Standard form: writing numbers and calculating with them

What is standard form?

Standard form writes a number as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number. The power of 10 says how many places the digits move.

Ordinary number	Standard form
32,000	3.2 × 104
6,000,000	6 × 106
450	4.5 × 102
7	7 × 100
0.03	3 × 10−2
0.00071	7.1 × 10−4

A positive power means a big number, and a negative power means a number smaller than 1. 10⁰ is 1, so numbers from 1 to 10 have a power of 0. The laws of indices explain why 10⁻² is 0.01.

32 × 10³ is the right size, but it isn't standard form, because 32 is bigger than 10. Neither is 0.32 × 10⁵, because 0.32 is less than 1.

How to write a large number in standard form

To write a large number in standard form, put the decimal point after the first digit, then count how many places the digits have moved.

Worked example: 32,000

1. The first number has to be between 1 and 10, so it's 3.2.
2. To get from 3.2 back to 32,000, the digits move 4 places to the left on a place value chart. That's multiplying by 10 four times.
3. So 32,000 is 3.2 × 10⁴.

Checking the other way: 3.2 × 10⁴ = 32,000, because 3.2 × 10,000 = 32,000.

How to write a small number in standard form

For a number smaller than 1, the power is negative. Count how many places the digits move to get a number between 1 and 10.

Worked example: 0.00071

1. The first number between 1 and 10 is 7.1.
2. To get from 7.1 to 0.00071, the digits move 4 places to the right. That's dividing by 10 four times.
3. So 0.00071 is 7.1 × 10⁻⁴.

A quick check: the power should be negative, because 0.00071 is less than 1. The decimals guide covers place value after the decimal point.

Comparing numbers in standard form

To compare two numbers in standard form, look at the powers first. The bigger power means the bigger number, whatever the first part says. 4.1 × 10⁵ (410,000) is bigger than 9.8 × 10⁴ (98,000). Only when the powers are the same do you compare the first numbers: 6.2 × 10³ is bigger than 5.9 × 10³.

How to multiply standard form

To multiply in standard form, multiply the first numbers, then add the powers of 10.

Worked example: (3 × 10⁴) × (2 × 10⁵)

• Multiply the first numbers: 3 × 2 = 6.
• Add the powers: 10⁴ × 10⁵ is 10⁹, because 4 + 5 = 9.
• The answer is 6 × 10⁹.

Sometimes the first number comes out bigger than 10, and you need one more step. For (4 × 10³) × (5 × 10⁶), 4 × 5 = 20, which gives 20 × 10⁹. But 20 isn't between 1 and 10. 20 is 2 × 10, so the answer is 2 × 10¹⁰.

Small numbers work the same way: (2 × 10⁻³) × (4 × 10⁻²) is 8 × 10⁻⁵, because 2 × 4 = 8 and −3 + (−2) is −5.

How to divide standard form

Dividing standard form is the opposite: divide the first numbers, then subtract the powers of 10.

Worked example: (8 × 10⁷) ÷ (2 × 10³)

• Divide the first numbers: 8 ÷ 2 = 4.
• Subtract the powers: 10⁷ ÷ 10³ is 10⁴, because 7 − 3 = 4.
• The answer is 4 × 10⁴.

The same tidy-up step can be needed in division in standard form. For (3 × 10⁵) ÷ (6 × 10²), 3 ÷ 6 = 0.5, which gives 0.5 × 10³. But 0.5 is less than 1. 0.5 is 5 × 10⁻¹, so the answer is 5 × 10², which is 500.

How to add standard form

Adding standard form is easiest if you change the numbers back to ordinary numbers, add, then change the answer back. You can't just add the first numbers unless the powers are the same.

Worked example: 3.2 × 10⁴ + 5 × 10³

• 3.2 × 10⁴ is 32,000, and 5 × 10³ is 5,000.
• Add them: 32,000 + 5,000 = 37,000.
• Back into standard form: 3.7 × 10⁴.

Subtracting works the same way. 6.5 × 10⁵ take away 4 × 10⁴ is 650,000 − 40,000 = 610,000, which is 6.1 × 10⁵. The written methods are the same as column addition and column subtraction.

Common mistakes with standard form

• A first number that's too big or too small. 32 × 10³ and 0.32 × 10⁵ aren't standard form. The first number must be at least 1 and less than 10.
• Counting zeros instead of places. 3.2 × 10⁴ is 32,000, which has three zeros, not four. The power counts how far the digits move.
• The wrong sign on the power. Numbers less than 1 have a negative power.
• Multiplying the powers. When multiplying in standard form, add the powers of 10: 10⁴ × 10⁵ is 10⁹, not 10²⁰.
• Adding the powers when adding numbers. 3 × 10⁴ + 2 × 10⁴ is 5 × 10⁴. The power stays the same.
• Forgetting the tidy-up step after multiplying or dividing.

How to help at home

• Find big and small numbers. News stories, food labels and science books are full of millions and thousandths. Ask your child to write one in standard form.
• Practise multiplying and dividing by 10, 100 and 1,000. This is the place value skill underneath standard form, and it comes from primary school. The multiplying decimals and dividing decimals guides cover it.
• Ask "is that a big number or a small one?" before your child writes the power. It catches most sign mistakes.
• Ask the school about the order. Key stage 3 covers Years 7 to 9 and schools decide when each topic comes up. Their scheme of work or curriculum map will show when standard form is taught.

Where standard form fits

Standard form is in the key stage 3 programme of study, for Years 7 to 9: reading, writing and comparing numbers in standard form, with positive, negative and zero powers. Calculating with standard form is in the key stage 4 programme of study, for GCSE in Years 10 and 11, as content for all pupils. Rounding to significant figures often goes with it, because very large and very small answers are usually rounded. For the other topics in these years, see the KS3 maths guide.

Standard form questions to try

1. Write 7,400,000 in standard form.
2. Write 0.0062 in standard form.
3. Write 2.05 × 10³ as an ordinary number.
4. Which is bigger: 3.9 × 10⁶ or 8.1 × 10⁵?
5. Work out (2 × 10³) × (3 × 10⁴). Give your answer in standard form.
6. Work out (9 × 10⁸) ÷ (3 × 10²). Give your answer in standard form.

Answers

1. 7.4 × 10⁶
2. 6.2 × 10⁻³
3. 2,050
4. 3.9 × 10⁶, because it has the bigger power (3,900,000 against 810,000)
5. 6 × 10⁷ (2 × 3 = 6, and 3 + 4 = 7)
6. 3 × 10⁶ (9 ÷ 3 = 3, and 8 − 2 = 6)