Real Numbers
Every number you meet this year is either rational — a fraction — or irrational. Knowing which tells you how it behaves.
Draw a square one unit wide. How long is the diagonal across it?
By Pythagoras it is √2. So write that as a decimal.
It is not 1.4, because 1.42 = 1.96. Not 1.41 either — that squares to 1.9881.
Keep going and the decimals get closer but never arrive. There is no fraction that gives it exactly.
And yet the diagonal is right there on the page. You can measure it with string.
So some perfectly real lengths cannot be written as a fraction. Those are the irrational numbers, and together with the fractions they fill the number line completely — with no gaps left over.
The families, nested inside each other
Counting numbers: 1, 2, 3, …
Whole numbers: the counting numbers plus 0.
Integers: whole numbers and their negatives.
Rationals: anything expressible as a fraction — which includes every integer, since 5 = 51.
Reals: the rationals together with the irrationals.
Each family sits inside the next, so -3 is an integer and a rational and a real, all at once.
How to tell rational from irrational
Write it as a decimal and watch what happens.
Stops — like 0.75 — rational.
Repeats a block forever — like 0.3333… or 0.142857142857… — rational.
Goes on forever with no repeating block — like π = 3.14159… or √2 = 1.41421… — irrational.
The root test
The square root of a perfect square is rational: √49 = 7. The square root of any other whole number is irrational: √50 never resolves.
Turning repeating decimals into fractions
0.3 means 0.333… forever. Let x = 0.333…, then 10x = 3.333…
Subtract: 10x - x = 3, so 9x = 3 and x = 13.
The trick is choosing the multiplier so the repeating tails line up and cancel.
Comparing and ordering real numbers
Convert everything to decimals, then compare. 78 = 0.875, √0.81 = 0.9, and 0.87 — so the order is 0.87 < 78 < √0.81.
For irrationals, use a good estimate: π ≈ 3.14 and √10 ≈ 3.16, so π < √10.
Careful with negatives: -34 > -0.8, because -0.75 sits to the right of -0.8.
One form at a time
Never compare a fraction against a decimal against a root in your head. Put them all in the same costume — decimals are usually easiest — and then compare.
Why π is not 227
227 = 3.142857… and repeats, so it is rational. π = 3.14159… and never repeats.
227 is a convenient approximation, close to π but not equal to it. The same goes for 3.14.
Worked examples
Classify
Classify each: -7, 25, √16, √20, 0.6.
- -7: integer, and rational since -7 = -71. Every integer is rational.
- 25 = 0.4: rational, terminating. Already a fraction.
- √16 = 4: integer and rational. 16 is a perfect square, so the root is whole.
- √20 ≈ 4.472…: irrational. 20 is not a perfect square.
- 0.6 = 23: rational. A repeating decimal is always a fraction in disguise.
All rational except √20, which is irrational.
Repeating to fraction
Write 0.27 as a fraction in lowest terms.
- Let x = 0.272727… Name the number so you can operate on it.
- The block is two digits, so multiply by 100: 100x = 27.272727… Now the tails match exactly.
- Subtract: 100x - x = 27, so 99x = 27 The infinite tails cancel — that is the whole trick.
- x = 2799 = 311 Divide top and bottom by 9.
311
Order them
Put in order from least to greatest: 56, 0.8, √0.64, 45.
- Convert everything to decimals. One costume makes comparison safe.
- 56 ≈ 0.833, 0.8, √0.64 = 0.8, 45 = 0.8 Three of them are exactly equal.
- So 0.8 = √0.64 = 45 < 56 Only 56 is larger.
0.8 = √0.64 = 45 < 56
Words to know
- Rational
- Any number writable as ab with integers a and b, b ≠ 0.
- Irrational
- A real number that cannot be written as a fraction — its decimal never repeats and never ends.
- Integer
- A whole number, positive, negative or zero.
- Terminating / repeating
- A decimal that stops, or one that repeats a block forever.
Mistakes to watch for
Calling every square root irrational. √25 is 5 — perfectly rational. A root is irrational only when the number under it is not a perfect square.
Treating 227 or 3.14 as equal to π. They are approximations, and good ones, but π has no exact fraction at all. Writing = where you mean ≈ quietly claims something false.
Mis-ordering negatives. -0.8 is less than -0.75, because it lies further left on the number line. With negatives, bigger digits mean a smaller number.
Assuming a long decimal must be irrational. 17 = 0.142857142857… runs forever but repeats, and anything that repeats is a fraction. Irrational means never repeating.
The Pythagoreans believed every length was a ratio of whole numbers. The diagonal of a square destroyed that belief, and the proof is short enough to fit on a postcard.
Legend says the discoverer, Hippasus, was thrown overboard at sea for revealing it — though the story may have been invented centuries later.
π was proved irrational only in 1761, by Johann Lambert. Nobody had managed it in the previous two thousand years of trying.