Standard form is a way of writing very large or very small numbers as a number between 1 and 10 multiplied by a power of 10. It is used across science, engineering, and maths to make extreme values manageable and to compare them easily.
What is standard form?
A number is in standard form (also called standard index form) when it is written as:
A × 10ⁿ
where 1 ≤ A < 10 and n is an integer (a positive or negative whole number, or zero).
The number A is called the coefficient. The power n tells you how many places the decimal point moves:
- A positive n means the original number is large (decimal point moves right).
- A negative n means the original number is small (decimal point moves left).
How to convert a large number to standard form
Step 1 — Write the number with only the significant digits, placing the decimal point after the first non-zero digit
Step 2 — Count how many places the decimal point has moved from its original position. That count is the positive power n
Step 3 — Write the number as A × 10ⁿ
Worked example 1: convert 340 000 to standard form
The first non-zero digit is 3. Write 3.4.
The decimal point started after the final zero and has moved 5 places to the left.
So n = 5.
Answer: 3.4 × 10⁵
Check: 3.4 × 10⁵ = 3.4 × 100 000 = 340 000 ✓
Worked example 2: convert 7 200 000 000 to standard form
First non-zero digit is 7. Write 7.2.
The decimal point moves 9 places to the left.
Answer: 7.2 × 10⁹
This is approximately the population of the world — a typical science application of standard form.
How to convert a small number to standard form
For small decimals, the power is negative because the decimal point moves to the right.
Worked example 3: convert 0.000 045 to standard form
First non-zero digit is 4. Write 4.5.
Count the places from the original decimal point to the new one: 0.000 045 → the point moves 5 places to the right.
Answer: 4.5 × 10⁻⁵
Check: 4.5 × 10⁻⁵ = 4.5 ÷ 100 000 = 0.000 045 ✓
Worked example 4: convert 0.003 07 to standard form
First non-zero digit is 3. Write 3.07 (include the zero between 3 and 7 — it is significant).
The point moves 3 places to the right: n = −3.
Answer: 3.07 × 10⁻³
How to convert from standard form back to an ordinary number
Reverse the process: multiply by the power of 10.
- Positive power: move the decimal point right by n places (or add zeros).
- Negative power: move the decimal point left by |n| places (or add leading zeros).
Worked example 5: convert 6.02 × 10⁴ to an ordinary number
Move the decimal point 4 places right: 6.02 → 60 200
Answer: 60 200
Worked example 6: convert 8.5 × 10⁻³ to an ordinary number
Move the decimal point 3 places left: 8.5 → 0.0085
Answer: 0.0085
Multiplying numbers in standard form
To multiply two numbers in standard form:
- Multiply the A values.
- Add the powers of 10.
- Adjust if the A value falls outside 1 ≤ A < 10.
Worked example 7
Calculate (3 × 10⁴) × (2 × 10³).
Multiply coefficients: 3 × 2 = 6
Add powers: 4 + 3 = 7
Result: 6 × 10⁷ — already valid (6 is between 1 and 10).
Answer: 6 × 10⁷
Worked example 8
Calculate (5 × 10⁶) × (4 × 10²).
Multiply coefficients: 5 × 4 = 20
Add powers: 6 + 2 = 8
Result: 20 × 10⁸ — but 20 is not between 1 and 10. Write 20 = 2 × 10¹.
Adjust: 2 × 10¹ × 10⁸ = 2 × 10⁹
Answer: 2 × 10⁹
Dividing numbers in standard form
To divide two numbers in standard form:
- Divide the A values.
- Subtract the powers of 10.
- Adjust if necessary.
Worked example 9
Calculate (8.4 × 10⁷) ÷ (2 × 10³).
Divide coefficients: 8.4 ÷ 2 = 4.2
Subtract powers: 7 − 3 = 4
Answer: 4.2 × 10⁴
Ordering numbers written in standard form
To put numbers in standard form in order:
- Compare the powers first — a higher power means a larger number.
- If powers are equal, compare the A values.
Worked example 10: arrange in ascending order
5 × 10³, 2.1 × 10⁵, 8 × 10³, 3 × 10⁴
Powers: 10³, 10⁵, 10³, 10⁴. Sorted by power: 10³ (two of them), 10⁴, 10⁵.
For the two 10³ numbers: 5 × 10³ < 8 × 10³.
Ascending order: 5 × 10³, 8 × 10³, 3 × 10⁴, 2.1 × 10⁵
Summary table
| Original number | Standard form | Notes |
|---|---|---|
| 340 000 | 3.4 × 10⁵ | Large, positive power |
| 7 200 000 000 | 7.2 × 10⁹ | Very large |
| 0.000 045 | 4.5 × 10⁻⁵ | Small, negative power |
| 0.003 07 | 3.07 × 10⁻³ | Small, include significant zero |
Common mistakes to avoid
Mistake 1 — A value outside the 1–10 range.
34 × 10⁴ is not standard form. Convert 34 to 3.4 × 10¹ and adjust: 3.4 × 10⁵.
Mistake 2 — Wrong sign on the power.
0.0005 = 5 × 10⁻⁴, NOT 5 × 10⁴. A small number needs a negative power.
Mistake 3 — Adding powers when multiplying A values.
Keep operations separate: multiply the A values and add the powers.
Mistake 4 — Losing significant figures.
3.07 × 10⁻³ — the zero between 3 and 7 is significant and must be kept.
How standard form fits the KS3 national curriculum
The Department for Education's KS3 mathematics programme of study requires pupils to "interpret and compare numbers in standard form A × 10ⁿ, 1 ≤ A < 10, where n is a positive or negative integer or zero." BBC Bitesize's KS3 maths resources position standard form as a Year 9 topic that links directly to powers and indices work, and it recurs throughout GCSE Physics and Chemistry when pupils deal with atomic masses, astronomical distances, and nanometre-scale measurements.
Frequently asked questions
Why is standard form used in science?
Scientific measurements often involve extremely large numbers (the distance from Earth to the Sun is approximately 1.5 × 10¹¹ metres) or extremely small numbers (a hydrogen atom has a radius of about 5.3 × 10⁻¹¹ metres). Standard form lets scientists write these values concisely, compare them directly, and perform calculations without counting dozens of zeros.
How do I add or subtract numbers in standard form?
You need to convert both numbers to the same power of 10 first. For example, to add 3 × 10⁴ + 5 × 10³, rewrite the second as 0.5 × 10⁴, then add: 3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴. At KS3 you are primarily expected to multiply, divide, and order numbers in standard form rather than add or subtract them.
What does a power of zero mean in standard form?
10⁰ = 1, so a number in standard form with n = 0 is just the coefficient A itself. For example, 3.7 × 10⁰ = 3.7. This is a valid standard form for any number between 1 and 10.
Is standard form the same as scientific notation?
Yes. "Standard form" is the term used in UK schools at KS3 and GCSE. "Scientific notation" is the equivalent term used in the United States, in many scientific journals, and on some calculators (shown as "SCI" mode). The format A × 10ⁿ is identical in both cases.
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