# Thread: PI is a Rational

1. Firstly, I must state I know perfectly well that PI is irrational; I am not trying to disprove that, or claim that PI is indeed a rational.

However, I have come across something odd about PI which I have found rather puzzling.

We all know that pi is equal to 22/7. The first five digits are 3.14156. Both of these are correct as to the value of pi, unless I am mistaken.

However, on actually dividing 22 by 7, I seem to getting the nonsensical answer 3.142857 bar, with a bar over 142857.

I am well aware that this would imply that pi is a rational, and I woud like to know why my answer seems to be coming out to be so.

Thank you. 2.

3. Originally Posted by Liongold
We all know that pi is equal to 22/7.
Pi is not equal to 22/7.

Pi is approximately equal to 22/7, a fact useful for many quick-fix practical calculations.

22/7 is a rational number because it is a ratio of two integers: 22 and 7.

Pi is not.

Hope this helps. 4. You sure its not a calculator issue? 5. Originally Posted by Leszek Luchowski Originally Posted by Liongold
We all know that pi is equal to 22/7.
Pi is not equal to 22/7.

Pi is approximately equal to 22/7, a fact useful for many quick-fix practical calculations.

22/7 is a rational number because it is a ratio of two integers: 22 and 7.

Pi is not.

Hope this helps.
Also pi is not approximately 3.14156 It is approximately 3.14159

And for more decimals than you can stand, in a particularly annoying presentation check this web site http://pi.ytmnd.com/

Pi is not only irrational it is transcendental, meaning not algebraic, and therefore not the root of any polynomial with rational coefficients. 6. Originally Posted by Leszek Luchowski Originally Posted by Liongold
We all know that pi is equal to 22/7.
Pi is not equal to 22/7.

Pi is approximately equal to 22/7, a fact useful for many quick-fix practical calculations.

22/7 is a rational number because it is a ratio of two integers: 22 and 7.

Pi is not.

Hope this helps.
Every value is rational. You may have to express the number in some other, base. Like base three, base nine, base thirteen, or some prime number base. But if it has an exact ratio. And any real value would. We can express it with integers exactly.

I know the definitions have changed over the years. So the young people probably think I am insane. I wish I could argue that point, with more affirmed and convincing facts. Ha-ha.

I just recount how I learned it. I make no claim that today it is still so.

In base ten there are values that you cannot express. It has almost nothing to do with the actual value. Rather the integers needed to express the value, and base ten. Base ten representing certain values, does not offer those increments of precision, to correctly express many values.

In base three, a base ten ratio of 1/3, would equal 0.1 expressed in base three. 2/3 in base ten would be expressed as 0.2 in base three.

Edited: Just disregard this information it is incorrect. Age. Ha-ha.

1/9 in base ten would be 0.023 in base three. 1/7 in base ten would be 0.021 in base three.

We used to have to do this for hours and hours in elementary school. With all the bases. Ahhhhh! I wanted to be out in space in a Grumman Space ship.

In other words if you say a wheel will roll in one turn some length any length. I can express it in some base exactly.

The computer would have no trouble spitting out an exact total using a different base. If there is an exact total. Or an exact measurement.

Sincerely,

William McCormick 7. Originally Posted by DrRocket Originally Posted by Leszek Luchowski Originally Posted by Liongold
We all know that pi is equal to 22/7.
Pi is not equal to 22/7.

Pi is approximately equal to 22/7, a fact useful for many quick-fix practical calculations.

22/7 is a rational number because it is a ratio of two integers: 22 and 7.

Pi is not.

Hope this helps.
Also pi is not apaproximately 3.14156 It is approximately 3.14159

And for more decimals than you can stand, in a particularly annoying presentation check this web site http://pi.ytmnd.com/

Pi is not only irrational it is transcendental, meaning not algebraic, and therefore not the root of any polynomial with rational coefficients.
Wow, that site was annoying.

Sincerely,

William McCormick 8. I use:  9. Originally Posted by William McCormick
In base three, a base ten ratio of 1/3, would equal 0.1 expressed in base three. 2/3 in base ten would be expressed as 0.2 in base three. 1/9 in base ten would be 0.023 in base three. 1/7 in base ten would be 0.021 in base three.
I know I shouldn't... but I can't help it.

How do you get that 1/9 (which, expressed as a fraction, isn't exactly in any base, BTW) = 0.023 in base 3?

0.023 base 3 = 0 * 1/3 + 2 * 1/(3*3) + 3 * 1/(3*3*3) = 0 + 2/9 + 1/9 = 1/3. (Not to mention that the digit 3 doesn't exist in base 3.) 1/9 in base 3 is 0.01. 1/7 in base 3 is not expressible in any finite number of digits. (It'd be 0.010212 010212 010212...)

Also, cannot be expressed in a finite number of digits in any integer base, nor as a ratio of any two integers, nor as the root of any integer polynomial.

Oh yeah. If you really want base independence, try continued fractions. Any rational number has a finite representation as a continued fraction. Any algebraic number (such as the square root of 2) has a repeating continued fraction. Unfortunately, still requires infinitely many numbers to represent as a continued fraction. 10. Originally Posted by MagiMaster Originally Posted by William McCormick
In base three, a base ten ratio of 1/3, would equal 0.1 expressed in base three. 2/3 in base ten would be expressed as 0.2 in base three. 1/9 in base ten would be 0.023 in base three. 1/7 in base ten would be 0.021 in base three.
I know I shouldn't... but I can't help it.

How do you get that 1/9 (which, expressed as a fraction, isn't exactly in any base, BTW) = 0.023 in base 3?

0.023 base 3 = 0 * 1/3 + 2 * 1/(3*3) + 3 * 1/(3*3*3) = 0 + 2/9 + 1/9 = 1/3. (Not to mention that the digit 3 doesn't exist in base 3.) 1/9 in base 3 is 0.01. 1/7 in base 3 is not expressible in any finite number of digits. (It'd be 0.010212 010212 010212...)

Also, cannot be expressed in a finite number of digits in any integer base, nor as a ratio of any two integers, nor as the root of any integer polynomial.

Oh yeah. If you really want base independence, try continued fractions. Any rational number has a finite representation as a continued fraction. Any algebraic number (such as the square root of 2) has a repeating continued fraction. Unfortunately, still requires infinitely many numbers to represent as a continued fraction.
As I said it is all how you see it. How you were taught about it. In what field you were taught about it for.

You can and we did express base two as 1,2

For computing we used 0,1 as well.

You do not need a zero to express base two.

You do not need a zero to express base three.

Base ten is kind of funny. It breaks some of the rules of a base as I learned them.

In base ten we only have zero to nine to work with. We then use a zero to hold the place of the ones column and give the next column the tens column the value of the integer times ten. Not nine. Even though we only have nine digits. It is just the way our minds have been conditioned. We are screwed up.

Something more accurate might be.

1,2,3,4,5,6,7,8,9,@

If you wanted to write ten then you would write @, if you wanted to write eleven you would write 11. To write one hundred you would write 9@. If you wanted to write one thousand you would put Edited: 99@

If you look at all the other raw bases, other then binary and base ten in popular use, you will see that is not true of the other bases. At least the way I was taught to use them. Base ten breaks some rules. But it was designed that way. If you try to work with other bases you would go mad using zeros.

In real base two. Using one and two as the two symbols. You can easily count and keep track of things with it.

Computer base two is really modified base one. Just like our modified base nine system that we call base ten.

In base two, one would be shown as 1. In base two, two would be represented as 2. To represent three you need a second digit, so 11 would represent three. 12 would represent four. 21 would represent 5. 22 would represent six and so on and so forth.

I showed base three modified using zero's as place holders in the decimal form, in the other post. There was confusion about whether or not you could use it like that.

I just absorbed all this stuff. It does work, and you can represent any given value, in any increment you supply. In any ratio with bases.

23 in base three would be 9

I will give the base ten and then the base three as I learned them.

1 = 1
2 = 2
3 = 3
4 = 11
5 = 12
6 = 13
7 = 21
8 = 22
9 = 23
10 = 31
11 = 32
12 = 33
13 = 111
14 = 112
15 = 113
16 = 121
17 = 122
18 = 123
19 = 131
20 = 132
21 = 133
22 = 211
23 = 212
24 = 213
25 = 221
26 = 222
27 = 223
28 = 231
29 = 232
30 = 233
31 = 311
32 = 312
33 = 313
34 = 321
35 = 322
36 = 323
37 = 331
38 = 332
39 = 333
40 = 1111

Each column to the left in base three is three times the column to the right of it.

I understand how these bases are used and modified in real life and how they are used in computing. But this is how it was in mathematics in my day.

If you go over that definition below, you will see our base ten system is really a modified base nine system. Because of the number of digits it has.

Computer binary, follows base two to some extent. However they modified it as well, and you can see that it does not meet the criteria for a true base two system. Computer binary is actually a modified base one system.

Some real mathematicians threw their hands up years ago and went fishing. They did not care if you documented what was, what could be and what you did or wanted to change. It was not about that though. It was about people putting their names on new formulas that did not work within the bounds of real mathematics. At least by the rules of the day.

base

base (bas) noun
1.In mathematics, a number that is raised to the power specified by an exponent. For example, in 2^3 = 2 × 2 × 2 = 8, the base is 2.
2.In mathematics, the number of digits in a particular numbering system. With microcomputers, four numbering systems are commonly used or referred tobinary, octal, decimal, and hexadecimaland each is based on a different number of digits. The binary, or base-2, numbering system, which is used to discuss the states of a computer's logic, has two digits, 0 and 1. Octal, or base-8, has eight digits, 0 through 7. The familiar decimal system, or base-10, numbering system, has ten digits, 0 through 9. Hexadecimal, or base-16, has sixteen digits, 0 through 9 and A through F. When numbers are written in a particular base, the base is often subscripted and enclosed in parentheses after the number, as in 24AE(16) = 9390. See also binary, decimal, hexadecimal, octal. Also called radix.
3.One of three terminals (emitter, base, collector) in a bipolar transistor.The current through the base controls the current between the emitter and the collector. See also transistor.
4.The insulating foundation of a printed circuit board. See also circuit board.

One finale note. I was never an expert at this. I just absorbed the basics. My teachers claimed I had a photographic memory. I claimed I could not remember I had a class that day. Ha-ha.

I am not opting we change anything. I just suggest that we face what has been and what is possible.

Sincerely,

William McCormick 11. I see, more or less, what you're talking about, but when most people say base 3 they mean:

base 10 : base 3
0 = 0
1 = 1
2 = 2
3 = 10
4 = 11
5 = 12
6 = 20
7 = 21
8 = 22
9 = 100

This makes extending things into fractional amounts and negative amounts easier, among other things. 12. You do realize that any ratio, no matter if it is comprised of a prime or two primes or an even number and a prime.

Can only give you an exact ratio. There is no irrational number except in the misunderstanding of base ten.

If a computer was lets say using a polygon to compute the radius of a circle. If the computer is using binary code. There is no irrational solution produced. Or the test is flawed by the inaccuracy from the many calculations. There is an exact output based upon the input.

The problem comes when a human wants to see it in base ten. If he wanted to see it in base two, it would be no problem.

To the smallest increment you can conceive, and the computer can deal with, the output will give you the exact ratio. Unless the computer is not programed properly to begin with. And not utilizing the fact that the computer can use binary to do math.

The computer can calculate any increment however small as long as the computer has the decimal places to make use of such a small increment.

If the real ratio you want to express in base ten is 3 1/7 increments to one. Your output will be 3.1428571428571428571428571428571.........

In base seven it would be 3.1

But it is all just a language barrier. Not an accuracy problem. And my inaccuracies. Ha-ha.

Sincerely,

William McCormick 13. Originally Posted by MagiMaster Originally Posted by William McCormick
In base three, a base ten ratio of 1/3, would equal 0.1 expressed in base three. 2/3 in base ten would be expressed as 0.2 in base three. 1/9 in base ten would be 0.023 in base three. 1/7 in base ten would be 0.021 in base three.
I know I shouldn't... but I can't help it.

How do you get that 1/9 (which, expressed as a fraction, isn't exactly in any base, BTW) = 0.023 in base 3?

0.023 base 3 = 0 * 1/3 + 2 * 1/(3*3) + 3 * 1/(3*3*3) = 0 + 2/9 + 1/9 = 1/3. (Not to mention that the digit 3 doesn't exist in base 3.) 1/9 in base 3 is 0.01. 1/7 in base 3 is not expressible in any finite number of digits. (It'd be 0.010212 010212 010212...)

Also, cannot be expressed in a finite number of digits in any integer base, nor as a ratio of any two integers, nor as the root of any integer polynomial.

Oh yeah. If you really want base independence, try continued fractions. Any rational number has a finite representation as a continued fraction. Any algebraic number (such as the square root of 2) has a repeating continued fraction. Unfortunately, still requires infinitely many numbers to represent as a continued fraction.

You are absolutely right about the decimals in base three. I definitely had that wrong. I will go edit that.
It was a long time ago that I fooled with them. I knew there was something I did not remember. It is that you have to create a new decimal for each amount. You cannot just work it like base ten.

Thanks for finding that and letting me know.

It is not easy to represent anything other then the base you are using, with decimals in base three. I believe the idea was that you could use a programmable calculator to change bases, and find an exact answer. But I think this all kind of died. Around that time.

Sincerely,

William McCormick 14. Originally Posted by MagiMaster
I see, more or less, what you're talking about, but when most people say base 3 they mean:

base 10 : base 3
0 = 0
1 = 1
2 = 2
3 = 10
4 = 11
5 = 12
6 = 20
7 = 21
8 = 22
9 = 100

This makes extending things into fractional amounts and negative amounts easier, among other things.
I am not arguing with you. You are absolutely right.

I am just saying that originally it was taught, and honestly technically, was and might still be something else.

I like working with base ten the way it is.
We went over this for almost six months everyday. I can say I did not enjoy it anymore. Ha-ha.
I saw no way for it to be useful in the atmosphere that existed at the time. Even though it may be very useful with the right computer system, to store and display numbers, very accurately.

I still like remainders for accuracy.

We also studied other countries and ancient countries methods of counting and storing information. It crossed all the boundaries. Out of necessity of the times. Making a zero with your hand the way I was taught was symbolizing the whole thing. What ever the whole thing was. I believe it meant a half dozen.

Single fingers meant one each, and circles made with your fingers meant six each. So a man could bid up to twelve whatever's, with hand signs. But I am not going to be using it anytime soon. Ha-ha.

I never saw any text on it.

Sincerely,

William McCormick 15. Originally Posted by William McCormick
You do realize that any ratio, no matter if it is comprised of a prime or two primes or an even number and a prime.

Can only give you an exact ratio. There is no irrational number except in the misunderstanding of base ten.
Sorry but i have to call BS here. 16. @William: If you've read the article on continued fractions, then can you tell me what the exact ratio of the number [1; 1, 1, 1, 1, 1, ...] is?

If you haven't read that, another way to write that would be:  17. Originally Posted by river_rat Originally Posted by William McCormick
You do realize that any ratio, no matter if it is comprised of a prime or two primes or an even number and a prime.

Can only give you an exact ratio. There is no irrational number except in the misunderstanding of base ten.
Sorry but i have to call BS here.

irrational number (i-rash??-n?l num?b?r) noun
Mathematics.
Any real number that cannot be expressed as an integer or as a ratio between two integers.

Excerpted from The American Heritage Dictionary of the English Language, Third Edition Copyright © 1992 by Houghton Mifflin Company. Electronic version licensed from Lernout & Hauspie Speech Products N.V., further reproduction and distribution restricted in accordance with the Copyright Law of the United States. All rights reserved.

Now, to me that is pure B.S. because I do not know of a ratio that I cannot express with two numbers. In fact you write down the two numbers and I will create the ratio. I will just put a line between them. Ha-ha.

Pi is an exact ratio of diameter to circumference. What ever the most perfectly round wheels diameter is, compared to the distance it rolls on the most perfect surface, in one revolution, is the exact ratio of diameter to circumference. Baring any variables like air between the wheel and surface, holding the wheel off the surface. Or some other variable.

I believe the problem is that you cannot get anywhere near a perfect output when using some formula to determine pi. Because the computer is constantly using numbers in the 1/7th range. They will always cause distortions in base ten.

If you actually measure a wheel, or conclude through some formula, to the highest accuracy you can achieve, or show me how you are coming to your conclusion of pi, I will show you the exact ratio.

Anything else is indeed, an irrational number.

Imagine a ratio between two horizontal rows of blocks, a lower row, and a higher row. The base row has seven blocks in the row. The top row has one.

All the blocks measure 12 inches by 12 inches by 12 inches. Exactly. When two are placed together they measure 24 inches exactly.

I know you cannot display that simple reality or the ratio between rows with base ten. It will always suffer from a flaw in representation. Because the reality is base seven, and the calculators mostly do base ten.

But there is an exact ratio. It can be represented by the ratio of two integers. Or it can be represented with one number in base seven.

Sincerely,

William McCormick 18. Here is another definition of irrational number. You can see that there is not much agreement on what an irrational number is.  Sincerely,

William McCormick 19. Originally Posted by MagiMaster
@William: If you've read the article on continued fractions, then can you tell me what the exact ratio of the number [1; 1, 1, 1, 1, 1, ...] is?

If you haven't read that, another way to write that would be: I must have been asleep or flying my, pen cap/interstellar space craft, around my desk. Ha-ha

What does that mean. What base?

Sincerely,

William McCormick 20. Originally Posted by William McCormick
irrational number (i-rash??-n?l num?b?r) noun
Mathematics.
Any real number that cannot be expressed as an integer or as a ratio between two integers.

Now, to me that is pure B.S. because I do not know of a ratio that I cannot express with two numbers. In fact you write down the two numbers and I will create the ratio. I will just put a line between them. Ha-ha.
You jumped over part of the definition, you have to use integers. Please write down the two integers you claim exist so that when their ratio is squared the answer is 2. 21. @William: I'd like to hear your answer to river_rat's question, but in answer to your question to me: whatever base you like. I'd accept an answer in any integer base. I'm sure river_rat would say the same. 22. Originally Posted by Liongold
Firstly, I must state I know perfectly well that PI is irrational; I am not trying to disprove that, or claim that PI is indeed a rational.

However, I have come across something odd about PI which I have found rather puzzling.

We all know that pi is equal to 22/7. The first five digits are 3.14156. Both of these are correct as to the value of pi, unless I am mistaken.

However, on actually dividing 22 by 7, I seem to getting the nonsensical answer 3.142857 bar, with a bar over 142857.

I am well aware that this would imply that pi is a rational, and I woud like to know why my answer seems to be coming out to be so.

Thank you.
It could be your calculator, or if using a pc, the software, or settings for either. 23. Originally Posted by phyti Originally Posted by Liongold
Firstly, I must state I know perfectly well that PI is irrational; I am not trying to disprove that, or claim that PI is indeed a rational.

However, I have come across something odd about PI which I have found rather puzzling.

We all know that pi is equal to 22/7. The first five digits are 3.14156. Both of these are correct as to the value of pi, unless I am mistaken.

However, on actually dividing 22 by 7, I seem to getting the nonsensical answer 3.142857 bar, with a bar over 142857.

I am well aware that this would imply that pi is a rational, and I woud like to know why my answer seems to be coming out to be so.

Thank you.
It could be your calculator, or if using a pc, the software, or settings for either.

His calculator is just fine. 22/7 = 3.142857 142857 142857 .......

It just is not all that good an approximation to pi. 24. Originally Posted by river_rat Originally Posted by William McCormick
irrational number (i-rash??-n?l num?b?r) noun
Mathematics.
Any real number that cannot be expressed as an integer or as a ratio between two integers.

Now, to me that is pure B.S. because I do not know of a ratio that I cannot express with two numbers. In fact you write down the two numbers and I will create the ratio. I will just put a line between them. Ha-ha.
You jumped over part of the definition, you have to use integers. Please write down the two integers you claim exist so that when their ratio is squared the answer is 2.

I believe you meant to say what square root of two will produce 2 as the answer, without a crazy decimal?

I think I get what you are saying. You are saying that the square root of two would be hard to express as a whole number.

In life we actually just change our scale. If we do not like dealing with an irrational decimal, with many places. We just take our object that is 2 feet. And we break down two feet into four parts. Instead of two parts. We change our tape measure, our base.

Now instead of 1.4142135623730950488016887242097 feet as the square root of an object two feet long.

The square root of our object is 2, because 2 times, 2 six inch segments, yields 24 inches or two feet. We break our scale down, to 4 segments 6 inches long. The square root of 4, six inch segments is 2 six inch segments.

In most cases you just use a smaller increment. Either custom or existing. But the actual size is anything but irrational. It is just difficult to express in base ten. Not irrational in other bases, other increments. Or in reality.

You can do almost anything without a calculator. It is just a lot more fun, and you don't get as hungry using the brain power.

What do you think built all the stuff we have. Calculator less calculations. It was all done with ratio, bases, and templates.

You do understand, I am only saying that a long base ten decimal is only difficult because, the actual base of the real object and what ever increment is chosen, is not in base ten. Usually base 3,7, or 9.

Sure calculating pi out that many places is an irrational act. But the value if accurate, and I have my doubts no matter what it is. Can be displayed more easily with other bases or increments.

Sincerely,

William McCormick 25. The square root for

Two feet equals 1.4142135623730950488016887242097 feet
24 inches equals 4.8989794855663561963945681494118 inches
24,000 Thousandths of an inch equals 154.9193338482966754071706159913 Thousandths of an inch.
4 Half foot segments equals 2

Me I am going with the half foot segments. Square root is just a ratio as well, based on the increments used. Nothing else.

Sincerely,

William McCormick 26. You didn't answer the question.

Give two integers such that the square of one is exactly twice the square of the other. (Which, in case it isn't clear, is just another way of asking what river_rat asked.)

BTW: 1.4142135623730950488016887242097 squared is not 2. It's only close to 2. Close enough for practical applications maybe, but still not actually 2. (Windows calculator says it comes out to 1.999999999999999999999999999998.) 27. How about six inches. That squares up to 36 inches.

That means that 18 inches is the number we have to find the square root of.

So we would have to divide 18 into nine increments. Each two inches long.

So the answer would be 3 times 3 two inch increments squared. That would give you 18 inches.

It is a cool way to find the square root of something. But to be honest today, with what accuracy you need in real life. Base ten can achieve it, very quickly.

And since we are almost always dealing with whole numbers and fractions and or irrational decimals. It does have limitations. With pencil and paper.

I always wanted to find a whambam way to get a root on paper. But you would have to break up the whole into different increments.

But I do love talking about math. It brings up some great stuff. Even stuff I never thought of.

You could also use 4.5 inches as an increment and go with 2 times 2, 4.5 inch increments. Giving you 18 inches.

Sincerely,

William McCormick 28. You still haven't answered the question... 29. 4 seems to work pretty good for just about any number. Even primes.

You can take any number you want the square root for.

And divide it by four. Then use the square root of 4, which is 2 times 2. Of whatever the quarter value of the whole number was.

35 divided by 4 equals, 8.75

So 2 times 2 equals four. So four times 8.75 equals 35.

Basically you just make believe your tape measure was a magic tape measure and it always comes out to four increments. You just plug in one of the four increments.

See it pays to discuss math. I had a misconception of the difficulty of dividing most common measurements. Getting 13/16ths is a bit rough on paper in the field, in the wind in the rain. But not something I could not do, even on paper. With a Fisher Space pen. I could do this on paper.

Thanks guys.

Sincerely,

William McCormick 30. Which still doesn't answer my question... 31. Ok. Since you insist on using units, how about this: Imagine I want to make a perfect square with an area of exactly 2 square inches. It's easy enough to measure out rational fractions of an inch, so can you give me a fraction (ratio of two whole numbers) so that using that fraction as the side of a square gives exactly two square inches? 3/2 is fairly close. 7/5 is even closer. Can you make it exact? 32. Mod note: My inclination is to lock this thread, since it seems to have run its course.

Any objections? 33. You should have locked it the moment the first post was posted. With an insane title like pi is rational, this thread clearly has no place in this subforum. If not lock it, at least whisk it away to Pseudoscience. 34. Not sure I agree - the OP was well-intentioned. But anyway....... 