Standard form writes a number as A × 10ⁿ, where A is at least 1 but less than 10 and n is a whole number. Large numbers have a positive power, so 53 000 = 5.3 × 10⁴, and numbers between 0 and 1 have a negative power, so 0.0014 = 1.4 × 10⁻³. To multiply or divide, deal with the numbers and the powers of 10 separately, then adjust so A is between 1 and 10.
Standard form
- A × 10ⁿ with 1 ≤ A < 10 and n an integer
- Large numbers: positive n; small numbers: negative n
- Multiplying: multiply the A values, add the powers
- Dividing: divide the A values, subtract the powers
- Adjust the answer so that 1 ≤ A < 10
What is standard form?
Standard form is a short way of writing very large or very small numbers. Every number is written as a value A, at least 1 and less than 10, multiplied by a power of 10. So 4500 = 4.5 × 10³ and 0.00062 = 6.2 × 10⁻⁴. Numbers such as 45 × 10² are not in standard form, because 45 is not less than 10.
Writing a number in standard form
Place the decimal point after the first non-zero digit, then count how many places the point has moved. For a large number the power is positive; for a number less than 1 it is negative. 53 000 becomes 5.3 with the point moved 4 places, giving 5.3 × 10⁴.
Converting back to an ordinary number
The power tells you how many places to move the decimal point. A positive power moves it to the right, making the number bigger: 3.6 × 10⁵ = 360 000. A negative power moves it to the left: 1.4 × 10⁻³ = 0.0014.
Ordering numbers in standard form
Compare the powers of 10 first: the bigger the power, the bigger the number. Only if the powers are equal do you compare the A values. So 3 × 10⁵ is bigger than 9 × 10⁴, even though 9 is bigger than 3.
Calculating in standard form
To multiply, multiply the A values and add the powers. To divide, divide the A values and subtract the powers. If the result for A is not between 1 and 10, adjust it: 15 × 10⁷ = 1.5 × 10⁸. For adding or subtracting, it is usually easiest to convert to ordinary numbers first.
Worked examples
Example 1. Write 53 000 in standard form.
- Place the point after the first digit: 5.3.
- The point has moved 4 places, and the number is large.
Answer: 5.3 × 10⁴.
Example 2. Write 1.4 × 10⁻³ as an ordinary number.
- The power is −3, so move the point 3 places left.
- 1.4 becomes 0.0014.
Answer: 0.0014.
Example 3. Write 0.000 72 in standard form.
- Place the point after the first non-zero digit: 7.2.
- The point has moved 4 places, and the number is small.
Answer: 7.2 × 10⁻⁴.
Example 4. Work out (5 × 10⁶) × (3 × 10²), giving your answer in standard form.
- Multiply the numbers: 5 × 3 = 15. Add the powers: 10⁸.
- 15 × 10⁸ = 1.5 × 10⁹.
Answer: 1.5 × 10⁹.
Example 5. Work out (8.4 × 10⁵) ÷ (2.1 × 10⁻²).
- Divide the numbers: 8.4 ÷ 2.1 = 4.
- Subtract the powers: 5 − (−2) = 7.
Answer: 4 × 10⁷.
Common mistakes
- Leaving A outside the range 1 to 10. 15 × 10⁸ must be written as 1.5 × 10⁹.
- Getting the sign of the power wrong. Numbers less than 1 have negative powers.
- Multiplying the powers. When multiplying, add the powers: 10⁶ × 10² = 10⁸.
- Ordering by the A values first. Compare the powers of 10 first.
- Miscounting places for small numbers. Count the moves to the first non-zero digit.
How standard form is examined
Standard form is tested at both tiers of Edexcel GCSE (1MA1) and in Edexcel International GCSE (4MA1), with and without a calculator. Questions ask you to:
- write a number in standard form, or convert back
- order numbers given in standard form
- multiply or divide numbers in standard form
- solve a real-world problem, such as distances in space or the mass of particles
Use the standard form worksheet, which has answers, and the GCSE number revision engine for practice.
Standard form relies on index laws and place value.
Frequently asked questions
What is standard form?
A number written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.
Why do small numbers have a negative power?
Because 10⁻ⁿ means 1 ÷ 10ⁿ, so a negative power makes the number smaller than 1.
How do you multiply numbers in standard form?
Multiply the A values, add the powers of 10, then adjust so that A is between 1 and 10.