Irrational Numbers

An Irrational Number is a real number that cannot be written as a simple fraction.

Irrational means not Rational

Let's look at what makes a number rational or irrational ...

Rational vs Irrational

rational vs irrational

Rational Numbers

A Rational Number can be written as a Ratio of two integers (ie a simple fraction).

Example: 1.5 is rational, because it can be written as the ratio 3/2

Example: 7 is rational, because it can be written as the ratio 7/1

Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

 

Irrational Numbers

But some numbers cannot be written as a ratio of two integers ...

...they are called Irrational Numbers.

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

 

Example: π (Pi) is a famous irrational number.

Pi

π = 3.1415926535897932384626433832795 (and more...)

You cannot write down a simple fraction that equals Pi.

The popular approximation of 22/7 = 3.1428571428571... is close but not accurate.

Another clue is that the decimal goes on forever without repeating.

Cannot Be Written as a Fraction

So you can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

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

9.5 = 19/2

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

Here are some more examples:

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

Square Root of 2

Let's look at the square root of 2 more closely.

square root 2 If you draw a square of size "1",
what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...

... in fact you cannot write the square root of 2 using a ratio of two numbers

... I explain why on the Is It Irrational? page,

... and so we know it is an irrational number

Famous Irrational Numbers

Pi

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

3.1415926535897932384626433832795 (and more ...)

e (eulers number)

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 (rational), and √9 = 3 (rational) ...

... so not all roots are irrational.

 

Note on Multiplying Irrational Numbers

Have a look at this:

  • π × π = π2 is irrational
  • But √2 × √2 = 2 is rational

So be careful ... multiplying irrational numbers might result in a rational number!

 

Fun Facts ....

Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to write 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 so it was irrational.

But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!