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Irrational Numbers

An Irrational Number is a number that cannot be written as a simple fraction - the decimal goes on forever without repeating.

Example: π (Pi) is an irrational number. The value of Pi is

3.1415926535897932384626433832795 (and more...)

There is no pattern to the decimals, and you cannot write down a simple fraction that equals Pi.

Values like 22/7 = 3.1428571428571... get close but are not right.


It is called irrational because it cannot be written as a ratio (or fraction),
not because it is crazy!

Rational vs Irrational

But if a number can be written as a simple fraction then it is called a rational number:

Example: 9.5 can be written as a simple fraction like this

19/2 = 9.5

So it is not an irrational number (and so is a rational number)

Here are some more examples:

Number As a Fraction Rational or
Irrational?
5 5/1 Rational
1.75 7/4 Rational
.001 1/1000 Rational
√2
(square root of 2)
? Irrational !


Example: Is the Square Root of 2 an Irrational Number?

My calculator says the square root of 2 is 1.4142135623730950488016887242097, but this is not the full story! It actually goes on and on, with no pattern to the numbers.

You cannot write down a simple fraction that equals the square root of 2.

So the square root of 2 is an irrational number

Famous Irrational Numbers

Pi

Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. The first few digits look like this:

3.1415926535897932384626433832795 (and more ...)

e

The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this:

2.7182818284590452353602874713527 (and more ...)

phi

The Golden Ratio is an irrational number. The first few digits look like this:

1.61803398874989484820... (and more ...)

radical symbol

Many square roots, cube roots, etc are also irrational numbers. Examples:

√3 1.7320508075688772935274463415059 (etc)
√99 9.9498743710661995473447982100121 (etc)

But √4 = 2, and √9 = 3, so not all roots are irrational.

History of Irrational Numbers

Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and it was was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!