C1.8 – Standard Form

Table of Contents

IGCSE Mathematics  |  Core Topic

1. What is Standard Form? #

Standard form is a way of writing very large or very small numbers in a short, clear way. Scientists and engineers use it a lot.

Standard Form A number in standard form is written as: $$A \times 10^n$$ where:
  • $A$ is a number such that $1 \leq A < 10$
  • $n$ is a positive or negative integer (a whole number)
Which of these are in standard form?
  • $3.2 \times 10^5$   ✓   (3.2 is between 1 and 10, power is a whole number)
  • $0.5 \times 10^3$   ✗   (0.5 is less than 1 — not allowed)
  • $12 \times 10^4$   ✗   (12 is not less than 10 — not allowed)
  • $7 \times 10^{-3}$   ✓   (7 is between 1 and 10, negative power is fine)

2. Converting to Standard Form #

To write a number in standard form, you move the decimal point until you have a number between 1 and 10. The number of places you move tells you the power of 10.

Quick rule:
  • Moving the decimal point left → positive power
  • Moving the decimal point right → negative power

Large Numbers (positive power) #

Worked Example — Write 45 000 in standard form
  1. Find where the decimal point is now:   $45000.$
  2. Move it left until you have a number between 1 and 10:   $4.5$
    You moved it 4 places to the left.
  3. The power of 10 is $+4$: $$45000 = 4.5 \times 10^4$$
Worked Example — Write 307 000 000 in standard form
  1. The decimal point is after the last zero:   $307000000.$
  2. Move it left to get a number between 1 and 10:   $3.07$
    You moved it 8 places to the left.
  3. Write the answer: $$307000000 = 3.07 \times 10^8$$

Small Numbers (negative power) #

Worked Example — Write 0.00062 in standard form
  1. Find where the decimal point is:   $0.00062$
  2. Move it right until you have a number between 1 and 10:   $6.2$
    You moved it 4 places to the right.
  3. The power of 10 is $-4$: $$0.00062 = 6.2 \times 10^{-4}$$

IMAGE NEEDED: Number line or arrow diagram showing decimal point moving left (large numbers, positive power) and right (small numbers, negative power)

Google Images Search: “standard form decimal point moving IGCSE maths diagram educational”

3. Converting from Standard Form to Ordinary Numbers #

To reverse the process, move the decimal point in the direction the power tells you.

Quick rule:
  • Positive power → move decimal point right (makes number bigger)
  • Negative power → move decimal point left (makes number smaller)
Worked Example — Write $2.9 \times 10^6$ as an ordinary number
  1. The power is $+6$, so move the decimal point 6 places to the right.
  2. $2.9 \rightarrow 2\,900\,000$   (fill empty spaces with zeros)
  3. $$2.9 \times 10^6 = 2\,900\,000$$
Worked Example — Write $5.4 \times 10^{-3}$ as an ordinary number
  1. The power is $-3$, so move the decimal point 3 places to the left.
  2. $5.4 \rightarrow 0.0054$   (fill empty spaces with zeros)
  3. $$5.4 \times 10^{-3} = 0.0054$$

4. Calculating with Standard Form #

Multiplying #

Method Multiply the $A$ values together, then add the powers of 10. $$(A_1 \times 10^m) \times (A_2 \times 10^n) = (A_1 \times A_2) \times 10^{m+n}$$ If the result is not in standard form, adjust it.
Worked Example — $(3 \times 10^4) \times (2 \times 10^3)$
  1. Multiply the $A$ values:   $3 \times 2 = 6$
  2. Add the powers:   $10^4 \times 10^3 = 10^{4+3} = 10^7$
  3. Write the answer: $$(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7$$
Worked Example — $(4 \times 10^5) \times (5 \times 10^2)$   (answer needs adjusting)
  1. Multiply the $A$ values:   $4 \times 5 = 20$
  2. Add the powers:   $10^{5+2} = 10^7$
  3. Result so far:   $20 \times 10^7$ — but 20 is not between 1 and 10, so adjust: $$20 \times 10^7 = 2.0 \times 10^1 \times 10^7 = 2.0 \times 10^8$$

Dividing #

Method Divide the $A$ values, then subtract the powers of 10. $$(A_1 \times 10^m) \div (A_2 \times 10^n) = \frac{A_1}{A_2} \times 10^{m-n}$$
Worked Example — $(8 \times 10^6) \div (2 \times 10^2)$
  1. Divide the $A$ values:   $8 \div 2 = 4$
  2. Subtract the powers:   $10^{6-2} = 10^4$
  3. Write the answer: $$(8 \times 10^6) \div (2 \times 10^2) = 4 \times 10^4$$

Adding and Subtracting #

Method You can only add or subtract standard form numbers directly if the powers of 10 are the same. If they are different, convert one number first so the powers match.
Worked Example — $(5 \times 10^4) + (3 \times 10^4)$   (same power)
  1. The powers are the same, so add the $A$ values:   $5 + 3 = 8$
  2. Keep the same power of 10: $$(5 \times 10^4) + (3 \times 10^4) = 8 \times 10^4$$
Worked Example — $(4.2 \times 10^5) + (3 \times 10^4)$   (different powers)
  1. Convert the smaller power to match the larger one: $$3 \times 10^4 = 0.3 \times 10^5$$
  2. Now both have $10^5$, so add the $A$ values: $$(4.2 + 0.3) \times 10^5 = 4.5 \times 10^5$$
Tip: For adding and subtracting, it is often easiest to convert both numbers into ordinary numbers, do the calculation, then convert back to standard form.

Syllabus Reference — C1.8 Standard Form #

  1. Use the standard form $A \times 10^n$ where $n$ is a positive or negative integer and $1 \leq A < 10$.
  2. Convert numbers into and out of standard form.
  3. Calculate with values in standard form. (Core candidates are expected to calculate with standard form only on Paper 3.)
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