When we write a fraction or a number as a decimal, different things can happen. Sometimes the decimal stops. Sometimes a group of digits repeats in a loop. And sometimes the digits continue forever without settling into any repeating pattern.
You do not need to know advanced algebra for this lesson. A few sections mention square roots and proofs; these are explained broadly, and the more technical parts are labelled as extensions.
Three kinds of decimal
The digits come to an end. For example, 1/4 = 0.25. After the 5, there are no more non-zero digits to write.
A digit or block of digits repeats forever. For example, 1/3 = 0.333… and 1/7 = 0.142857142857….
The digits continue forever, but no fixed block repeats. Examples include √2 = 1.41421356… and π = 3.14159265….
A rational number is any number that can be written exactly as one whole number divided by another, such as 3/4, 7/2 or 5 = 5/1.
When written as a decimal, a rational number either stops or eventually falls into a repeating pattern.
An irrational number cannot be written exactly as such a fraction. Its decimal keeps going forever and does not repeat in a fixed cycle.
Use a calculator where helpful. Before opening the answer, work out or inspect each decimal and sort it into terminating, repeating, or non-repeating.
For example: 1/7, 3/8, sqrt(2) or pi. A calculator can show lots of digits, but it cannot by itself prove that a decimal never repeats.
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Reveal sorting
Terminating: 1/2 = 0.5, 3/8 = 0.375, √9 = 3, 0.125.
Repeating: 1/3 = 0.333…, 2/7 = 0.285714285714…
Non-repeating: √2 and π.
Use a calculator or long division to test these fractions:
1/2, 1/4, 1/5, 1/8, 1/10, 1/20, 1/3, 1/6, 1/7
Can you spot anything special about the denominators of the fractions that stop?
Reveal pattern
The terminating examples have denominators built only from factors of 2 and 5.
Why these two numbers? Because our decimal system is based on 10, and 10 = 2 × 5. If a denominator can be turned into 10, 100, 1000 and so on by multiplying by whole numbers, the decimal can stop.
For example, 1/8 terminates because 8 = 2 × 2 × 2, and 8 × 125 = 1000, so 1/8 = 125/1000 = 0.125.
For a fraction in lowest terms, if the denominator contains another prime factor, such as 3 or 7, its decimal repeats instead of terminating.
The diagonal of a square
Now we meet an irrational number in a very ordinary shape: a square.
The symbol √2 means “the positive number which, when multiplied by itself, gives 2”. Since 1 × 1 = 1 and 2 × 2 = 4, √2 must lie somewhere between 1 and 2. Its decimal begins 1.41421356….
If a square has side length 1, its diagonal forms a right-angled triangle with sides 1 and 1. Pythagoras tells us that the square of the diagonal is
1² + 1² = 2
So the diagonal is the number whose square is 2: √2.
A square with side length 1 has diagonal length √2.
This is surprising because the diagonal is a perfectly ordinary length that we can draw with a ruler, yet √2 cannot be written exactly as a fraction of two whole numbers. In other words, a simple geometric length can still be irrational.
Is 0.999… really 1?
This is one of the strangest-looking facts about infinite decimals. At first, 0.999… can feel as though it ought to be just a tiny bit smaller than 1.
Which is larger: 0.999… or 1?
Remember that the dots mean the 9s continue forever. There is no final 9 and no “last little gap” left over.
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Reveal answer
0.999… and 1 are exactly the same number.
An algebraic way to show this is to let x = 0.999… . Then 10x = 9.999… . Subtracting the first equation from the second gives 9x = 9, so x = 1.
The equality feels odd only because we are used to finite decimals. With an infinite decimal, there is no last digit after which a gap could remain.
The repeating magic of 1/7
1/7 = 0.142857142857…
The repeating block 142857 has a remarkable property. When you multiply it by 2, 3, 4, 5 or 6, the same six digits appear again, just in a different order.
Because each digit keeps its colour, it becomes easier to see what is happening: the same six digits are not changing, they are just cycling round.
This proof is much more advanced than the main lesson. It is here for learners who enjoy formal reasoning; it is completely fine to skip it.
Show the proof
Suppose, for the sake of argument, that √2 could be written as a fraction a/b in lowest terms.
If a²/b² = 2, then a² = 2b². This means a² is even, which forces a to be even. So we can write a = 2k.
Substituting that back gives 4k² = 2b², so b² = 2k². Therefore b is even as well.
But then both a and b are even, so the fraction was not really in lowest terms after all. That contradiction shows that √2 cannot be written as a fraction.
What have we learned?
- Decimals can terminate, repeat, or continue forever without repeating.
- Rational numbers can be written exactly as fractions, and their decimals terminate or repeat.
- Irrational numbers cannot be written exactly as fractions, and their decimals never settle into a repeating cycle.
- √2 is a simple geometric example of an irrational number.
- Infinite decimals can represent exact values, even when they look surprising: 0.999… = 1.
Next we will encode irrational numbers using fractions inside fractions.
Lesson 4: Numbers That Go On Forever
Download the Lesson 4 printable handout to explore terminating and repeating decimals, irrational numbers and patterns in decimal expansions.
Get the Lesson 4 resources →What did you think?
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