# Pi **URL:** \ **Category:** Factual Questions\ **Created:** [July 23, 2017, 8:47am UTC](https://boards.straightdope.com/t/pi/791924 "2017-07-23T08:47:09Z")\ **Posts on this page:** 20\ **Page:** 5
**Author:** ![abashed](https://avatars.discourse-cdn.com/v4/letter/a/c6cbf5/32.png) [@abashed](https://boards.straightdope.com/u/abashed)\ **Post date:** [July 24, 2017, 12:36pm UTC](https://boards.straightdope.com/t/pi/791924/81 "2017-07-24T12:36:24Z")
> [@DPRK](#): > > If I understood you correctly, you are considering two, or possibly three, sets of real numbers. > > One (the largest one!) consists of numbers which cannot be computed. > > Two, numbers which can be computed in the sense that a program running forever can print out more and more decimal digits, but it cannot terminate. This class includes numbers like 1/3 and π. > > Three, rational numbers with a finite decimal expansion, like 1.23345. > > Do you have a philosophical problem with a simple fraction like 1/3 being in class 2 rather than 3 and therefore decimal notation sucks? Some people agree with you, which is why one pound used to be divided into 240 pennies instead of 100 cents. On the other hand, your computer probably uses binary floating-point arithmetic, which means that it cannot precisely represent a number like 0.1 any more precisely than it can an irrational number. Does that mean your computer sucks? I take your point and I think my confusion arises out of regarding anything not ‘perfect’ as in sqrt of 2 = an irrational number, less accurate than say sqrt of 4 = 2. I guess because in the former case there is always more work to do while in the latter case it’s all tied up and finished. Maybe it’s just a psychological prejudice. ---
**Author:** ![abashed](https://avatars.discourse-cdn.com/v4/letter/a/c6cbf5/32.png) [@abashed](https://boards.straightdope.com/u/abashed)\ **Post date:** [July 24, 2017, 12:46pm UTC](https://boards.straightdope.com/t/pi/791924/82 "2017-07-24T12:46:55Z")
[quote=“Francis\_Vaughan, post:76, topic:791924”] 0 Bolding mine. Your use of the word “calculated” is the core problem. What do you think you mean by this word? It _seems_ you mean “expressible as a decimal number.” Which as I noted above is identical to saying “expressible as a rational number.” So why? Why insist that _calculation_ of a value requires that it be expressed as a rational number? Apart from being taught in school that numbers are written as decimals there is absolutely no reason to insist on this. Decimal numbers, or any fractions (ie rationals) are just a convenience that are easy to use in many applications. There absolutely no reason to insist that these are somehow the “right” form of number, or that they are anything other than an accident of history. Now we get to the core problem. What do you mean buy _true_? You seem to again believe that the only true values are those that are expressible as decimals. Why? We have discussed earlier, almost all numbers are not so expressible, and only an infinitesimally small set of numbers are so expressible. By your logic almost no number has a true value. By calculation you mean express as a decimal value. Most mathematicians would not regard this as calculating its value. We might determine a quick approximation to its decimal representation for engineering purposes, but the value of the square root of two is just that - the square root of two. Anyone doing mathematics (as opposed to arithmetic) will carry that value around in expressions until they (might) be able to combine terms in some manner to make it vanish (or not, as the case may be.) There may be a point where some arithmetic is performed to determine a useful expression of the value for some purpose. But that isn’t mathematics. > [@](#): > > In fact, how could you possibly _prove_ it has an exact value when it is impossible to calculate it? It must always be an\* approximation\*, surely. > Easy. You have already seen proof by reductio-ad-absurdum. Typically you assume that there is more than one (different) value as a solution and show that if you assume this, some simple algebraic manipulation leads to a contradiction. Indeed proving the: existence of, number of, or non-existence of, such solutions is a core part of mathematics. This is a vastly more powerful a concept. > > Again, this problem with _calculate_. The answer to the above is trivially yes, you can use non-rational bases. But it doesn’t actually help your main issue. Such a base cannot represent all other numbers, so it is just moving the deckchairs. > > The resolution is to let go of the idea that decimal numbers (or any other form of rational numbers) are the basic true representation of numbers. They are not. It is sad in a way that some schools seem to ingrain people with this notion, and it is hard to unlearn, and perhaps counter intuitive. Numbers are much more interesting than this. Sure the Greeks had problems with this, but they also realised that the rationals were not the basic true form one they discovered the square root of two. They had a lot of soul searching to do, and it took a long time (centuries) for a solid understanding of numbers to be built, but it has been built, and the Greeks are regarded as a bit of history, not as having some deep knowledge. You’ve really got me thinking now. I want to go over this some more and I appreciate your politeness and patience with me. I think you might be right about how schooling places too much emphasis on rational numbers, possibly because they are more likely the ones the average person will use in life. 🙂 ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 24, 2017, 1:06pm UTC](https://boards.straightdope.com/t/pi/791924/83 "2017-07-24T13:06:35Z")
Put it this way: There are many different ways of expressing numbers, some of which are better-suited for some purposes than others. For instance, when you’re counting things, you often use tick marks. Like, if I’m tracking the number of eggs my mom’s chickens have produced each day, then every time I take an egg out, I make a mark. For convenience, I make every fifth mark go through a group of four. So if the chickens are having a good day and give us a total of seven eggs, the final tally will look something like [del]||||[/del] || . This is a perfectly valid way of representing numbers. And when what I’m counting is the eggs produced by the chickens in my mom’s backyard, it’s a very convenient and appropriate way of representing them. But it’s not well-suited to all sorts of numbers. If instead of my mom’s backyard chickens, I were dealing with all of the eggs produced in a year by Cal-Maine Foods, well, I _could_ count the eggs with tick-marks, but it’d be horribly inconvenient, because the numbers are so large. And if I want to measure anything that’s not a positive integer, tick marks don’t really work at all. If I can’t easily represent 1.3 billion using tick marks, and I can’t represent four and a half using tick marks at all, does that mean that those numbers aren’t exact? Of course not; it just means that I’m not using the most appropriate system of representing those numbers. Well, place value numbers with radix can represent many more numbers. They still don’t work well for all numbers, though. In any base, there will be some rational numbers that can be represented in a finite amount of space, and some that can’t be, with no fundamental distinction between the two (there’s no fundamental reason, for instance, why “one fifth” is better or more important than “one third”). And there are some numbers, like pi and sqrt(2), that can’t be finitely represented in a place value system with any base. Does that mean that those numbers aren’t exact? No, it just means that we’re using an inconvenient notation for them. But there are notations for those numbers that are much more convenient, and which can be used to represent them exactly in a finite amount of space. In fact, we’ve been using those notations throughout this thread. I don’t have to call that number “3.14159265358979323846…” . I can just call it “pi”, and I’m done. All it takes is a single character (or two, if I don’t have a Greek keyboard), or a single syllable when spoken, and I’ve told you the number. The notation works just fine. Warning: Deep into the mathematical weeds, for this next paragraph. Read at your own risk: It turns out, though, that in _any_ sort of notational system, there are always some numbers that can’t be represented in a finite amount of space. Which numbers? I can’t tell you, of course: That’s the whole point. But not only do such numbers exist, but they’re infinitely more numerous than the numbers that we can express. Fortunately, no such number is ever of interest (because if it were of interest, we could use the property that makes it interesting to define it). ---
**Author:** ![abashed](https://avatars.discourse-cdn.com/v4/letter/a/c6cbf5/32.png) [@abashed](https://boards.straightdope.com/u/abashed)\ **Post date:** [July 24, 2017, 1:56pm UTC](https://boards.straightdope.com/t/pi/791924/84 "2017-07-24T13:56:28Z")
> [@Chronos](#): > > Put it this way: There are many different ways of expressing numbers, some of which are better-suited for some purposes than others. For instance, when you’re counting things, you often use tick marks. Like, if I’m tracking the number of eggs my mom’s chickens have produced each day, then every time I take an egg out, I make a mark. For convenience, I make every fifth mark go through a group of four. So if the chickens are having a good day and give us a total of seven eggs, the final tally will look something like [del]||||[/del] || . > > This is a perfectly valid way of representing numbers. And when what I’m counting is the eggs produced by the chickens in my mom’s backyard, it’s a very convenient and appropriate way of representing them. But it’s not well-suited to all sorts of numbers. If instead of my mom’s backyard chickens, I were dealing with all of the eggs produced in a year by Cal-Maine Foods, well, I _could_ count the eggs with tick-marks, but it’d be horribly inconvenient, because the numbers are so large. And if I want to measure anything that’s not a positive integer, tick marks don’t really work at all. If I can’t easily represent 1.3 billion using tick marks, and I can’t represent four and a half using tick marks at all, does that mean that those numbers aren’t exact? Of course not; it just means that I’m not using the most appropriate system of representing those numbers. > > Well, place value numbers with radix can represent many more numbers. They still don’t work well for all numbers, though. In any base, there will be some rational numbers that can be represented in a finite amount of space, and some that can’t be, with no fundamental distinction between the two (there’s no fundamental reason, for instance, why “one fifth” is better or more important than “one third”). And there are some numbers, like pi and sqrt(2), that can’t be finitely represented in a place value system with any base. Does that mean that those numbers aren’t exact? No, it just means that we’re using an inconvenient notation for them. > > But there are notations for those numbers that are much more convenient, and which can be used to represent them exactly in a finite amount of space. In fact, we’ve been using those notations throughout this thread. I don’t have to call that number “3.14159265358979323846…” . I can just call it “pi”, and I’m done. All it takes is a single character (or two, if I don’t have a Greek keyboard), or a single syllable when spoken, and I’ve told you the number. The notation works just fine. > > Warning: Deep into the mathematical weeds, for this next paragraph. Read at your own risk: > > It turns out, though, that in _any_ sort of notational system, there are always some numbers that can’t be represented in a finite amount of space. Which numbers? I can’t tell you, of course: That’s the whole point. But not only do such numbers exist, but they’re infinitely more numerous than the numbers that we can express. Fortunately, no such number is ever of interest (because if it were of interest, we could use the property that makes it interesting to define it). Very good. I’m getting a fantastic education and now I realize why mathematics can be so interesting. TYVM. 😉 ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [July 24, 2017, 2:54pm UTC](https://boards.straightdope.com/t/pi/791924/85 "2017-07-24T14:54:24Z")
> [@abashed](#): > > But just to clarify then, my misconception is to assume irrational numbers are inferior to rational numbers (i.e. a numbers that can be written as a fraction) because they go on forever? I think what you’re missing is that the _numbers_ don’t go on forever, the decimal representations of those numbers go on forever. Pi is an exact number, so is the square root of two, so is e. But if you insist on writing the number as a decimal, that decimal representation goes on forever. In summary, I think in this discussion you should use different terms for the number and for its decimal representation. In your writing, be careful to use “pi” and the “decimal representation of pi” and it will help you clear this up. > [@](#): > > But, one further point: how can we be certain we will never arrive at a termination with an irrational number? I know it will never happen but to the uninitiated it seems mysterious. If the decimal representation of a number ended, then that number is rational. Any decimal representation that either stops (ends with repeating zeros) or repeats is provably a rational number. Pi and the square root of two are proved NOT to be rational, therefore we know they can’t repeat or stop. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [July 24, 2017, 3:11pm UTC](https://boards.straightdope.com/t/pi/791924/86 "2017-07-24T15:11:19Z")
As an aside, is there a good reason that there is so much fascination about the number pi, and much less about the number e? Pi is the rock star, e is its little-known brother, but I can’t think of any reason that pi is favored to the public. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 24, 2017, 3:23pm UTC](https://boards.straightdope.com/t/pi/791924/87 "2017-07-24T15:23:23Z")
Are there statistics supporting your claim? It never occurred to me that one could rank numbers by popularity. In any case, one important fact is that pi is an ancient concept, while e is not. Also, everybody studies geometry, while perhaps analysis is less generally taught. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [July 24, 2017, 3:28pm UTC](https://boards.straightdope.com/t/pi/791924/88 "2017-07-24T15:28:05Z")
How do we know that pi is an exact number? Because it appears in millions of equations with exact answers. If pi were an approximation, there would be millions of answers and those equations would be useful except to the degree that outside measurements were brought in. You may argue that pi cannot be used in an equation that forms an integral. That’s not correct. What’s called [Euler’s Identity](https://en.wikipedia.org/wiki/Euler%27s_identity), named after the 18th century mathematician Leonhard Euler, combines three different components with no exact decimal representation and yet yields an integral. e[sup]i\*pi[/sup] + 1 = 0 This is often called the most beautiful equation in mathematics. It uses no more than high school algebra. It utterly depends upon pi and e and i to have exact values on the complex number plain. Most people neither understand it or believe it at first. It’s worth study, though. It forever breaks your addiction to arithmetic and opens up the world of mathematics as something larger and more spectacular. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 24, 2017, 3:29pm UTC](https://boards.straightdope.com/t/pi/791924/89 "2017-07-24T15:29:27Z")
It may be worth pointing out that irrational numbers were discovered before decimal notation was. The ancient Greeks knew about irrational numbers, and that the square root of two was one, but they didn’t use decimal notation to write down or imagine their numbers. They thought of numbers as corresponding to lengths of line segments (according to which sqrt(2) is a number), but also as ratios of whole numbers (according to which it is not), which is why the irrationality of something like sqrt(2) bothered them. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 24, 2017, 3:35pm UTC](https://boards.straightdope.com/t/pi/791924/90 "2017-07-24T15:35:41Z")
> [@DPRK](#): > > Also, everybody studies geometry, while perhaps analysis is less generally taught. Right, almost everybody encounters (and “uses”) pi fairly early on in their schooling, since it shows up in formulas for the circumference of a circle, the area of a circle, etc. People don’t encounter e until much later, if at all; and it really requires calculus to see how it’s defined or what’s so special about it. Other famous irrational numbers include the [“golden ratio” phi](https://en.wikipedia.org/wiki/Golden_ratio), and [Euler’s gamma](https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant). ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 24, 2017, 3:38pm UTC](https://boards.straightdope.com/t/pi/791924/91 "2017-07-24T15:38:46Z")
> [@abashed](#): > > But, one further point: how can we be certain we will never arrive at a termination with an irrational number? Because if we did arrive at a termination, we could write it as a ratio of integers (i.e. a fraction) and it would therefore be rational. For example, if pi were exactly equal to 3.14159, we could write it as 314159/100000. If there were more decimal places (but still only finitely many), the denominator would just be a bigger power of 10. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [July 24, 2017, 3:42pm UTC](https://boards.straightdope.com/t/pi/791924/92 "2017-07-24T15:42:09Z")
Which of course gets us to another class of numbers. The Constructible Numbers. Given a straight edge and a compass, and finite steps, any length you can construct is a constructible number. Which terns out to mean any number you can make starting with the numbers 0 and 1, and the operations, add, subtract, multiply, divide, and square root. Now here you can still buy an argument about what constitutes a number. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 24, 2017, 3:46pm UTC](https://boards.straightdope.com/t/pi/791924/93 "2017-07-24T15:46:32Z")
> [@abashed](#): > > I take your point and I think my confusion arises out of regarding anything not ‘perfect’ as in sqrt of 2 = an irrational number, less accurate than say sqrt of 4 = 2. I guess because in the former case there is always more work to do while in the latter case it’s all tied up and finished. Maybe it’s just a psychological prejudice. Probably so, although arguably, some mathematicians have had similar psychological prejudices. I’m thinking of Leopold Kronecker, to whom is attributed the quote “God made the integers, all else is the work of man,” and who is apocryphally supposed to have said “What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational numbers do not even exist?”—see discussion [here](https://mathoverflow.net/questions/34445/meaning-of-kroneckers-comment-to-lindemann). ---
**Author:** ![RickJay](https://avatars.discourse-cdn.com/v4/letter/r/bb73d2/32.png) [@RickJay](https://boards.straightdope.com/u/RickJay)\ **Post date:** [July 24, 2017, 4:26pm UTC](https://boards.straightdope.com/t/pi/791924/94 "2017-07-24T16:26:21Z")
> [@abashed](#): > > Okay but I’m confused. You say it’s an exact value like 1/3, right? What is that value please? Pi. [QUOTE=CurtC] As an aside, is there a good reason that there is so much fascination about the number pi, and much less about the number e? Pi is the rock star, e is its little-known brother, but I can’t think of any reason that pi is favored to the public. [/QUOTE] What pi is is obvious; it’s how you get the circumference of a circle. People understand circles; circles are, well, all around us. They’re important. We deal with circular things all the time. The comparison between how far a circle is around, and how far it is from one side to the other, is an obvious one to think about. Kids get circles, they like circles. Pi is a natural thing to want to know based on the most simple observations of the world. e is “the base of the natural logarithm.” I have a degree in economics and so had to take and pass calculus and statistics, and even I am still not super clear on what e is. You really have no reason to ever think about e until you actually get to that level of understanding mathematics; it’s not something that is a visually obvious concept, like pi. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [July 24, 2017, 4:58pm UTC](https://boards.straightdope.com/t/pi/791924/95 "2017-07-24T16:58:00Z")
Fair enough, about pi being more popular than e. Next question - do the things we know about pi also apply to e? Is it normal for example? ---
**Author:** ![Marvin\_the\_Martian](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/marvin_the_martian/32/2898_2.png) [@Marvin\_the\_Martian](https://boards.straightdope.com/u/Marvin_the_Martian)\ **Post date:** [July 24, 2017, 5:00pm UTC](https://boards.straightdope.com/t/pi/791924/96 "2017-07-24T17:00:16Z")
> [@RickJay](#): > > I have a degree in economics and so had to take and pass calculus and statistics, and even I am still not super clear on what e is. You really have no reason to ever think about e until you actually get to that level of understanding mathematics; it’s not something that is a visually obvious concept, like pi. In any field where you solve differential equations (like electrical or mechanical engineering) “e” is an extremely important constant. Any basic circuit theory student (should) be able to visualize “e” from the settling time of a first-order system. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 24, 2017, 5:16pm UTC](https://boards.straightdope.com/t/pi/791924/97 "2017-07-24T17:16:51Z")
> [@CurtC](#): > > Next question - do the things we know about pi also apply to e? Is it normal for example? Both are suspected to be, but neither one has been proved to be. From Wikipedia: > [@](#): > > While a general proof can be given that almost all real numbers are normal (in the sense that the set of exceptions has Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown to be normal. For example, Chaitin’s constant is normal (and uncomputable). It is widely believed that the (computable) numbers √2, π, and e are normal, but a proof remains elusive. ---
**Author:** ![RickJay](https://avatars.discourse-cdn.com/v4/letter/r/bb73d2/32.png) [@RickJay](https://boards.straightdope.com/u/RickJay)\ **Post date:** [July 24, 2017, 5:40pm UTC](https://boards.straightdope.com/t/pi/791924/98 "2017-07-24T17:40:31Z")
> [@Marvin\_the\_Martian](#): > > In any field where you solve differential equations (like electrical or mechanical engineering) “e” is an extremely important constant. Any basic circuit theory student (should) be able to visualize “e” from the settling time of a first-order system. Yes, I know that, Marvin. What CurtC wanted to know is why Pi is more popular. How many people have to solve differential equations? A lot in total terms, but it’s a small percentage of all people. How many people see CIRCLES everywhere? Everyone who can see. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [July 24, 2017, 5:48pm UTC](https://boards.straightdope.com/t/pi/791924/99 "2017-07-24T17:48:24Z")
> [@Derleth](#): > > (Now, as for an infinite number of _nonzero_ terms to the _left_ of the decimal point, those numbers don’t exist in any of the familiar number systems, since infinite values are not especially useful or intuitive to work with.) Oh yes they do exist! Consider the number represented by “…999.000…”. Let’s add _one_, which we will represent as “…0001.000…”, to it. And the answer is “…000.000…”. What do we usually call the number which, if we add _one_ produces _zero_? Yes, “…999.000…” is another representation of _negative one_. But wait, there’s more! Another representation for _one_ is “…000.999…”. Let’s add that to our representation of _negative one_ “…999.000…”. And, you see where this is going? _One_ added to _negative one_ is always _zero_, or equivalently “…999.999…”. While this may seem bizarre, it’s actually related to how signed integers are often stored on a computer. See [Two's complement - Wikipedia](https://en.wikipedia.org/wiki/Two%27s_complement) . And it’s also comes up in physics. See [Negative temperature - Wikipedia](https://en.wikipedia.org/wiki/Negative_temperature) . ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 24, 2017, 5:56pm UTC](https://boards.straightdope.com/t/pi/791924/100 "2017-07-24T17:56:39Z")
Yes you can consider -1 = …999 as a sort of 10-adic expansion, but beware that p-adic numbers are not the real numbers you learned about in your introductory analysis class. [Previous page](https://boards.straightdope.com/t/pi/791924.md?page=4) [Next page](https://boards.straightdope.com/t/pi/791924.md?page=6)