IGCSE Maths · Number · Core & Extended

Standard Form: Writing Very Large and Very Small Numbers

Standard form is where the index laws you’ve just learned earn their keep. It’s a compact, exact way of writing numbers so big or so small that writing them out in full would be clumsy and error-prone. Almost everything here applies equally to Core and Extended — the one real difference is noted where it comes up.

Why this matters The distance from Earth to the Sun is about 150,000,000,000 metres. The mass of a single water molecule is about 0.0000000000000000000000299 grams. Try reading either of those aloud without losing count of the zeros. Scientists, engineers, and astronomers don’t write numbers like this out in full — they use standard form, which turns both of those into something you can actually read and compare at a glance: 1.5 × 10¹¹ m, and 2.99 × 10⁻²³ g.

1. What Is Standard Form?

A number is in standard form when it’s written as:

A × 10ⁿ, where 1 ⩽ A < 10 and n is an integer

The letter A must be a number between 1 and 10 (it can include decimals, like 3.6, but never 10 or more, and never less than 1). The power n tells you how many places to shift the decimal point — a positive n for large numbers, a negative n for small numbers.

Is it in standard form? Check each one 45 × 10³ — No (45 is not between 1 and 10)
4.5 × 10³ — Yes
0.45 × 10⁴ — No (0.45 is less than 1)
7 × 10⁻² — Yes (A can be a whole number, as long as it’s 1–9)
Try it — Warm-up
  1. Is 3.2 × 10⁵ in standard form?
  2. Is 62 × 10⁴ in standard form?
  3. Is 0.9 × 10⁻³ in standard form?
  4. Is 8 × 10⁷ in standard form?
Show answers
  1. Yes
  2. No — 62 is not between 1 and 10.
  3. No — 0.9 is less than 1.
  4. Yes

2. Converting Large Numbers Into Standard Form

Count how many places the decimal point moves to land just after the first non-zero digit — that count becomes a positive power of 10.

Worked example Write churches 68,000,000 in standard form.
The decimal point starts after the last 0. Move it so only one digit (6) is in front: 6.8
Count the places moved: 7 places.
So 68,000,000 = 6.8 × 10⁷
Worked example Write 425,000 in standard form.
4.25, moved 5 places → 4.25 × 10⁵
Try it — Large Numbers
  1. Write 7,200,000 in standard form.
  2. Write 93,000 in standard form.
  3. Write 500,000,000 in standard form.
  4. Write 6,050,000 in standard form.
Show answers
  1. 7.2 × 10⁶
  2. 9.3 × 10⁴
  3. 5 × 10⁸
  4. 6.05 × 10⁶

3. Converting Small Numbers Into Standard Form

For numbers less than 1, the decimal point moves the other way — count the places to land just after the first non-zero digit, and that count becomes a negative power of 10.

Worked example Write 0.00047 in standard form.
Move the decimal point until only one non-zero digit is in front: 4.7
Count how many places you moved it: 4 places.
So 0.00047 = 4.7 × 10⁻⁴
Worked example Write 0.0000039 in standard form.
3.9, moved 6 places → 3.9 × 10⁻⁶
Try it — Small Numbers
  1. Write 0.006 in standard form.
  2. Write 0.000081 in standard form.
  3. Write 0.00000205 in standard form.
  4. Write 0.5 in standard form.
Show answers
  1. 6 × 10⁻³
  2. 8.1 × 10⁻⁵
  3. 2.05 × 10⁻⁶
  4. 5 × 10⁻¹

4. Converting Out of Standard Form

To go back to an ordinary number, do the reverse: move the decimal point the number of places the power tells you — right for a positive power, left for a negative power.

Worked examples Write 3.6 × 10⁵ as an ordinary number.
Move the decimal point 5 places right (adding zeros as needed): 360,000

Write 5.14 × 10⁻³ as an ordinary number.
Move the decimal point 3 places left: 0.00514
Try it — Back to Ordinary Numbers
  1. Write 2.9 × 10⁴ as an ordinary number.
  2. Write 7 × 10⁶ as an ordinary number.
  3. Write 8.13 × 10⁻² as an ordinary number.
  4. Write 6 × 10⁻⁵ as an ordinary number.
Show answers
  1. 29,000
  2. 7,000,000
  3. 0.0813
  4. 0.00006

5. Comparing and Ordering Numbers in Standard Form

The power of 10 tells you the size of a number at a glance — a bigger power always means a bigger number (for positive powers), so you very rarely need to convert back to ordinary numbers just to compare.

Worked example Order from smallest to largest: 3 × 10⁴, 2.5 × 10⁻², 6 × 10⁴, 9 × 10⁻³.
Compare the powers first: 10⁻² is bigger than 10⁻³, and both are far smaller than 10⁴.
Order: 9 × 10⁻³, 2.5 × 10⁻², 3 × 10⁴, 6 × 10⁴
Try it — Comparing
  1. Which is larger: 4 × 10⁵ or 3.9 × 10⁶?
  2. Which is smaller: 5 × 10⁻⁴ or 5 × 10⁻⁶?
  3. Order from smallest to largest: 6 × 10³, 6 × 10⁻³, 6 × 10⁰, 6 × 10⁻¹.
Show answers
  1. 3.9 × 10⁶ (larger power of 10)
  2. 5 × 10⁻⁶ (more negative power means a smaller number)
  3. 6 × 10⁻³, 6 × 10⁻¹, 6 × 10⁰, 6 × 10³

6. Multiplying and Dividing in Standard Form

Core note Core candidates are only expected to calculate with standard form on the calculator paper — but the method below is worth knowing either way, since it’s really just the index laws from the last note in action.

Worked example — multiplying Work out (3 × 10⁴) × (2 × 10³), giving your answer in standard form.
Multiply the A-values: 3 × 2 = 6. Add the powers: 4 + 3 = 7.
Answer: 6 × 10⁷
Worked example — multiplying, with a re-adjustment Work out (5 × 10⁴) × (4 × 10³), giving your answer in standard form.
5 × 4 = 20, and 4 + 3 = 7, giving 20 × 10⁷ — but 20 isn’t between 1 and 10.
Rewrite 20 as 2 × 10¹, so 20 × 10⁷ = 2 × 10¹ × 10⁷ = 2 × 10⁸
Worked example — dividing Work out (9 × 10⁷) ÷ (3 × 10²), giving your answer in standard form.
Divide the A-values: 9 ÷ 3 = 3. Subtract the powers: 7 − 2 = 5.
Answer: 3 × 10⁵
Try it — Multiplying & Dividing
  1. (2 × 10³) × (4 × 10²)
  2. (6 × 10⁵) × (3 × 10²) — remember to check A is still between 1 and 10
  3. (8 × 10⁶) ÷ (2 × 10²)
  4. (9 × 10³) ÷ (3 × 10⁶)
Show answers
  1. 8 × 10⁵
  2. 18 × 10⁷ = 1.8 × 10⁸
  3. 4 × 10⁴
  4. 3 × 10⁻³

7. Adding and Subtracting in Standard Form

Unlike multiplying and dividing, you cannot just add the A-values and add the powers. The powers of 10 must match first — convert one number so both have the same power, then add or subtract the A-values.

Worked example Work out (3 × 10⁵) + (4.5 × 10⁴), giving your answer in standard form.
Rewrite 4.5 × 10⁴ with a power of 5: 4.5 × 10⁴ = 0.45 × 10⁵
Now add: 3 × 10⁵ + 0.45 × 10⁵ = 3.45 × 10⁵
Answer: 3.45 × 10⁵
Worked example — subtracting Work out (7 × 10⁻²) − (2 × 10⁻³), giving your answer in standard form.
Rewrite 2 × 10⁻³ with a power of −2: 2 × 10⁻³ = 0.2 × 10⁻²
Now subtract: 7 × 10⁻² − 0.2 × 10⁻² = 6.8 × 10⁻²
Answer: 6.8 × 10⁻²
Try it — Adding & Subtracting
  1. (2 × 10⁴) + (3 × 10³)
  2. (6 × 10⁶) − (5 × 10⁵)
  3. (1.2 × 10⁻³) + (8 × 10⁻⁴)
Show answers
  1. 2 × 10⁴ + 0.3 × 10⁴ = 2.3 × 10⁴
  2. 6 × 10⁶ − 0.5 × 10⁶ = 5.5 × 10⁶
  3. 1.2 × 10⁻³ + 0.8 × 10⁻³ = 2 × 10⁻³
Common mistakes to watch for
  • Leaving the final answer with A outside the range 1–10, e.g. writing 20 × 10⁷ instead of correcting it to 2 × 10⁸.
  • Adding or subtracting the A-values directly without first matching the powers of 10 — this only works for multiplying and dividing, never for adding or subtracting.
  • Miscounting decimal-point shifts by one place — always double-check by estimating: 68,000,000 is roughly 7 × 10⁷, so if your answer came out as 6.8 × 10⁶, that’s a red flag.
  • Forgetting the negative sign on the power for small numbers — 0.0004 is 4 × 10⁻⁴, not 4 × 10⁴.

8. Quick-Fire Challenge Round

True or False?
  1. 35 × 10⁴ is a valid way to write a number in standard form.
  2. A number in standard form with a negative power is always less than 1.
  3. (2 × 10³) + (3 × 10³) = 5 × 10⁶
  4. The bigger the power of 10, the bigger the number (assuming positive powers and positive A).
Show answers
  1. False — 35 is not between 1 and 10.
  2. True — a negative power means dividing by a power of 10, which always shrinks the number below 1.
  3. False — when the powers already match, you only add the A-values, not the powers: (2+3) × 10³ = 5 × 10³.
  4. True.
Spot the Error
  1. “0.0032 = 3.2 × 10⁴”
  2. “(4 × 10³) × (2 × 10²) = 8 × 10⁶”
  3. “70 × 10⁵ is already in standard form because 10⁵ is a power of 10.”
Show answers
  1. The power should be negative for a small number: 0.0032 = 3.2 × 10⁻³, not 10⁴.
  2. Powers should be added, not the result guessed: 3 + 2 = 5, so it’s 8 × 10⁵, not 10⁶.
  3. 70 is not between 1 and 10, so this isn’t standard form — it should be rewritten as 7 × 10⁶.

9. Mixed Practice — Bringing It All Together

  1. Write 340,000 in standard form.
  2. Write 0.000029 in standard form.
  3. Write 9.4 × 10⁶ as an ordinary number.
  4. Work out (6 × 10⁵) × (5 × 10²), giving your answer in standard form.
  5. Work out (7 × 10⁴) − (4 × 10³), giving your answer in standard form.
  6. Order from smallest to largest: 8 × 10⁻², 3 × 10⁻¹, 9 × 10⁻³.
Show answers
  1. 3.4 × 10⁵
  2. 2.9 × 10⁻⁵
  3. 9,400,000
  4. 30 × 10⁷ = 3 × 10⁸
  5. 7 × 10⁴ − 0.4 × 10⁴ = 6.6 × 10⁴
  6. 9 × 10⁻³, 8 × 10⁻², 3 × 10⁻¹

Practice at Home

Another large set — take it in stages rather than all at once.

Part A — Recognising and writing standard form (large numbers)
  1. Is 8.4 × 10⁶ in standard form? Is 84 × 10⁵?
  2. Write 51,000,000 in standard form.
  3. Write 730,000 in standard form.
  4. Write 2,000,000,000 in standard form.
Part B — Small numbers
  1. Write 0.0071 in standard form.
  2. Write 0.00000602 in standard form.
  3. Write 0.00033 in standard form.
Part C — Converting back to ordinary numbers
  1. Write 4.6 × 10⁵ as an ordinary number.
  2. Write 1.05 × 10⁷ as an ordinary number.
  3. Write 9 × 10⁻⁴ as an ordinary number.
  4. Write 3.3 × 10⁻¹ as an ordinary number.
Part D — Comparing and ordering
  1. Which is larger: 6 × 10⁻³ or 6 × 10⁻²?
  2. Order from smallest to largest: 5 × 10², 5 × 10⁻², 5 × 10⁰, 5 × 10⁻¹.
Part E — Multiplying and dividing
  1. (3 × 10²) × (3 × 10⁴)
  2. (7 × 10⁵) × (2 × 10³) — check whether A needs adjusting
  3. (9 × 10⁸) ÷ (3 × 10³)
  4. (6 × 10²) ÷ (1.5 × 10⁵)
Part F — Adding and subtracting
  1. (4 × 10⁵) + (6 × 10⁴)
  2. (9 × 10⁻²) − (3 × 10⁻³)
  3. (2.5 × 10³) + (7.5 × 10²)
Part G — Applied / challenge
  1. A computer’s hard drive holds approximately 2 × 10¹² bytes. A photo file is approximately 4 × 10⁶ bytes. Roughly how many photos could the drive hold? Give your answer in standard form.
  2. The population of a country is 6.4 × 10⁷. The population of a city within it is 9.8 × 10⁶. Roughly how many times bigger is the country’s population than the city’s? (Estimate, then check with a calculation.)
  3. Write these three numbers in standard form, then order them smallest to largest: 0.00081, 810,000, 0.081.

Key Vocabulary Recap

Standard form — A × 10ⁿ, where 1 ⩽ A < 10 and n is an integer
Ordinary number — a number written out in full, not using powers of 10
Positive power — used for numbers 10 or greater
Negative power — used for numbers less than 1

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C1.8 (Core subject content) and section E1.8 (Extended subject content) — Standard form. The content is shared almost entirely between tiers; the only distinction the syllabus makes is that Core candidates are expected to calculate with standard form only on the calculator paper.

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