How many digits should you write in a physics answer?

A number alone means nothing

Say a table is "2 long". 2 what? Metres? Hand-spans? Measuring means comparing a quantity with a reference standard called a unit. NCERT describes that standard as "arbitrarily chosen and internationally accepted". So every result is a number with a unit.

Some units are base units. Others, like the metre per second for speed, are derived by combining them. Older systems include CGS (centimetre, gram, second), FPS or the British system (foot, pound, second) and MKS (metre, kilogram, second). The system accepted internationally today is SI, short for the French name of the International System of Units.

SI has seven base units:

The base unit of mass is the kilogram, not the gram. With the mole, you must specify the entities, such as atoms or molecules. The book also gives two more units, the radian for plane angle and the steradian for solid angle. Both are dimensionless. A footnote says the long definitions in Table 1.1 "need not be remembered or asked in a test".

Where significant figures come from

Every measurement has some error, so a result must show how precise it was.

This example is our own. Lay a ruler with millimetre marks along a pencil. You can read 14.3 cm for certain. The end falls between two marks, so you estimate one more digit, say 6, and write 14.36 cm. The 1, 4 and 3 are reliable. The 6 is a careful guess. NCERT defines significant figures as the reliable digits plus the first uncertain digit, so 14.36 cm has four.

Five rules for counting

1. All non-zero digits are significant.
2. Zeros between non-zero digits are significant. 2.308 has four.
3. In a number less than 1, zeros before the first non-zero digit are not. 0.002308 still has four.
4. Trailing zeros in a number without a decimal point are not significant. 123 m = 12300 cm, and both have three.
5. Trailing zeros in a number with a decimal point are significant. 3.500 has four.

So 4.7 m has two significant figures and 4.700 m has four. The extra zeros show a more precise measurement.

Now the catch. Convert 4.700 m to 4700 mm and rule 4 seems to say two significant figures, but changing units can't change the precision, so it must still be four. The way out is scientific notation: 4.700 × 10³ mm. The power of ten doesn't count, and every digit in 4.700 does.

Calculating with significant figures

Multiplying or dividing: keep as many significant figures as the number with the fewest. Mass 4.237 g (four) divided by volume 2.51 cm³ (three) gives 1.688047… on a calculator. Keep three: 1.69 g cm⁻³.

Adding or subtracting: count decimal places instead. Adding 436.32 g + 227.2 g + 0.301 g gives 663.821 g. The number 227.2 has just one decimal place, so you report 663.8 g.

Rounding off

Round to three significant figures:

• 2.746 becomes 2.75. The dropped 6 is more than 5, so raise the digit before it.
• 1.743 becomes 1.74. The dropped 3 is less than 5, so leave it.

When the dropped digit is exactly 5, look at the digit before it. If that digit is even, drop the 5 and leave it alone: 2.745 becomes 2.74. If it's odd, raise it: 2.735 also becomes 2.74.

In a long calculation, carry one extra digit through the middle steps and round only at the end.

Check yourself

A 5.74 g sample occupies 1.2 cm³. Write its density with the correct number of significant figures. (Answer at the bottom.)

Questions students ask

Answer: 5.74 ÷ 1.2 = 4.783…, and 1.2 has only two significant figures, so the density is 4.8 g cm⁻³.