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What causes irrational numbers such as pi to never form patterns? Answered

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here it is:

Let's suppose √2 were a rational number. Then we can write it √2 = a/b where a, b are whole numbers, b not zero. We additionally make it so that this a/b is simplified to the lowest terms, since that can obviously be done with any fraction.

It follows that 2 = a2/b2, or a2 = 2 * b2. So the square of a is an even number since it is two times something. From this we can know that a itself is also an even number. Why? Because it can't be odd; if a itself was odd, then a * a would be odd too. Odd number times odd number is always odd. Check if you don't believe that!

Okay, if a itself is an even number, then a is 2 times some other whole number, or a = 2k where k is this other number. We don't need to know exactly what k is; it won't matter. Soon is coming the contradiction:

If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:

2 = (2k)2/b2
2 = 4k2/b2
2*b2 = 4k2
b2 = 2k2.

This means b2 is even, from which follows again that b itself is an even number!!!
WHY is that a contradiction? Because we started the whole process saying that a/b is simplified to the lowest terms, and now it turns out that a and b would both be even. So √2 cannot be rational.

That is a trivial consequence of the definition of irrational. A rational number is one which is expressed in the form of a ratio of integers (a fraction). The numerator and denominator may be arbitrarity large, such as

12345678901234567890 / 9999998888888777777766666665555553

but they are both still integers. All rational numbers, when expressed in the form of a "decimal" (in any base, not just base ten) will either terminate with a finite number of digits (followed by zeros :-), or will form a repeating pattern.

The converse is also true. Any decimal expression which forms a terminating or repeating pattern can be expressed as a fraction

All numbers which cannot be expressed as fractions, and which therefore must have an infinite series of non-repeating decimal digits, is called irrational.

No, 1/3 is rational -- trivially, as you just wrote it as a ratio (that's where the word "rational" comes from) of two whole numbers (1 and 3).  Did you actually read what I wrote before replying to it?

Pi is not rational precisely because you cannot find any two integers with which you can write a ratio equal to pi.

this is going to take a while. i will use the square root of 2 as an example.
Let's suppose √2 were a rational number. Then we can write it √2 = a/b where a, b are whole numbers, b not zero. We additionally make it so that this a/b is simplified to the lowest terms, since that can obviously be done with any fraction.

It follows that 2 = a2/b2, or a2 = 2 * b2. So the square of a is an even number since it is two times something. From this we can know that a itself is also an even number. Why? Because it can't be odd; if a itself was odd, then a * a would be odd too. Odd number times odd number is always odd. Check if you don't believe that!

Okay, if a itself is an even number, then a is 2 times some other whole number, or a = 2k where k is this other number. We don't need to know exactly what k is; it won't matter. Soon is coming the contradiction:

If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:

2 = (2k)2/b2
2 = 4k2/b2
2*b2 = 4k2
b2 = 2k2.

This means b2 is even, from which follows again that b itself is an even number!!!
WHY is that a contradiction? Because we started the whole process saying that a/b is simplified to the lowest terms, and now it turns out that a and b would both be even. So √2 cannot be rational.

It has been proved that pi is irrational (it cannot be written as a fraction of two whole numbers) and that it is trancendental (it cannot be written as a polynomial with rational coefficients). For highschool-level math classes, this is generally sufficient to assume that the digits are 'as good as random'. If you are interested in the proofs, a little calculus is needed.

However, whether the digits appear with equal frequency is a very tricky question that has yet to be solved definitively.

Still, given how many digits of pi have been calculated so far, I think it is fairly safe to say that we will not be finding any simple pattern to pi.