Significant figures.

Step 1 of 5

The measured digits

Significant figures are the digits in a number that the measurement actually settled. The bar on the board ends exactly on the 12 cm mark, and the ruler reads to millimetres. So you write 12.0 cm: the zero after the point is not decoration. It says the millimetres were checked and there was a zero there.

This is how you count significant figures:

• Every digit that is not zero counts.
• A zero between digits counts: $305$⁠ has three significant figures.
• Zeros at the start of a number do not count: they only show where the point is.
• Zeros at the end count when the number has a decimal point.

Step 2 of 5

Zeros at the start

The same measurement in metres is 0.120 m. Changing the unit moved the decimal point and put a zero at the start, but it did not change what was measured.

Step 3 of 5

Scientific notation

In a whole number like $1200$⁠ the zeros at the end are unclear: perhaps it was measured to the metre, perhaps only to the nearest hundred and the zeros just hold places. Scientific notation leaves no doubt, because every digit before the $10$⁠ was measured: $1.2 \times 10^3$⁠ has two significant figures, and $1.200 \times 10^3$⁠ has four.

Step 4 of 5

Multiplying

A card was measured with the millimetre ruler: 7.3 cm by 4.6 cm. On the board the calculation gives an area of $33.58$⁠ square centimetres, but four digits is more than was measured.

The rule for multiplying and dividing: the result has as many significant figures as the number with the fewest. The weakest link decides.

Step 5 of 5

Adding

For adding and subtracting the rule is different: the result has as many decimal places as the number with the fewest decimal places. What matters here is which place was measured, not how many digits.

A stack of books 12.3 cm tall was measured with a ruler. A card 0.27 cm thick, measured with a finer instrument, is laid on top.