How do you write an irrational number?

Answer 1

It is easier to answer the opposite: an irrational number cannot be written as:

• a ratio of two integers
• a terminating decimal
• a recurring decomal.

It can be written as an expression containing Irrational Numbers, such as 2*pi or 3*√2 or sin(37°) or log(3) or e2 or a combination of these or similar functions.

Answer 2

The short answer is "with great difficulty".

You cannot write an irrational number in decimal or fractional form. At least, you cannot write an exact value although you will be able to get a very close approximation. For example, pi has been calculated to over 10 trillion digits (Oct 2012) whereas 40 digits are enough to give a measure of the entire known universe to an accuracy of the width of a hydrogen atom!

Some irrationals can be expressed exactly - written in their radical form. For example, sqrt(2) or cuberoot(5) but expressions such as 1 + sqrt(3) cannot be simplified so it could be necessary to write out long expressions. The Golden ratio (phi) is one such number.

Even worse are irrationals that are transcendental. There is no way that mathematical constants such as pi or e can be written in a form described above.

There are infinite series for pi or e but obviously it is not possible to write out an infinite series. The main advantage is that, although the infinite sequence of decimal digits are more or less random, the infinite series can be defined in simple and elegant ways.

For example:

e = 1 + 1/1! + 1/2! + 1/3! + 1/4! ... where n! represents 1*2*3*... *n

there are many series for pi.

One is

pi = 4*(1 - 1/3 + 1/5 + 1/7 ... )

Note though that elegant series may be very slow to reach their destination (slow to converge) whereas there may be "ugly" series that converge much more rapidly.