Standard form

Foundation and Higher

The idea

Standard form writes a number as $A \times 10^n$ , where $1 \le A < 10$ and $n$ is an integer: $45\,000 = 4.5 \times 10^4$ and $0.0032 = 3.2 \times 10^{-3}$ .

Place the decimal point after the first non-zero digit.
Count how many places it moved: big numbers give a positive power, small numbers a negative one.
To multiply, multiply the numbers and add the powers, then tidy so that $1 \le A < 10$ .

Watch out $32 \times 10^4$ is not standard form: rewrite it as $3.2 \times 10^5$ .

See it worked

Worked exampleNon-calculator3 marks

Answer each part without a calculator.

(a) Write $45\,000$ in standard form.

(b) Write $3.2 \times 10^{-3}$ as an ordinary number.

(c) Which is larger, $6 \times 10^5$ or $5.9 \times 10^6$ ?

(a) Place the decimal point after the first digit: $4.5$ . Moving it back to $45\,000$ multiplies by $10$ four times.
(b) $10^{-3}$ means divide by $1000$ : $3.2 \div 1000 = 0.0032$ .
(c) Compare the powers of 10 first: $10^6$ is bigger than $10^5$ , so $5.9 \times 10^6$ is larger ( $5\,900\,000 > 600\,000$ ).

Answer (a) $4.5 \times 10^4$ ; (b) $0.0032$ ; (c) $5.9 \times 10^6$

Your turn

Question 1Calculator allowed3 marks

The population of a country is $6.7 \times 10^7$ . Its area is $2.4 \times 10^5\ \text{km}^2$ .

(a) Work out the average number of people per $\text{km}^2$ . Give your answer to the nearest whole number.

(b) Write $6.7 \times 10^7$ as an ordinary number.

Question 2Non-calculator3 marks

The distance from the Earth to the Sun is about $1.5 \times 10^8$ km. Light travels about $3 \times 10^5$ km each second.

(a) Work out the time, in seconds, for light to travel from the Sun to the Earth.

(b) Write your answer to part (a) in standard form.

Question 3Calculator allowed3 marks

Work out ((3.6 $\times$ $10^{-4}$ ) $\times$ (8 $\times$ $10^{7}$ )) $\div$ (1.2 $\times$ $10^{2}$ ). Give your answer in standard form.

In past papers

106 official questions in 100 papers since 2017. The latest:

All GCSE Maths past papers

GCSE Maths Pro

£119 once

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.

Start free

No VAT is charged.