Significant Figures & Scientific Notation (SQA National 5 Physics): Revision Note

Significant figures

• Significant figures are the digits in a numerical value that show how precise a measurement is

• The significant figures in a measurement include all the certain digits and one uncertain digit

  • The uncertain digit is the last significant digit

• A pencil is measured using a ruler

• Its length is measured to be between 5.3 and 5.4 cm

• Therefore, it can be said that the length is 5.35 ± 0.05 cm

• The digits 5 and 3 in 5.3 are certain

  • The pencil is definitely at least 5.3 cm in length

• The last digit is uncertain

  • It is an estimate

  • The pencil is somewhere between 5.3 cm and 5.4 cm

  • It is estimated to be 5.35 cm

• Therefore, the measurement is given to 3 significant figures (3 s.f.)

Using significant figures

• When calculating using measurement values, the final answer can only be given to the same precision as the least precise input value

• This means that the final answer can have no more significant figures than the value with the least number of significant figures used in the calculation

Which digits are significant?

• Non-zero digits are always significant

  • 123 is 3 s.f.

  • 1.78 is 3 s.f.

• Any zeros between two significant digits are significant

  • 108 is 3 s.f.

  • 10003 is 5 s.f.

  • 1.006 is 4 s.f.

• Only a final zero or trailing zeros in the decimal portion (after the decimal point) are significant

  • 0.183 is 3 s.f. (the zero is before the decimal, so it is not significant)

    • 1, 8, and 3 are the significant figures

  • 1390 is 3 s.f. (the final zero is not after a decimal point, so it is not significant)

    • 1, 3, and 9 are the significant figures

  • 1.40 is 3 s.f. (the final zero is after the decimal point, so it is significant)

    • 1, 4, and 0 are all the significant figures

  • 0.012 is 2 s.f. (the zeros are either before the decimal point or are not the final zero, so not significant)

    • 1 and 2 are the significant figures

  • 1.9000 is 5 s.f (the trailing zeros are after the decimal point - so is significant)

    • 1, 9, 0, 0 and 0 are all the significant figures

Rules for rounding

• When rounding to a certain number of significant figures, use the following procedure:

1. Find the number of significant figures to round to

2. Go to the digit for this significant figure

3. Look at the value after this digit

• If the value is 5 or greater, round this significant digit up

• If the value is less than 5, leave this significant digit as it is

• Examples:

  • The value 7.8 is 2 s.f

    • To 1 s.f this is equal to 8

  • The value 9.12 is 3 s.f

    • To 2 s.f this is equal to 9.1

  • The value 3.65 × 10^-4 is equal to 3 s.f

    • To 2 s.f this is equal to 3.7 × 10^-4

  • The value 1020 is equal to 3 s.f

    • To 2 s.f this is equal to 1000

Scientific notation

• Standard form is a system of writing large and small numbers which is useful for working with very large or very small numbers

  • This also means writing whole lines of zeros can be avoided

• Numbers in standard form are in written as:

a × 10ⁿ

• They follow these rules:

  • a is a number between 1 and 10

  • n > 0 for large numbers

    • i.e how many times a is multiplied by 10

  • n < 0 for small numbers

    • i.e how many times a is divided by 10

• For example:

  • 3 × 10⁸ = 300 000 000 (3 multiplied by 10, 8 times)

  • 2 × 10^-5 = 0.00002 (2 divided by 10, 5 times)

• When rounding a number in standard form to a certain number of significant figures, only the value of a is rounded (the × 10ⁿ value will not be significant)

  • For example, 5.18 × 10⁶ to 2 s.f. is 5.2 × 10⁶