Irrational number question?

Question

I know they are numbers that cannot be expressed as a ratio but there is often an underlying pattern to them, ie gregories series which adds up to pi. What I was wondering is, is there a pattern to all irrational numbers or not.

Answer 1

Is there a pattern to all irrationals? No, there is not.

Having said that, there are many, many irrationals that do have a pattern. For example, 1.01001000100001000001... is irrational and has an obvious pattern.

Yours is an important question since it goes to the very heart of computational cryptography and the creation of pseudo-random numbers. Encryption algorithms usually require a string of random digits (to be used as the encryption key, say), and irrationals are often offered as a source for them. Choosing an irrational with a "pattern" like the one above gets a cryptographer into trouble very quickly.

So, your question leads straight to a very deep part of mathematics, namely, are there irrationals whose pattern is "random"? Take pi, for example, its digit sequence will pass all or almost all tests for randomness, yet any given digit of pi can be calculated. So, is an irrational like pi whose digits can be calculated "random?" There are irrationals --called Blum Blum Shub numbers-- that have the following property: given any digit your chances of guessing the digit before or the digit after is exactly 1 in 10, yet any given digit can be calculated. In other words, if I say "The current digit of this number is 7, can you guess the digit that came immediately after it?" your chances of guessing would be 1 in 10. Is this number "random?" I can calculate the next number and see if you are right, even though neither of have any good chances of guessing what it might be.

If you keep asking questions like "Is there a pattern to all irrationals?" and searching for answers you will be a full-fledged mathematician before you know it!

HTH

Charles

Answer 2

Rational numbers can be written as a decimal with an infinite number of digits after the decimal point, but they will always repeat after a bit. For example, 1/3 = 0.3333333..., 5/6 = 0.833333, 1/7 = 0.142856142856142856...

All irrational numbers can be expressed as a decimal with an infinite number of non-repeating digits after the decimal point. For example:

Pi = 3.14159265...
e = 2.7182818284...
Sqrt(2) = 1.414213562...

There are an infinite number of irrationals, and they are far more common than the rationals. There's no pattern to their occurrence within the set of all numbers (R).

It's very simple to construct an irrational number which has a pattern in itself. Here's one:

0.12345678910111213141516171819202122...

It's easy to see the patten, but since it never repeats, it is irrational.

Answer 3

The concept of their being NO pattern is not perfectly defined.

That said, if you were to list all the patterns you could think of, an irrational number could be found that satisfied none of them. The reason for this is that by the very act of describing a pattern, you almost certainly limit the number of numbers that meets it to being "countably infinite", whereas the real numbers are "uncountably infinite". And "uncountably infinite" is strictly more than "countably infinite".

If you're describing a set of things, and the way to specify an infinite number of them is to make a FINITE number of choices of integers, then the set of countably infinite. This applies even if the finite number itself is a choice. For example, the number of polynomials with integer coefficients in one variable is countably infinite. So is the number of polynomials of any degree. So is the number of polynomials of any degree, of any finite number of variables.

Answer 4

The square roots of those numbers that are not perfect squares .such as the square roots of 2 , 3 ,5 ,6 ,7 8 ,10.The square root of 2 is 1.414 square this number and it is not 2 is a number tha approaches 2 but it is not 2.Try for the rest o the list

Answer 5

I'm not 100% sure on this, but I am pretty sure I have read somewhere that if the square root of a number is not a whole number then it is irrational, but like I say I may be wrong.

Answer 6

sqrt(2) and -sqrt(2) are the two irrational, yet their sum is rather rational. extra normally, in case you have a rational extensive form q and an irrational extensive form r, (q-r) would be irrational, and r and (q-r) would be 2 irrational numbers with a rational sum.

Answer 7

rational numbers can be expressed
in the form p/q where p and q are
integers

for instance, sqrt2 cannot be
expressed in the form of two
integers p/q,therefore, it is
irrational

whereas, 2=2/1=4/2=6/3 etc
can be expressed in the form
p/q and is therefore rational

i hope that this helps

Answer 8

I doubt it, but x^0.5 is irrational unless x is a square number

Answer 9

I don't yet have the answer to that question, but I liked it so much I thought it deserved a star.

Answer 10

I reckon there is, i was once told that all numbers have an intrinsic pattern.....