No loop, ever

The idea

The decimal expansion of an irrational number never ends and never repeats; there is no repeating block.

Hunt for a loop

The decimals of rational numbers stop or loop. What do the decimals of √2 and π do?

Will the digits of √2 = 1.41421356… ever settle into a repeating block?

• Yes, after enough digits
• No, never
• We cannot tell

Look for a loop

Choose a block length n and press Look for a loop. Does every digit equal the digit n places before it? Test all four numbers.

Never ends, never repeats

The decimal expansion of an irrational number never ends and never repeats: there is no repeating block.

√2 = 1.4142135623730950488… and π = 3.1415926535897932384… A decimal that stops or repeats belongs to a rational number. A decimal with no end and no loop is the signature of an irrational number.

Notes

The decimal expansion of an irrational number never ends and never repeats; it has no repeating block.

Check yourself

Which decimal could be the decimal of an irrational number?

Which statement about √2 is true?

Which of these has a decimal with a repeating block?

5/11 = 0.4545… How many digits are in the repeating block?

Answer: 2

The block 45 has 2 digits.

• 1.4142135623730950488… (it never ends and never repeats) — correct. Yes. A decimal that neither stops nor repeats cannot be a rational number.
• 0.125. This decimal stops, so it is a rational number: 0.125 = 1/8.
• 0.272727… (27 repeats). This decimal repeats, so it is rational: 0.2727… = 3/11.
• Its decimal repeats after about 20 digits. No block ever repeats in the decimal of √2.
• Its decimal never ends and never repeats — correct. Yes. There is no repeating block.
• Its decimal stops after 19 digits. The digits 1.4142135623730950488… go on forever.
• π. The decimal of π never repeats.
• √2. The decimal of √2 never repeats.
• 5/11 — correct. 5/11 = 0.4545…: the block 45 repeats forever.