How do we know that irrational numbers such as pi never end or repeat?

Question

I'm not very good at math, but I remember that ever since the concept of irrational numbers and repeating fractions was presented to me, I was unable to comprehend it.

If irrational numbers go on forever- infinitely- then we can bring them out to billions and billions of digits without ever seeing them repeat or end rationally- but how do we know that this is the case? How can we be sure that pi doesn't end at the trillionth digit or something? or maybe it begins to repeat at some point. It just doesn't seem logical or scientific to say that "these numbers go on forever" without having any actual PROOF of this.

Answer 1

First, if a decimal ends then the number can be written as a fraction:

3.14=314/100.

Also, if the decimal repeats, the number can be written as a fraction:

If x=3.14141414....(repeating)
100x=314.141414.. (repeating),
so
99x=314-3=311,
so x=311/99.

Hence every number with a finite or repeating decimal expansion can be written as a fraction with two whole numbers. The way to show that the decimal expansion of a number neither ends nor repeats is to show that it cannot be written as such a fraction.

The proof that pi is irrational is rather tricky. On the other hand, the proof that sqrt(2) is irrational is actually easy:

Suppose that sqrt(2)=a/b with a and b whole numbers and assume that the fraction is in lowest terms.
Then 2=(a/b)^2 =a^2 /b^2, so
2b^2=a^2.
This means that a^2 is even. But the square of an odd number is odd, so a must be even. So we can write a=2k. Plug in:
2b^2=(2k)^2=4k^2.
Cancel a 2:
b^2=2k^2.
But, now, for the same reason as above, b must be even. But then 2 is a common factor of both a and b, so a/b was not in lowest terms (as we assumed it was!).

This shows that sqrt(2) is not a fraction. This shows that the decimal expansion of sqrt(2) neither ends nor repeats.

Answer 2

What an excellent question!

For a number to terminate (in base 10), it must be able to be written as a fraction that reduces to a prime factorization with factors of 2 or 5, only.
As an example, 71 / 40 can be written as 1.775

Because of our place-value system, every decimal place has a value of 10 to some power. To the left of the decimal point, we have ones, tens, hundreds, and so on. To the right, tenths, hundredths, thousandths, and so on. The reason a terminating decimal has to have a denominator of 2's and 5's only is because those are the only numbers that divide powers of ten evenly. The 1.775 in the above example is 1775 / 1000, which reduces to 71 / 40.

Any other rational number will be a repeating decimal. This is easy to see when you actually use long division. The fraction 10 / 3 is a simple example. 3 goes into 10 three times, with a remainder of 1. "Bringing down" the next zero in the division gives 3 into 10 again. It's going to be the same answer as the last time, and the process goes on ad infinitum... it repeats forever.

For a fraction like 14 / 11, the process changes a bit. Grab a piece of paper and perform the long division as we go. Eleven goes into 14 once, leaving 3. The next division is eleven into 30, going twice and leaving 8. The next division is eleven into 80, going seven times and leaving 3... the same remainder we had before. 14 / 11 = 1.27272727..., forever. The same pattern of digits repeat, rather than just one digit.

Now consider the non-terminating rational number n/d. When you go through the division algorithm, d will go into n a certain number of times, leaving a remainder. It will then go into the next partial dividend a certain number of times, leaving what could be a different remainder. The thing is, since the decimal doesn't terminate, it can't leave a remainder of 0, so at most there are (d - 1) possible remainders it can have. Even if you divide it out this many times, the next division will leave a remainder you've already seen, causing a repeating pattern. Therefore, all rational numbers must terminate or repeat. The contrapositive of this also means that any non-terminating, non-repeating decimal must be irrational.

There are a number of proofs (some have been mentioned above) that certain numbers, such as π or √2, are irrational... that there is no way to express them as a ratio of an integer to a natural number. The difficulty in proving a number like π or e is irrational lies with the fact that the maths behind many of them are very difficult to prove in a typical high-school or basic undergraduate level class. The proofs are out there, but they can be difficult to follow.

What else is weird is that there are some numbers that no one has been able to prove are rational or not. No one knows if the Euler constant is rational or not... no one knows whether (π + e) is rational or not... even though each has been calculated to millions of decimal places. That's what makes your question so interesting. Heavy-duty math geeks are still trying to figure these one out!

Answer 3

OK, any number that terminates or repeats can be written as a fraction. This is synonymous with being rational. Actuall, the definition of rational is that it can be written as the ratio of 2 integers.

Theorem: The sqrt(2) is irrational.

Proof by contradiction:
Assume that sqrt(2) IS rational and therefore can be written as a/b for 2 integers a and b. Further, since all fractions have a completely reduced form, we can assume that a and b are relatively prime.

Before we get started, lets do a lemma.

Lemma: If a*a is even, then a must be even.
Proof: Assume a is odd. Then a*a is odd. So, a*a is even, if and only if a is even.

Continuation of proof that sqrt(2) is irrational.

a/b = sqrt(2). Squaring both sides gives us

a*a/(b*b) = 2.

a*a = 2*b*b. Therefore, a*a is even. So, a must be even.

Replace a with 2*k for some k an integer using the definition of even.

(2*k)(2*k) = 2*b*b. Or,
4*k = 2*b*b. Mulitply both sides by 1/2 we get
2*k = b*b. This tells us that b*b is even.

Hey, remember that lemma that says if b*b is even that b must be even? So, we conclude that b is even.

We have now arrived at a contridiction because both a and b have to be even for this to work out. But, wait, a and b are supposed to be relatively prime.

The only assumption that we made was that sqrt(2) = a/b. It lead to something that was impossible. This means that you can't write sqrt(2) as a fraction therefore it must never terminate or repeat.

Answer 4

Suppose that a number has the form

x = aaaaa.aaaaaa bbbbb bbbbb bbbbb bbbbb ...

where the block of b's repeats. Let m be the number of digits of a after the decimal point, and n the length of block b. We can now write x as

x = [a * p + b] / [p * q]

with p = 10^n - 1 and q = 10^m. This is a fraction, so x is a rational number.

For instance, let x = 135.792 5466 5466 5466 ...
Then a = 135792, b = 5466, m = 3 and n = 4. We find

p = 10^4 - 1 = 9999; q = 10^3 = 1000

x = [135792 * 9999 + 5466] / [9999 * 1000]
= 1357789674 / 9999000.

This proves that every repeating decimal number is rational -- and therefore, every irrational number is non-repeating.



The fact that pi is irrational is much harder to prove! A theorem by Hermite-Lindemann states that

if x<>0 is algebraic over the rational numbers then exp(x) is transcendental (i.e. non-algebraic).

"Algebraic" numbers can be irrational, but rational numbers are for sure algebraic. Now suppose that pi were algebraic. Then the number 2*pi*i would also be algebraic. The conclusion of the theorem woud be that exp(2*pi*i) is transcendental. This is not true, because in fact exp(2*pi*i) = 1. Therefore, the assumption that pi is algebraic does not work, and pi must be transcendental. All transcendental numbers are irrational.

Answer 5

Real numbers are either rational, that is, can be expressed as a quotient of two integers, with the denominator being nonzero; or not, irrational, numbers whose decimal representation results in either a repeating sequence or not.

One proof approach is to assume pi is rational. The resulting contradiction leads to the rejection of the assumption, which leaves only one place for pi, in the irrational subset of the Reals.

Answer 6

How To Do Irrational Numbers

Answer 7

Because if the numbers ever ended or repeated, they would be rational numbers, not irrational numbers.

If the decimal ended, it would be rational, with all of those digits divided by some power of 10.

If the decimal repeated, it would be rational, with all of those repeated digits divisible by some multiple of 9.

Answer 8

Most accurate value of pi
As continuation of a long-running project, Yasumasa Kanada of the University of Tokyo has calculated the number pi to 1,241,100,000,000 decimal places. It has been done using an HITACHI SR8000/MPP computer

http://www.guinnessworldrecords.com/content_pages/record.asp?recordid=47055

Answer 9

Many of these issues have been tested out to million and millions of digit past the decimal point, but for issues like pi we do not know for sure they never end. On the other hand dividing 10 by 3 is obviously never going to end.

Answer 10

There are proofs for pi being irrational and square root of 2 being irrational. Your teachers would not prove that pi is irrational since the proof is difficult. You may have seen the proof that square root of 2 is irrational.

proofs that pi is irrational:
http://www.mathpages.com/home/kmath313.htm
http://www.mcs.csuhayward.edu/~malek/Mathlinks/Pi.html

proof for square root of 2 is irrational:
http://www.cut-the-knot.org/proofs/sq_root.shtml