). These supplements will likely not help you lose weight or reverse disease. At least there’s no evidence for that.65Learn more in our ketosis guideBottom line: To get into ketosis, restrict carbs to very low levels, preferably below 20 net carbs per day. That’s a ketogenic diet, and it’s by far the most important thing for ketosis to occur.Should you need to increase the effect, implement more steps from the list above, starting from the top. Got questions? Our Facebook group has answers.5. How to know you’re in ketosisHow do you know if you’re in ketosis? It’s possible to measure it by testing urine, blood or breath samples. But there are also telltale symptoms that require no testing: Dry mouth and increased thirst. Unless you drink enough water and get enough electrolytes like sodium, you may feel a dry mouth. Try a cup of bouillon or two daily, plus as much water as you need. You may also feel a metallic taste in your mouth.6667Increased urination. A ketone body, acetoacetate, may end up in the urine. This makes it possible to test for ketosis using urine strips. It also – at least when starting out – can result in having to go to the bathroom more often. This may be the main cause of the increased thirst (above).Keto breath. This is due to a ketone body called acetone escaping via our breath.68 It can make a person’s breath smell “fruity,” or similar to nail polish remover. This smell can sometimes also come from sweat, when working out. It’s often temporary. Learn moreOther, less specific but more positive signs include: Reduced hunger. Many people experience a marked reduction in hunger on a keto diet.69 In fact, many people feel great when they eat just once or twice a day, and may automatically end up doing a form of intermittent fasting. This saves time and money, while also speeding up weight loss.70Possibly increased energy. After a few days of feeling tired (the “keto flu“) many people experience a clear increase in energy levels. This can also be experienced as clear thinking, a lack of “brain fog,” or even a sense of euphoria.71There are three ways to measure for ketones, which all come with pros and cons. For a detailed comparison, see our full guide to the best way to test ketones.For a shorter version, use the links below to our ketosis guide. Note that we have no affiliations with any of the brands shown here.726. Practical keto diet guidesA keto diet is simple, but it helps to learn some basic new skills. How do you prepare easy keto breakfasts? Have you shunned fat for years and don’t know how to get more in your diet? How do you eat out and still stay on plan?These tips and guides answer common keto questions.How should you start your day? If you love bacon and eggs, dig in! If you don’t, some awesome keto breakfasts have no eggs at all.Have you been told that “breakfast is the most important meal of the day”? That’s likely not true.73 If you’re not hungry when you wake up, feel free to skip breakfast or just have a cup of coffee. Reduced hunger is common on a keto diet, so don’t worry about skipping any meal.74If you’re hungry when you wake up but are short on time, many keto breakfasts are tasty, filling and fast. All keto breakfastsHmmm, what to eat for lunch or dinner? Daily meal planning can be as simple as meat, fish or chicken mains with a salad, or vegetable side – with melted butter, cheese, or a yummy full-fat sauce. We have hundreds of choices for delicious keto mealsMany people believe that a keto diet is expensive, and it can be. After all, good-quality food costs more than unhealthier options. But there are many ways to stay budget-friendly, and in this guide you’ll learn all about themFor decades we have been told to fear fat, a position that we have evidence to seriously question.75 We now have reason to believe that fat is likely not harmful, plus it is satiating and makes food taste great.Do you need advice on how to add fat back into your food? What fats should you use, olive oil or butter? And just how much fat do you need each day? Tip: if you are constantly feeling hungry on a keto diet, you may need more protein or fat, or both.Bread is one of the most common things that people miss on a ketogenic diet. Fear not! There are plenty of good keto bread options.How do you eat keto at a buffet, a friend’s house, or a fast­-food restaurant? Avoid starchy foods (like bread, rice, or pasta) and ask for additional natural fat, like butter or olive oil, if you need it. Dining out on ketoDon’t be fooled by the creative marketing of special “low-carb” products. Remember: An effective keto diet for weight loss does not include refined and industrially processed foods.Low-carb products like chocolate, candy, pasta, and bread often use all kinds of deceptive marketing, while being just junk food – including carbs – in disguise. Learn more7. Potential side effects of a keto dietWhen you suddenly switch your body’s metabolism from burning carbs (glucose) to fat and ketones, you may have some side effects as your body gets used to its new fuel, especially during days two through five.Symptoms may include headache, tiredness, muscle fatigue, cramping, and heart palpitations. These side effects are short-term for most people, and there are ways to minimize or cure them (see below).76To reduce potential side effects, you may decide to gradually decrease your consumption of carbs over a few weeks. But with a slower start you’ll likely not see results as quickly. While the short-term results may differ, the long-term results should remain the same.77We recommend you stop sugar and starches all at once. You will likely lose a number of pounds within days. While much of the initial rapid weight loss is water weight (from reduced swelling), it’s still a highly motivating way to start your keto journey.Most people who start a ketogenic diet will experience some symptoms of the “keto flu.” This is what you may feel, more or less, a few days after you’ve started a keto diet: FatigueThese initial symptoms often disappear within a week or two, as your body adapts to increased fat burning.78The main cause of the keto flu is that carb-rich foods can result in water retention (swelling) in the body.79 When you start a low-carb diet much of this excess fluid is lost. You may notice increased urination, and with that some extra salt is lost too.Before your body adapts, this can result in dehydration and a lack of salt. These appear to be behind most of the symptoms of the keto flu.You can reduce or even eliminate these symptoms by making sure you get enough water and salt. One simple way to do this is to drink a cup of bouillon or broth, once or twice a day.8081Most side effects of a keto diet are minor and temporary. But there are a lot of controversies and myths that scare people.Have you heard that your brain will cease functioning unless you eat lots of carbs? It’s a myth, based on a lack of understanding of the way the body works in ketosis (switching the fuel supply of the brain to ketones).Another common misunderstanding is mixing up normal ketosis – resulting from a keto diet – with the dangerous medical emergency ketoacidosis. Don’t worry! They are two very different things. Ketoacidosis does not happen just from eating a keto diet.82The keto diet controversies don’t stop there. Will keto kill your kidneys or destroy your bones? Will it stop your thyroid from working? Will saturated fats give you a heart attack? Is keto bad for the environment? Can it make you depressed?While the environmental effect of a keto diet depends a lot on how it is formulated, the short answer to most of these questions is “no”. For all the details and the scientific support click the link below.8. Keto FAQ and other resourcesThere are many common questions about keto, and we do our best to answer them all. Feel free to check out our full keto diet FAQ, or choose below: How much weight will I lose on a keto diet?83Results vary widely. Most people lose 2-4 pounds (1-2 kg) during the first week. This is mainly water weight. After that, it’s common to lose about 1 pound (0.5 kg) of excess weight per week. However, some lose much faster (often younger men), some a bit slower (often women over 40).You can speed up the process or break a weight loss plateau by following our top tips.When you approach your normal body weight, the weight loss will slow. Just remember, a “normal” body weight differs from person to person depending on our genetics and environmental exposures and may not fit what we see in the popular media. The weight loss won’t go on forever. As long as you follow the advice to eat when you are hungry, you will eventually stabilize your weight.Read more in our guide on striking the balance between weight, health and happiness.If you use our keto recipes and keto meal plans you’ll stay under 20 net grams of carbs per day, with no need to count.Using our keto foods guidelines and visual guides will make it simple to estimate roughly how many carbs you eat in a day.If you want to count carbs exactly, the most popular way is with apps like MyFitnessPal, Chronometer, Senza, Carb manager or others.Once you reach your goals you can either keep eating keto (to maintain the effect), or you can try adding a bit more carbs. In the latter case the effect of the keto diet will be slightly weaker, and you may or may not regain some weight.If you revert to your old habits, you’ll slowly return to the weight and health situation you had before. It’s like exercising – if you stop doing it, you’ll slowly lose the benefits. As you may expect, a keto diet, like exercise, only works when you do it.It’s a common misconception that all of the body only runs on ketones when on a keto diet. The truth is that most of the body – including the muscles – can burn fat directly. It would be a waste to have the liver first convert them into ketones. This is mostly necessary for the brain.The brain consumes about 20% of the body’s required energy every day, despite only representing 2% of the body’s mass.This could be a bonus for weight loss, if your hungry brain burns fat for you.Another note: ketosis is why the old idea that “the brain needs carbohydrates” is wrong – the brain requires limited amounts of glucose even in ketosis, but the body can create that glucose from protein and fat in a process called gluconeogenesis.As a sidenote, the increased fat burning can use either body fat, or dietary fat, or (usually) both. This means that eating more fat than you need to stay satisfied can slow down the burning of body fat stores. If you’re aiming to lose weight, only eat when you’re hungry. Learn more ↩This is something that is often reported by people on a keto diet [very weak evidence]. According to our expert panel, this happens in approximately 70% of the patients they treat with keto diets.The scientific support isn’t very strong. Here are references showing minor signs of improvements: The ketosis experienced after a longer period of fasting is called starvation ketosis. But since starvation is neither healthy nor sustainable, we focus more on nutritional ketosis. However, intermittent fasting may have a role as an adjunct to diet for health gains and weight loss. Read more in our guides on intermittent fasting and time restricted eating. ↩The ketosis experienced on a ketogenic diet is often called nutritional ketosis. This seems to be sustainable long term.Studies have demonstrated that ketosis can be maintained for years; for instance, in children treated with a ketogenic diet for epilepsy, and for at least a year or two when treating obesity or type 2 diabetes.It may even be possible to stay on a ketogenic diet indefinitely. There is no obvious or proven upper limit for how long nutritional ketosis can be maintained. This is based on clinical experience of low-carb practitioners and was unanimously agreed upon by our low-carb expert panel. You can learn more about our panel here [weak evidence]. ↩The main fear about lower-carb and higher-fat diets has always been an increase in the risk of heart disease. However, interventional studies so far indicate that if anything the risk appears to decrease: Subtracting fiber from total carbs to calculate net carbs is useful for natural foods, such as vegetables, fruits, and nuts. However, it may be less helpful for processed or packaged foods.Since we recommend eating mostly natural, minimally processed foods, we focus on net carbs. However, if you find yourself eating low-carb packaged foods, then we suggest using total carbs rather than net carbs as your guide. ↩This is mainly based on the consistent experience of experienced practitioners, and stories from people trying different levels of carb restriction [weak evidence].One trial showed that diets providing 20 grams and 50 grams of carbs equally helped healthy volunteers maintain ketosis. However, we don’t know if the same is true for people with obesity, insulin resistance, or diabetes.In addition, there is not yet an RCT that tests the health benefits of two low-carb diets of varying strictness head-to-head. But RCTs of strict low-carb diets appear to show better results compared to RCTs of more moderate or liberal low-carb diets: However, you can “overdo it” with fat calories and protein as well. Too much dietary fat means your body won’t use body fat for energy.Also, adequate protein is important, but protein above what your body needs can likewise contribute to excess calories. Excessive amounts of protein (such as over 2.0 grams of protein per kg of ideal body weight) provides extra calories and may, under certain circumstances, be converted to blood sugar.Remember that the levels at which carbohydrate and protein exert their effects on the body seem to reflect thresholds, not a percentage of calories. Therefore, the key is to restrict carbohydrates, ensure adequate protein intake, and adjust fat calories as needed for satiety and weight loss goals.Even zero-calorie sweeteners may have negative effects in some people, including maintaining a preference for sweet tastes and potentially increasing the risk of overeating and even food addiction. This is mainly based on clinical experience [weak evidence].There is also one RCT study showing weight loss from avoiding artificial sweeteners: If you ate three of the represented meal per day, you will likely be within the specified carb range. The image is not meant to imply that one meal has the specified number of carbs ↩There is not yet any RCT that has actually tested two low-carb diets of varying strictness head-to-head. But RCTs of strict low-carb diets appear to often show better results, compared to RCTs of more moderate or liberal low-carb diets: These 30 studies are randomized controlled trials, considered the gold standard of scientific research on diet and health. They include Shai 2008 and Bazzano 2014. [moderate evidence]Several meta-analyses of RCTs, an even higher level of evidence, like Mansoor 2016 [strong evidence] have confirmed greater weight loss with low carb diets.One word of caution. As with most weight loss trials, these studies show that weight loss tends to lessen at the 6-12 month mark. Some trials, such as the DIETFITS trial, liberalize carb intake after the first few months. The increased carb intake could easily explain the decrease in weight loss over time. Therefore, we believe most people should maintain low-carb long term to maintain weight loss benefits.And there is strong clinical experience that keto diets make it easier to implement intermittent fasting. This is based on clinical experience of low-carb practitioners and was unanimously agreed upon by our low-carb expert panel. You can learn more about our panel here [weak evidence]. ↩This is mainly based on consistent experience from experienced practitioners and is also a very common report from people trying a keto diet [weak evidence]. But it’s also supported by the fact that keto diets generally reduce appetite: However, there aren’t randomized trials studying keto diets for food addiction. A keto diet’s effect on food addiction is based on consistent experience from experienced clinicians along with droves of anecdotal reports support the belief. [weak evidence]Some disagree with the use of the word “reverse” when it comes to type 2 diabetes. The concern is that it may imply the disease is completely gone, never to return.At Diet Doctor, we use the term “reverse” to indicate that the diagnosis of diabetes is no longer present at that moment. However, we acknowledge that the diagnosis of diabetes will likely return if a patient goes back to their prior high-carb eating habits. Therefore, “reverse” does not imply a forever cure.The advantage of a low-carb diet has been shown in these reviews of RCTs: There’s a controversial but compelling theory that chronically high insulin levels may increase the risk of diseases like obesity, heart disease, and several common forms of cancer.Learn more about hyperinsulinemia and insulin resistance in our evidence-based guide. ↩There are currently no RCTs testing this hypothesis on this group of people, so the direct support is still only anecdotes and clinical experience. [weak evidence] ↩A recent systematic review finds that a low-carb diet results in significantly larger improvements in body weight (compared to a low-fat diet) and also improved HDL and triglycerides. LDL went up by, on average, a modest 0.16 mmol/l or 6 mg/dl: While LDL is marginally elevated on average, there are exceptions. Some people, estimated anywhere between 5-20%, may see significant LDL elevations.While this might not have the same implication as elevated LDL in other circumstances, such as with insulin resistance and small dense LDL particles, there is no data to prove this is the case. Here’s our guide to possible ways to reduce this effect: This is based on clinical experience of low-carb practitioners and was unanimously agreed upon by our low-carb expert panel. You can learn more about our panel here [weak evidence] ↩Athletic performance on a keto diet remains a controversial subject with conflicting data. Issues of adaptation time, type of exercise, differences of training and racing diets, and baseline fitness all play a role in measuring response, and thus the level of evidence is weak. But there are anecdotal reports and case series documenting increased performance for endurance athletics.For background and discussions about this potential effect, see these articles: While success stories generally can’t prove the benefits of a lifestyle intervention, they can add potential value in other ways. ↩Fiber is digested by bacteria in the colon, and some of it is transformed into a fat called short-chain triglycerides. This fat can be absorbed by the body and potentially turned into ketones. Thus eating more fiber (but still very low carb) could result in higher ketone levels in the blood. This is based on hypothesis and mechanistic data [very weak evidence] ↩How sustainable is the keto diet? It’s not known in detail. But there are a number of studies of ketogenic diets that are at least 2 years in duration. This study looks at children who have been treated with a ketogenic diet for a period of six years or more: There’s a lack of clear scientific evidence about exactly what level of protein intake is most beneficial on a keto diet. Quite likely it depends on your goals.Here are some thoughts on individualization of protein intake, and details about the views of different low-carb experts. ↩In the beginning of a keto diet, especially with high levels of insulin resistance, some anecdotal reports find “too much” protein can slow down the progression to ketosis and blood sugar control. This is likely less of a concern as insulin resistance improves and should not dissuade people from eating protein.Greater than 2 grams per kg of reference body weight may be an example of too much protein per day, if you want to stay in ketosis.Adding extra calories that you don’t need quite likely slows down weight loss. Eating carbohydrates is likely to reduce ketosis, while adding more fat may increase ketosis (but still slow down weight loss). This is based on clinical experience of low-carb practitioners and was unanimously agreed upon by our low-carb expert panel. You can learn more about our panel here [weak evidence]. ↩One study of MCT oil for mild cognitive impairment is sometimes mentioned, but it did not show any clear effect. Out of only six participants two quit during the study, one saw some improvement and one got worse: This recommendation is based on positive experience from experienced clinicians. [weak evidence]However, there’s also some support from a study that found only minor increases in these side effects in people who were advised to drink bouillon while eating a ketogenic diet: Could the added salt be dangerous? Likely not, as long as your blood pressure is under control, as it’s mostly about replacing salt that is being lost.