Standard Form

1. Write very large and very small numbers in standard form, and convert back to ordinary numbers

   Standard form writes every number as A × 10ⁿ where A is at least 1 but less than 10. So 3 400 000 = 3.4 × 10⁶, and 0.00052 = 5.2 × 10⁻⁴ - a negative power means a small number. To convert back, move the decimal point n places.

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2. Order and compare numbers written in standard form

   Compare the powers of 10 first - a bigger power means a bigger number, so 2 × 10⁷ beats 9 × 10⁶. Only when the powers are equal do you compare the front numbers: 7.1 × 10⁵ is more than 6.9 × 10⁵.

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3. Multiply and divide numbers in standard form, with and without a calculator

   Multiply the front numbers and ADD the powers: (3 × 10⁴) × (2 × 10⁵) = 6 × 10⁹. When dividing, divide the fronts and subtract the powers. If the front lands outside 1–10, adjust: 40 × 10⁶ must be rewritten as 4 × 10⁷.

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4. Add and subtract numbers in standard form

   Adding and subtracting needs matching powers of 10, so rewrite one number first: 3.2 × 10⁵ + 4 × 10⁴ becomes 3.2 × 10⁵ + 0.4 × 10⁵ = 3.6 × 10⁵. Alternatively, convert both to ordinary numbers, calculate, then convert back.

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5. Solve real-world problems with standard form - populations, distances in space, sizes of cells

   Exam questions dress standard form up in context: how many times further, how many cells fit, total mass of a population. Set up the calculation as usual (often a division), keep the numbers in standard form, and sense-check whether the answer should be huge or tiny.

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• Leaving the front number outside 1 to 10, as in 40 × ${10}^6$ or 0.4 × ${10}^8$.

  The front number must be at least 1 and less than 10. Dividing it by 10 adds 1 to the power, so 40 × ${10}^6$ = 4 × ${10}^7$.

• Giving a small number a positive power, so 0.00052 becomes 5.2 × ${10}^4$.

  Numbers less than 1 have a negative power. The point moves 4 places to reach 5.2, so 0.00052 = 5.2 × ${10}^{-4}$.

• Multiplying the powers as well as the front numbers, so 3 × ${10}^4$ times 2 × ${10}^5$ becomes 6 × ${10}^{20}$.

  Multiply the front numbers and add the powers: 3 × 2 = 6 and 4 + 5 = 9, so the answer is 6 × ${10}^9$.

• Adding the front numbers when the powers are different, so 3.2 × ${10}^5$ + 4 × ${10}^4$ becomes 7.2 × ${10}^5$.

  Make the powers match first: 4 × ${10}^4$ = 0.4 × ${10}^5$, so the total is 3.6 × ${10}^5$. Or convert both: 320 000 + 40 000 = 360 000.

• Comparing the front numbers first, so 9 × ${10}^6$ looks bigger than 2 × ${10}^7$.

  Compare the powers first: 2 × ${10}^7$ is 20 million, while 9 × ${10}^6$ is only 9 million. Look at the front numbers only when the powers match.

More Number topics

• Fractions
• Percentages
• Surds
• Indices
• Bounds & Rounding
• Factors & Multiples

Keep reading

• How do I revise for GCSE maths?
• What's on the GCSE maths syllabus?
• What are the GCSE Maths grade boundaries?
• The GCSE Maths syllabus, explained