Some decimals stop. Some repeat forever. And some go on forever without ever settling into a repeating pattern.
Three kinds of decimal
Terminating: 1/4 = 0.25.
Repeating: 1/3 = 0.333… and 1/7 = 0.142857142857…
Non-repeating: √2 = 1.41421356… and π = 3.14159265…
A rational number can be written as a fraction of two whole numbers. Its decimal terminates or repeats. An irrational number cannot be written that way, and its decimal never terminates or repeats.
The diagonal of a square
A unit square has side length 1. By Pythagoras, its diagonal d satisfies d² = 1² + 1² = 2, so d = √2.
The ancient discovery that √2 is irrational was shocking: even this simple geometric length cannot be expressed as an exact fraction.
Is 0.999… really 1?
Which is larger: 0.999… or 1?
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Reveal answer
Let x = 0.999… . Then 10x = 9.999… . Subtract the first equation from the second: 9x = 9, so x = 1.
Therefore 0.999… = 1. There is no tiny gap hiding at the end.
The repeating magic of 1/7
1/7 = 0.142857142857…
The repeating block 142857 has a remarkable property:
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
The same digits keep cycling around.
Suppose √2 = a/b in lowest terms. Can you show this would force both a and b to be even?
Reveal answer
If a²/b² = 2, then a² = 2b², so a² is even and therefore a is even. Write a = 2k. Then 4k² = 2b², so b² = 2k², making b even too.
But then a/b was not in lowest terms. Contradiction. So √2 is irrational.
What have we learned?
- Rational decimals terminate or repeat.
- Irrational decimals continue without repeating.
- √2 is a simple geometric example of an irrational number.
- Infinite decimals can behave in surprising but exact ways, such as 0.999… = 1.
Next we will encode irrational numbers using fractions inside fractions.