Standard form
Section: Number 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Standard Form Notation and Conversion
Standard form is a compact way to write very large or very small numbers. Every number in standard form has exactly the same shape: a single non-zero digit before the decimal point, multiplied by a power of 10.
- Standard form is written as A × 10^n, where 1 A < 10 and n is a positive or negative integer
- For numbers of 10 or more, n is positive - it counts how many places the decimal point moves left
- For numbers between 0 and 1, n is negative - it counts how many places the decimal point moves right
Worked Example: Converting Into Standard Form
- 4700 = 4.7 × 10^3 (decimal point moves 3 places left)
- 0.00052 = 5.2 × 10^-4 (decimal point moves 4 places right)
Worked Example: Standard Form in a Measurement Context
- Question: Write 3.6 km in standard form, in metres.
- Step 1: Convert km to m: 3.6 × 1000 = 3600 m
- Step 2: Write in standard form: 3600 = 3.6 × 10^3
- Answer: 3.6 × 10^3 m
- Question: Write 45 cm³ in standard form, in litres.
- Step 1: Convert cm³ to litres (1 litre = 1000 cm³): 45 ÷ 1000 = 0.045 litres
- Step 2: Write in standard form: 0.045 = 4.5 × 10^-2
- Answer: 4.5 × 10^-2 litres
Common Mistakes
MistakeLeaving A outside the range 1 to 10, e.g. writing 4700 as 47 × 10²
FixA must satisfy 1 ⩽ A < 10 - keep adjusting the decimal point and the power of 10 together until only one non-zero digit remains before the point: 4700 = 4.7 × 10³
MistakeUsing a positive power of 10 for a number smaller than 1, e.g. writing 0.00052 = 5.2 × 10⁴
Fixnumbers less than 1 always need a negative power of 10: 0.00052 = 5.2 × 10⁻⁴
Calculating with Standard Form
Multiplying and dividing standard-form numbers works directly with the two parts separately. Adding and subtracting is different - the powers of 10 must match before the leading numbers can be combined.
- To multiply, multiply the A-parts and add the powers of 10
- To divide, divide the A-parts and subtract the powers of 10
- After multiplying or dividing, check the result is still in standard form (1 ⩽ A < 10) - adjust it if not
- To add or subtract, first rewrite both numbers with the same power of 10, then add or subtract the A-parts
Worked Example: Multiplying and Dividing in Standard Form
- Question: Work out (i) (3 × 10^4) × (5 × 10^6), (ii) (9 × 10^8) ÷ (3 × 10^3).
- Step 1 (i): Multiply the A-parts and add the powers: 3 × 5 = 15, 10^4 × 10^6 = 10^10, giving 15 × 10^10
- Step 2 (i): 15 is not between 1 and 10, so adjust: 15 × 10^10 = 1.5 × 10^11
- Step 1 (ii): Divide the A-parts and subtract the powers: 9 ÷ 3 = 3, 10^8 ÷ 10^3 = 10^5, giving 3 × 10^5 (already valid)
- Answer: (i) 1.5 × 10¹¹, (ii) 3 × 10⁵
Worked Example: Subtracting in Standard Form
- Question: Work out (6.5 × 10^15) - (6.5 × 10^14), giving your answer in standard form.
- Step 1: Rewrite both terms with the same power of 10: 6.5 × 10^14 = 0.65 × 10^15
- Step 2: Subtract the A-parts, keeping the common power: (6.5 - 0.65) × 10^15 = 5.85 × 10^15
- Answer: 5.85 × 10^15
Common Mistakes
MistakeAdding or subtracting standard-form numbers without first matching their powers of 10
Fixrewrite one number so both share the same power of 10 before combining the A-parts - only then can they be added or subtracted directly
MistakeLeaving the A-part outside the range 1 to 10 after multiplying or dividing, e.g. leaving an answer as 15 × 10¹⁰
Fixalways check the final A-part is between 1 and 10, adjusting the power of 10 to compensate if it is not: 15 × 10¹⁰ = 1.5 × 10¹¹
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