Standard form, explained with examples
Standard form is a short way to write very large or very small numbers, such as the distance to the Sun or the size of a virus. It is used in GCSE Maths and throughout science.
On this page
- What standard form looks like
- Why scientists use standard form
- Writing large numbers in standard form
- Writing small numbers in standard form
- Converting back to ordinary numbers
- Multiplying and dividing
- Adding and subtracting
- Using a calculator
- Standard form in context
- Common mistakes
- Practice questions
- Frequently asked questions
- Key takeaways
- Related guides
What standard form looks like
A number in standard form is written as A × 10n, where
- 1 ≤ A < 10: A is at least 1 but less than 10, and
- n is an integer: a whole number, which can be positive, negative or zero.
So 3.2 × 104 is in standard form, but 32 × 103 and 0.32 × 105 are not, even though they all equal 32,000. Exam questions often check whether you know this rule.
A positive power of 10 means a large number. A negative power means a small number between 0 and 1. The power tells you how many places the decimal point moves.
Why scientists use standard form
Writing 0.000 000 000 000 000 000 16 for the charge on an electron in coulombs is slow and easy to miscopy. In standard form it is 1.6 × 10−19. Standard form makes very large and very small numbers easy to read, compare and calculate with, which is why it appears throughout physics, chemistry and biology as well as maths.
Writing large numbers in standard form
Move the decimal point to the left until you have a number between 1 and 10. The number of places you moved it is the power of 10.
Example 1: a large number
Write 45,000 in standard form.
- Move the decimal point 4 places left to get 4.5.
- 45,000 = 4.5 × 104.
Example 2: a very large number
The Sun is about 150,000,000 km from Earth. Write this in standard form.
- Move the point 8 places: 1.5.
- 1.5 × 108 km.
Writing small numbers in standard form
For numbers less than 1, move the decimal point to the right until you reach a number between 1 and 10. The power is negative, and its size is the number of places moved.
Example 3: a small number
Write 0.0036 in standard form.
- Move the point 3 places right to get 3.6.
- 0.0036 = 3.6 × 10−3.
Example 4: a very small number
A virus is 0.000 000 12 m across. Write this in standard form.
- Move the point 7 places right: 1.2.
- 1.2 × 10−7 m.
Converting back to ordinary numbers
Reverse the process. A positive power moves the decimal point right, making the number bigger. A negative power moves it left, making it smaller. Fill any gaps with zeros.
Example 5: converting back
Write 7.08 × 105 and 2.5 × 10−4 as ordinary numbers.
- 7.08 × 105: move the point 5 places right: 708,000.
- 2.5 × 10−4: move it 4 places left: 0.00025.
Multiplying and dividing
Use the laws of indices. Multiply (or divide) the number parts, and add (or subtract) the powers. Then check that the answer is still in standard form, adjusting if A is no longer between 1 and 10.
Example 6: multiplying
Work out (3 × 104) × (2 × 105) without a calculator.
- 3 × 2 = 6 and 104 × 105 = 109.
- Answer: 6 × 109.
Example 7: multiplying and adjusting
Work out (4 × 103) × (5 × 106).
- 4 × 5 = 20 and 103 × 106 = 109, giving 20 × 109.
- 20 is not between 1 and 10, so write it as 2 × 101: the answer is 2 × 1010.
Example 8: dividing
Work out (8 × 107) ÷ (4 × 102).
- 8 ÷ 4 = 2 and 107 ÷ 102 = 105.
- Answer: 2 × 105.
Example 9: dividing with adjustment
Work out (3 × 105) ÷ (6 × 10−2).
- 3 ÷ 6 = 0.5 and 105 ÷ 10−2 = 107, giving 0.5 × 107.
- 0.5 = 5 × 10−1, so the answer is 5 × 106.
Adding and subtracting
You cannot simply add the powers when adding. The safest method is to convert both numbers to ordinary form, add or subtract, then convert back. Alternatively, rewrite both numbers with the same power of 10 first.
Example 10: adding
Work out 3.2 × 104 + 5 × 103.
- Convert: 32,000 + 5,000 = 37,000.
- Answer: 3.7 × 104.
Using a calculator
Most scientific calculators have a ×10x or EXP button for entering standard form. To enter 3.2 × 104, type 3.2, press ×10x, then type 4. Avoid typing × 10 ^ 4 with brackets missing, which can give wrong answers in divisions. If your calculator shows a long answer, convert it to standard form yourself, rounding A as the question asks.
Standard form in context
Exam questions often set standard form in real situations: the mass of planets, populations of countries, sizes of cells, or the speed of light (3 × 108 m/s). Read carefully for units, and check whether the question wants the answer in standard form or as an ordinary number. Comparing numbers is easy in standard form: compare the powers first, and only if the powers are equal compare the A values.
Common mistakes
- Writing A outside the range 1 to 10, such as 45 × 103.
- Getting the sign of the power wrong for small numbers.
- Adding the powers when adding numbers, instead of when multiplying.
- Forgetting to adjust the answer after multiplying or dividing.
Practice questions
Try each one before opening the answer.
Q1. Write 6,300,000 in standard form.
Show answer
Q2. Write 0.00058 in standard form.
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Q3. Work out (6 × 104) × (3 × 10−2).
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Q4. Work out (9 × 108) ÷ (3 × 105).
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Q5. Which is larger: 4.1 × 10−3 or 9.8 × 10−4?
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Q6. Work out 6 × 105 − 4 × 104.
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Frequently asked questions
Is standard form on the Foundation paper?
Yes. Standard form is on both Foundation and Higher tiers, and calculations without a calculator are common.
Is standard form the same as scientific notation?
Yes. In the US and in many science courses it is called scientific notation. The rules are the same.
Key takeaways
- A × 10n with 1 ≤ A < 10 and n an integer.
- Large numbers have positive powers; small numbers have negative powers.
- Multiply: multiply the numbers, add the powers.
- Divide: divide the numbers, subtract the powers.
- Add or subtract by converting to ordinary numbers first.