Number: Standard Form and Rounding
Standard form is a way of writing very large or very small numbers as a value between 1 and 10 multiplied by a power of 10, such as 4.8 x 10^5. Rounding gives an approximate value to a stated degree of accuracy, for example the nearest 10, a number of decimal places, or a number of significant figures. Both skills are used throughout KS3 and GCSE maths for estimating and comparing measurements.
Before you start
Make sure you're comfortable with these topics first:
Method
- To round to a given place value, look at the next digit along; if it is 5 or more, round the previous digit up, otherwise leave it, then replace the remaining digits with zeros (for whole numbers) or remove them (for decimals).
- To round to a number of significant figures, count from the first non-zero digit and apply the same 5-or-more rule, keeping placeholder zeros so the number keeps its correct size.
- To write a number in standard form, place the decimal point after the first non-zero digit so the front number is between 1 and 10, then count how many places the point moved to find the power of 10.
- For numbers of 10 or more the power of 10 is positive; for numbers between 0 and 1 the power of 10 is negative.
- To multiply or divide numbers in standard form, multiply or divide the front numbers and add or subtract the powers of 10, then adjust back into standard form if the front number is no longer between 1 and 10.
- To add or subtract numbers in standard form, first convert both numbers to the same power of 10 (or to ordinary numbers), combine them, then convert the answer back to standard form.
Worked example
Write 0.00073 in standard form. Then work out (2 x 10^4) x (3.5 x 10^3), giving your answer in standard form.
- For 0.00073, place the decimal point after the first non-zero digit (7) to get the front number 7.3.
- Count how many places the point moved from 0.00073 to 7.3: it moved 4 places to the right, so the power of 10 is -4, giving 7.3 x 10^-4.
- For (2 x 10^4) x (3.5 x 10^3), multiply the front numbers: 2 x 3.5 = 7.
- Add the powers of 10: 4 + 3 = 7, giving 7 x 10^7.
- Check the front number 7 is between 1 and 10, so 7 x 10^7 is already in standard form.
Practice questions
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Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
The mass of a proton is approximately 1.673 x 10^-27 kg. A hydrogen atom contains 1 proton and 1 electron. The mass of an electron is approximately 9.11 x 10^-31 kg. Work out the total mass of one hydrogen atom, giving your answer in standard form.
A grain of sand has a diameter of about 5.8 x 10^-4 metres. Round this to 1 significant figure, then use it to estimate the total length, in metres, of 50000 grains of sand placed end to end in a line. Give your answer in standard form.
A number, when rounded to the nearest 1000, is 8000. Assuming the number is a positive whole number, (a) write down the smallest number it could have been, and (b) write down the largest number it could have been.
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