Look for the rhythm

The idea

π cannot be written as a ratio of two integers, so it is irrational; its digits go on forever with no repeating pattern, unlike the decimal expansions of fractions.

A rhythm in the digits

The digits of π go on for ever. Fractions give decimals with a rhythm. The fraction 1/3 gives 0.3333…, 1/11 gives 0.090909…, and 1/7 gives 0.142857 142857…

Does π settle into a repeating rhythm like these?

• Yes, a block of digits repeats, like in 1/7
• No, no block of digits repeats
• Only the first few digits repeat

Cut the digits into blocks

Pick a number and cut its digits into blocks. Choose the block length. A digit turns green when it matches the same place in the first block, and red when it does not.

No rhythm at all

π cannot be written as a ratio of two integers, so it is irrational. Its digits go on for ever with no repeating pattern, unlike the decimal expansions of fractions.

A rational number can be written as a fraction a/b, where a and b are integers and b is not 0. For example 12/3, −7/11 and 1.4. In 1761 the mathematician Lambert showed that π is irrational. Proving it needs more advanced mathematics, which we will learn much later.

Notes

π cannot be written as a ratio of two integers, so it is irrational. Its digits go on for ever with no repeating pattern, unlike the decimal expansions of fractions.

Check yourself

The fraction 1/11 gives 0.090909… How many digits are in the shortest block that repeats?

Answer: 2

The block 09 repeats, so it has 2 digits.

The fraction 1/7 gives 0.142857 142857…, and the block 142857 repeats for ever. What is the 8th digit after the decimal point?

Answer: 4

Digits 7 and 8 are the start of the block again: 1, 4. So the 8th digit is 4.

Which of these numbers is irrational?

A number's decimal has no repeating rhythm at all. What can we say?

• π — correct. Yes. π cannot be written as a ratio of two integers.
• −7/11. −7/11 is already a ratio of two integers, so it is rational.
• 1.4. 1.4 = 14/10, a ratio of two integers, so it is rational.
• It is a fraction with a very big denominator. Even a fraction with a big denominator falls into a repeating block.
• It cannot be a fraction: it is irrational — correct. Yes. Every fraction gives a decimal with a rhythm, like 1/3, 1/11 and 1/7.
• It must be a whole number. A whole number has no digits after the decimal point at all, so it has nothing to repeat.