Rounding to Significant Figures

The leading zeros (0.00) are not significant — they are just placeholders showing how small the number is. The first significant figure is 3 and the second is 7. To round to 2 s.f., look at the third significant figure: 2. Since 2 < 5, we round down, giving 0.0037.

A common mistake is to count the zeros and say the answer is 0.00 (confusing significant figures with decimal places) or to include the leading zeros in the count and think the first two significant figures are 0 and 0. The key principle: start counting significant figures from the first non-zero digit.

The first two significant figures of 4372 are 4 and 3. The next digit is 7 (≥ 5), so we round up: 43 becomes 44. But we must keep the number in the thousands — we need placeholder zeros to preserve its size. The answer is 4400, not 44.

Think of it on a number line: 4372 sits between 4300 and 4400. Since 4372 is closer to 4400, that’s where it rounds. Writing “44” would place us between 40 and 50 — a completely different part of the number line.

In 3.50, the digits 3, 5, and 0 are all significant. The trailing zero after the decimal point counts because it was deliberately included to show precision — it tells us the measurement is accurate to the nearest hundredth, not just the nearest tenth.

Compare 3.5 and 3.50: they have the same value, but 3.5 has 2 s.f. while 3.50 has 3 s.f. The difference matters in science and measurement: 3.50 says “we measured carefully enough to know the hundredths digit, and it was 0.” That trailing zero carries information about precision.

The first significant figure is 9 and the second is 9. The third significant figure is 6 (≥ 5), so we round up. But rounding 99 up gives 100 — the 9 in the second position becomes 10, which carries over. So 0.099… becomes 0.10.

The trailing zero in 0.10 is important — it shows the answer has been rounded to 2 significant figures. Writing just 0.1 would suggest only 1 significant figure. This is a case where rounding pushes a number across a “boundary” — from the thousandths into the tenths.

Example: 4.72

Another: 0.00305

Creative: 10.0 (trailing zero counts) or Standard Form: \(1.20 \times 10^4\) (the coefficient 1.20 has 3 s.f.).

Trap: 0.52 — a student might count “0, 5, 2” and think there are 3 significant figures. But the leading zero is not significant, so 0.52 has only 2 s.f.

Example: 4.72 and 4.68

Another: 4.74 and 4.66

Creative: 4.7049 and 4.6501 — pushing to opposite ends of the range that rounds to 4.7.

Trap: 4.75 and 4.65 — a student might pair these up symmetrically, but 4.75 to 2 s.f. gives 4.8 (the third s.f. is 5, which rounds up), not 4.7. Meanwhile 4.65 does correctly round to 4.7.

Example: 1490 → 1000 (change of 490)

Another: 5600 → 6000 (change of 400)

Creative: 14 999 → 10 000 (change of 4999) — the bigger the number and the closer the leading digits are to a boundary, the larger the possible change.

Trap: 5040 → 5000 (change of only 40). A student might think “it’s a big number so rounding must change it by a lot” but the change here is only 40, well under 100.

Example: 960 → 1000 (jumps from hundreds to thousands)

Another: 0.0097 → 0.01 (jumps from thousandths to hundredths)

Creative: 0.9996 → 1 — crosses from below 1 to exactly 1, jumping from the tenths into the units.

Trap: 470 → 500. A student might think “it jumped from 470 to 500 — that’s a big change!” but both numbers are still in the hundreds. No change in power of ten.

It’s true that 3.42 rounded to 2 s.f. gives 3.4, which is smaller. But 3.67 rounded to 2 s.f. gives 3.7, which is larger than the original.

Rounding can make a number larger (when rounding up), smaller (when rounding down), or leave it unchanged (when the dropped digits are all zeros, as in 3.40 → 3.4). The direction depends on the digit being examined, not on the act of rounding itself.

Leading zeros in decimals like 0.003 or 0.0742 are never significant. They are placeholders that show position — how far right of the decimal point the significant digits are. The number 0.003 has just 1 significant figure (the 3), not 4.

Students sometimes think more digits means more significant figures, but leading zeros exist only to indicate scale, not precision. You can verify this using standard form: 0.003 = 3 × 10−3. Converting to standard form strips away leading zeros entirely, confirming they carry no information about measurement accuracy.

By definition, 1 significant figure means keeping just one significant digit. Since significant figures start from the first non-zero digit, the result always has exactly one non-zero digit, plus any placeholder zeros needed to preserve magnitude.

Examples: 47 → 50 (one non-zero digit: 5), 0.0083 → 0.008 (one non-zero digit: 8), 6.7 → 7 (one non-zero digit: 7), 3200 → 3000 (one non-zero digit: 3). The placeholder zeros in results like 50 or 3000 are not non-zero digits — they simply hold the number’s position on the number line.

It works in many cases: 4.32 and 4.37 both round to 4.3 (2 s.f.) and both round to 4 (1 s.f.).

But consider 0.0348 and 0.0351. Both round to 0.035 to 2 s.f., but to 1 s.f., 0.0348 rounds to 0.03 while 0.0351 rounds to 0.04 — they go in different directions! This happens whenever two numbers straddle a 1 s.f. rounding boundary while agreeing at 2 s.f.

The student has confused significant figures with decimal places. They counted 3 digits after the decimal point (giving 3 d.p., not 3 s.f.). These are different things: “0.047” is 0.04736 rounded to 3 decimal places, and it only has 2 significant figures.

The first 3 significant figures of 0.04736 are 4, 7, 3 (leading zeros don’t count). The fourth significant figure is 6 (≥ 5), so we round up: the correct answer is 0.0474.

The student is confusing integer rules with decimal rules. In integers (like 500), trailing zeros are indeed often ambiguous or not significant. But in decimals (like 0.050), trailing zeros are always significant.

Why write the zero if it doesn’t add value? It’s there to show precision. The ‘5’ and the ‘0’ are both significant, so the answer is 2 significant figures.

The student has used cascade rounding (also called “double rounding” or “creeping rounding”) — rounding digit by digit from right to left. Each intermediate rounding changes the next digit, creating a chain reaction that inflates the final answer. This is one of the most dangerous rounding misconceptions because it looks systematic and careful.

The correct method: look only at the digit immediately after the rounding position. To round 2.449 to 1 s.f., the first significant figure is 2 and the next digit is 4. Since 4 < 5, round down. The correct answer is 2, not 3. Never round in stages — always go straight from the original number.

The student counted a leading zero as a significant figure. They treated the “0” in the hundredths column as the first significant figure and the “5” as the second. But leading zeros are never significant — they are just placeholders.

The first significant figure is 5 and the second is 2. Looking at the next digit (9 ≥ 5), we round up: the correct answer is 0.053.