Significant Figures
Every digit you write is a promise. Keep only the ones you can keep.
Lesson at a glanceShow the key points
- Record every digit you can read plus exactly one estimated (doubtful) digit; the finest scale division is the least count.
- Leading zeros are never significant; trailing zeros in a whole number are ambiguous unless written in powers of ten.
- Multiplication/division: keep as many significant digits as the least precise input.
- Addition/subtraction: round every number to the leftmost doubtful decimal place, then add.
- When the dropped part is exactly 5, round so that the last kept digit is even.
A measurement is a claim about the world. The digits you write tell the reader how precise that claim is, so writing too many is quietly dishonest.
Certain digits and one doubtful digit
Every measurement is read off some calibrated scale: a ruler, a stopwatch, the needle of an ammeter. You write down every digit the markings give you directly, then one more digit estimated by eye between the finest markings. The finest division of the scale is its least count.
On an ordinary metre rule the numbered marks are a centimetre apart, with ten small divisions in between, so the least count is 1 mm. Suppose an object’s edge falls between the 10.4 cm and 10.5 cm marks. Mentally slice that millimetre into tenths, judge where the edge sits, and you might record 10.46 cm. The 1, 0 and 4 are certain; the 6 is a careful guess.
All of these digits, the certain ones plus the single doubtful one, are called significant digits. The reading 10.46 cm has four. The rightmost (doubtful) digit is the least significant digit; the leftmost is the most significant digit.
The trouble with zeros
Zeros are slippery. If someone reports a length as 600 mm and says nothing about the instrument, are those zeros measured or just placeholders?
- With a scale marked only every metre, the 6 is already the eye estimate. Only one digit is significant.
- With centimetre markings, “60” was read and the last zero is a placeholder: two significant digits.
- With millimetre markings, all three digits 6, 0, 0 are significant.
Scientific notation removes the guesswork: write mm, mm or mm to state one, two or three significant digits. And when a number is less than 1, the zeros right after the decimal point are never significant. They just position the decimal point. So 0.0023 has two significant digits while 1.0023 has five.
Significant digits in calculations
Say body A has a measured mass of 12.0 kg and body B 7.0 kg. A calculator says the ratio is 1.714285… But the last zero of both measurements was already doubtful, so most of those decimals are fiction. Two rules decide how much to keep.
Rounding off
When dropping digits, look at the digit right after the last one you keep. If it is more than 5, round up. If less than 5, leave it. If it is exactly 5 (with nothing but zeros after it), round so the kept digit becomes even: bump an odd digit up, leave an even one. Over many calculations this keeps rounding from drifting systematically upward. Digits dropped to the left of the decimal point are replaced with zeros.
The next digit is exactly 5. The digit being rounded (6) is even, so it stays.
Round (a) 27384, (b) 18.635, (c) 18.650 and (d) 18.750 to three significant digits.
Evaluate , where every digit shown is significant.
Evaluate 31.47 + 0.0518 + 182.3.
- 1Record all certain digits plus exactly one estimated digit.
- 2Leading zeros never count; trailing zeros in whole numbers are ambiguous, so use powers of ten.
- 3× and ÷: keep as many significant digits as the least precise input.
- 4+ and −: round to the leftmost doubtful decimal column first, then add.
- 5Exact 5: round to make the kept digit even.
Check yourself
Keep exploring
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