The world runs on rounded numbers. The sign outside the petrol station says 149.9p, not 149.8763p. The news says the world's population is 8.1 billion — not 8{,}118{,}835{,}999, a figure that would be out of date before the newsreader finished saying it (three people are born every second). You say you're "one metre sixty," not 1.5983 metres — and nobody calls you a liar.

But here's the thing: all of those numbers are lies, technically. Tiny, useful, deliberate lies. Rounding is honest lying — you trade a little accuracy for a number a human brain can actually hold — and the rules on this page exist so that everyone, everywhere, tells exactly the same lie. If you and a scientist in Tokyo both round 3.47 to one decimal place, you must both get 3.5. Every time. No arguments.

This page gives you the two tools everyone uses: rounding to decimal places (how many digits after the point) and rounding to significant figures (how many digits that actually carry information). Same one rule underneath both.

Rounding to decimal places

Rounding a decimal works exactly like rounding a whole number: we swap an awkward number for a tidy nearby one. The only new question is how many digits after the point we want to keep — that is the number of decimal places (d.p.).

Here is the whole trick, and it never changes for the rest of your life. To round to a given place, look only at the very next digit — the one just after the place you are keeping:

• that next digit is 5 or more → round up;
• that next digit is 4 or less → round down (keep it as is).

One digit. One glance. That's the entire rule — everything else on this page is just about where you point that glance.

Worked slowly: find the cut, ask the neighbour

Turn the rule into a three-step recipe you can do half-asleep:

1. Find the cut. Mark where the digits you're keeping end.
2. Ask the next digit. Look at the single digit just after the cut — nothing beyond it.
3. Round and drop. 5 or more bumps the last kept digit up by one; 4 or less leaves it alone. Then everything after the cut vanishes.

Round 3.47 to 1 d.p. The cut goes after one digit past the point: 3.4\,|\,7. The neighbour just after the cut is 7 — that's 5 or more, so the 4 bumps up to 5:

Round 0.832 to 2 d.p. The cut goes after two digits: 0.83\,|\,2. The neighbour is 2 — that's 4 or less, so the 3 stays put and the 2 is dropped:

Notice what rounding down really means: it doesn't mean the digit gets smaller — it means you keep it exactly as it is and just chop off the tail. And a sanity check that catches most mistakes: the rounded answer must be close to the original. 3.5 sits right next to 3.47; if you'd somehow got 4.5, that distance is your alarm bell.

Why does "5 or more" round up?

Because a decimal always sits between two neighbours, and the next digit tells you which one it's closer to. Rounding to 1 d.p. snaps a number to the nearer tenth. The midpoint between two tenths is the 5 in the next place — anything from there on is closer to the tenth above. So the rule isn't an arbitrary convention someone dreamed up; it's just the honest answer to "which neighbour is nearer?" The one genuinely arbitrary bit is the exact-halfway case: a number ending in …5 is equally close to both neighbours, and the world agreed long ago to break that tie upwards so everyone gets the same answer.

Press play, then replay it — you get a fresh number each time. A two-digit decimal appears between its two tenths; we mark the halfway point, then snap the number to the nearer tenth — that is rounding to 1 decimal place.

The rounding staircase

Here is what rounding looks like if you do it to every number between 0 and 1 at once. The faint dashed line is each number left alone; the solid line is that number rounded to 1 d.p. Rounding turns the smooth ramp into a staircase: every number standing on the same stair snaps to the same tenth, and the jumps between stairs happen exactly at the halfway points — 0.05,\ 0.15,\ 0.25,\ \dots

Two things to spot. Each stair is a whole crowd of different numbers all telling the same rounded "lie" — 0.31, 0.28 and 0.345 all become 0.3. And a rounded number is never more than half a step from the truth — that's the guarantee that makes the lie an honest one.

Two traps

Only one digit gets a vote. Look at the one digit right after your cut — never the whole tail. Rounding 0.749 to 1 d.p. gives 0.7, because the neighbour digit is just 4. The 9 lurking after it does not get a vote — and it mustn't, because 0.749 really is closer to 0.7 than to 0.8 (check: 0.049 below versus 0.051 above).

Rounding up can carry. Rounding 2.96 to 1 d.p., the neighbour is 6, so the 9 must go up by one — and 9 + 1 rolls over, carrying into the units exactly like column addition:

And yes, you write 3.0, not 3 — the question asked for one decimal place, and that .0 is you answering it. (It also tells the reader something real: "accurate to the nearest tenth," which plain 3 doesn't promise.)

Significant figures: counting the digits that matter

Decimal places count from the decimal point. But that's a strange way to measure how precise a number is: 8{,}118{,}835{,}999 has zero decimal places, yet it's absurdly precise, while 0.5 has one decimal place and is quite rough. Significant figures (s.f.) fix this by counting from the first non-zero digit instead — the first digit that actually says something — and counting every digit from there on.

• 34{,}682 has 5 significant figures.
• 0.004718 has 4 — the leading zeros are just scaffolding showing how small the number is; the information starts at the 4.
• 405 has 3 — a zero sandwiched between non-zero digits is real information (it says "no tens"), so it counts.

Worked slowly: round 34{,}682 to 2 s.f. Start at the first non-zero digit: the 3 is significant figure number one, the 4 is number two. Cut there: 34\,|\,682. Ask the neighbour: it's a 6, which is 5 or more, so the 4 bumps up to 5. Now the crucial move — you cannot just write 35, because thirty-five is nowhere near thirty-four thousand. The dropped digits become zeros that hold the place value:

And on a small decimal: round 0.004718 to 2 s.f. Walk past the leading zeros — they don't count. The first significant figure is the 4, the second is the 7. Cut: 0.0047\,|\,18. The neighbour is a 1, which is 4 or less, so everything kept stays as it is and the tail is dropped:

Notice it's the same one rule as before — find the cut, ask the next digit, 5 or more rounds up. Significant figures never change the rule; they only change where the cut goes: you start counting at the first non-zero digit instead of at the decimal point.

Why scientists live by significant figures

Measure a leaf with a school ruler: 2.5 cm wide, 3.5 cm long. Multiply for the area and your calculator beams back 8.75\ \text{cm}^2. Write that down and you've just told a dishonest lie — the bad kind.

Here's why. A ruler reading of "2.5 cm" really means "somewhere between 2.45 and 2.55" — your eye can't do better than the nearest millimetre. Same for the 3.5. So the true area could be anywhere from 2.45 \times 3.45 \approx 8.45\ \text{cm}^2 up to 2.55 \times 3.55 \approx 9.05\ \text{cm}^2. Quoting 8.75 — three confident-looking figures — claims a precision your ruler never had. The calculator didn't lie; copying all its digits is the lie.

The fix is the scientists' rule of thumb: an answer is only as precise as the measurements that went into it. Two-significant-figure measurements in, two-significant-figure answer out:

This is why lab reports, engineering drawings and chemistry papers all quote significant figures — the number of digits is itself a message about how much to trust the number.

Want to see the number-line picture again with someone else's voice? Khan Academy rounds decimals to the nearest tenth here — watch for the same "find the cut, ask the neighbour" moves: