Some numbers end.
Some never do.

“Rational” just means “ratio.”

Forget the everyday meaning of rational (calm and sensible). In maths it comes from ratio — a comparison of two whole numbers. A number is rational if you can write it as a / b, where a and b are integers and b is not zero.

An integer is a whole number with nothing after the decimal point — the counting numbers, their negatives, and zero: … −2, −1, 0, 1, 2, 3 …. (Bottom can't be zero, because splitting something into zero pieces makes no sense.)

The clever part is how many disguises a rational number can wear. Whole numbers are rational, because every whole number is secretly itself-over-one:

Going the other way is just as easy. Any decimal that stops can be turned back into a fraction by reading its place value: 0.75 is "seventy-five hundredths", so 0.75 = 75/100, which tidies up to 3/4. Negative numbers and zero are card-carrying members of the family too — −3 = −3/1 and 0 = 0/1 — because the top is allowed to be any integer. The one rule you can never break is the bottom: it must not be zero.

So the rational family is huge: it scoops up every integer, every fraction, and — as you're about to see — every decimal that either stops or settles into a repeating rhythm. That's most of the numbers you'll ever write down.

Decimals that stop — or repeat — are rational too.

Turn any fraction into a decimal (just divide the top by the bottom) and one of exactly two things happens. Either the decimal stops, or it falls into a repeating loop. Both are rational, because both came from a fraction in the first place.

terminatingA terminating decimal is one that ends — it runs out of digits. 3/4 = 0.75 stops dead after two places. 1/8 = 0.125 stops after three.

recurringA recurring (or repeating) decimal never stops, but a fixed block of digits repeats forever. 1/3 = 0.333… — the 3 loops on and on. 1/7 = 0.142857142857… — the whole block 142857 repeats. We mark the repeating block with a bar over it, like 0.3.

Why does a fraction ever repeat? Watch what happens when you actually divide. To work out 1 ÷ 7 you carry remainders: 10 ÷ 7 leaves 3, then 30 ÷ 7 leaves 2, then 20 ÷ 7 leaves 6, and so on. But there are only seven possible remainders (0 through 6), so sooner or later one of them must come back around — and the instant a remainder repeats, the digits start repeating with it. That's the engine inside every recurring decimal: a limited supply of remainders, looping. A decimal can only stop when a remainder finally lands on zero.

And it runs backwards too, which is the real proof that repeating decimals are genuinely fractions. Let x = 0.333…. Multiply both sides by 10: 10x = 3.333…. Now subtract the first line from the second — the endless tails cancel perfectly — leaving 9x = 3, so x = 3/9 = 1/3. Any repeating decimal can be unwound into a fraction with this trick, so the entire repeating clan is rational, no exceptions.

So here's a great question: which fractions terminate and which ones repeat? It's not random — it's hidden in the bottom number. Drive the machine below and find the rule for yourself.

“Irrational” means it never settles.

irrationalA number is irrational when its decimal goes on forever and never falls into a repeating block. Not "hasn't repeated yet" — never, not in a million digits, not in a billion. And because of that, it can't be squeezed into any fraction a/b at all.

Think about what that means. Every fraction you tested in the machine eventually showed its hand — it stopped, or it locked into a loop. An irrational number does neither. It keeps inventing new digits with no pattern you could ever predict from a rule. The most famous members of this family are √2, √3, and π.

It helps to picture the number line. The rationals are sprinkled everywhere along it — between any two of them you can always squeeze in another, and another, forever. Yet even with all that crowding, the line still has tiny gaps the fractions never quite reach. The irrationals are exactly the numbers that fill those gaps. √2 isn't off in some strange place; it sits right there between 1.4 and 1.5, at a spot no fraction can ever name.

That's a bold claim — a number whose digits truly never repeat. The fastest way to feel it is to put a rational number and an irrational one side by side and reveal their digits one at a time. Watch what each one does.

Watch one lock — and one refuse.

Reveal more digits and keep your eye on the two streams. The rational one snaps into a repeating block (highlighted in teal). The irrational one keeps surprising you — no block, ever.

√2 is a real length you can draw.

It's tempting to think an irrational number must be some weird, made-up thing. But √2 is as real as the ruler on your desk. Take a square that is 1 unit on every side and draw a line corner to corner. That diagonal has a length of exactly √2 — about 1.414. You can draw it in one stroke.

So here's the strange, beautiful fact: that line is a perfectly definite length, yet no fraction equals it exactly. You can get close — 1.4, then 1.41, then 1.414 — but you'll never land on it with any a/b, however enormous you make the numbers. People proved this more than two thousand years ago; you don't need the proof to feel it, because the digit stream already showed you √2 never repeating.

Here's a friendly way to sense why no fraction can hit it (no scary algebra, promise). Squaring √2 has to give exactly 2. But when you square a neat fraction you only ever get another neat fraction — and 2 sits stubbornly between the perfect squares 1 and 4, with no whole-number root of its own. So every fraction you try comes out a hair too small or a hair too big; the harder you zoom in, the more the digits wriggle out of reach. √2 simply lives in the gaps between the fractions, where no a/b can follow.

The discovery that rattled the Greeks.

For a long time, people were sure every number was rational — that any length you could draw could be written as a ratio of two whole numbers. The followers of Pythagoras, the ancient Greek mathematicians, built their whole picture of the universe on that belief. Then someone looked hard at the diagonal of a plain little square and realised √2 could not be written as any fraction at all. Legend says the news was so unwelcome it was meant to be kept a secret.

Whether or not the dramatic stories are true, the discovery itself was real, and it changed mathematics forever. It meant the number line had gaps that fractions could never fill — and those gaps needed a brand-new kind of number to live in them. We call them the irrationals, and √2 was the very first one anyone ever proved. Not bad for a single line drawn across a square.

Perfect squares are the lucky ones.

Not every square root is irrational — and there's a tidy rule for which are which. A perfect square is a whole number made by multiplying an integer by itself: 1, 4, 9, 16, 25, 36 … Their roots come out as clean whole numbers, so they're rational: √4 = 2, √9 = 3, √16 = 4.

But the square root of any other whole number — one that isn't a perfect square — is irrational: √2, √3, √5, √7, √10, and so on. Slide through the numbers and watch which roots land exactly on a whole number, and which spill into endless digits.

There's a quick mental check. Ask yourself: is there a whole number I can multiply by itself to get this? For 49 the answer is 7, so √49 = 7 (rational). For 50 there's no such whole number — 7 × 7 is 49 and 8 × 8 is 64, so √50 falls between them and runs on forever (irrational). The perfect squares are surprisingly thin on the ground — 1, 4, 9, 16, 25, 36, 49, 64 — so most square roots you ever meet turn out to be irrational, with √2 the friendly face of the whole bunch.

Rational or irrational? Call it.

For each number, pick a family. Some are sneaky on purpose — a messy-looking decimal can still be rational, and a tidy-looking pattern can still be irrational. Tap to lock in your answer and see why.

Where each family lives.

The rationals run everyday life. Money goes to the exact cent (£3.75 is 375/100). Recipes split into halves and thirds. A test score of 18/20 is rational; so is your height in centimetres, your running time in seconds, the slope of a wheelchair ramp. Whenever you can count something or measure it to a fixed number of places, you're holding a rational number.

The irrationals hide in shapes and patterns. √2 is the diagonal of any square — and it's also why a sheet of A4 paper keeps its proportions when you fold it in half. π lives on the edge of every circle. The golden ratio shows up in spirals and design. These aren't errors or weird exceptions; they're exact answers that simply refuse to be fractions.

Here's a detail that surprises people: even your calculator and computer only ever store rational numbers. When a screen shows √2 as 1.4142136, it has quietly rounded — chopped the endless tail off to make it fit. An irrational number can be perfectly described (√2, π) and perfectly drawn (a diagonal, a circle), but it can never be fully written out in digits. That's not a bug in the maths; it's just the nature of the wild family.

Three traps to dodge.

Trap 1: “A messy-looking decimal must be irrational.” Nope. 0.142857142857… looks wild, but it's just 1/7 — the block 142857 repeats, so it's rational. And the long decimal your calculator shows is very often a rounded-off fraction, not a true irrational. Looks aren't proof; the pattern is.

Trap 2: “π equals 22/7.” This one trips up almost everyone. 22/7 is a handy approximation of π, but it is not π. In fact 22/7 = 3.142857142857… — it repeats, which means it's rational. The real π = 3.14159265… and never repeats. They're close near the start, then drift apart. Close is not equal.

Trap 3: “If a decimal has a pattern, it's rational.” Careful — "rational" needs a repeating block, not just any pattern. Look at 0.101001000100001… (one more zero each time). There's a clear, obvious pattern, yet no fixed block ever repeats — so it's irrational. Repeating means the exact same chunk, over and over, forever.