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Cambridge IGCSE Mathematics — 0580 Extended

Topic 1.7: Number — Standard Form

Standard form notation

A number is in standard form when written as \(A \times 10^n\), where:

  • \(A\) is a number with one non-zero digit before the decimal point: \(1 \le A < 10\)
  • \(n\) is an integer (positive, zero or negative)

Examples: \(3.2 \times 10^5 = 320\,000\); \(4.7 \times 10^{-3} = 0.0047\).

Method

  1. Move the decimal point so there is exactly one non-zero digit before it — this gives \(A\).
  2. Count how many places you moved; that count is \(n\) (right = positive, left = negative).
  3. Check: \(1 \le A < 10\).

Write \(0.00056\) in standard form.

Worked solution: 0.00056 written as 5.6 times 10 to the power negative 4
Move the decimal 4 places right: \(0.00056 = 5.6 \times 10^{-4}\)

Calculations in standard form

Multiply and divide

  1. Multiply or divide the \(A\) values.
  2. Add the powers when multiplying; subtract them when dividing.
  3. If \(A\) is not in range, adjust by moving the decimal and correcting \(n\).

Calculate \((3 \times 10^4) \times (2 \times 10^5)\).

Worked solution: 3 times 10 to the 4 multiplied by 2 times 10 to the 5 equals 6 times 10 to the 9
Multiply the \(A\) values and add the powers: \(6 \times 10^9\)

Calculate \((8 \times 10^6) \div (2 \times 10^3)\).

Worked solution: 8 times 10 to the 6 divided by 2 times 10 to the 3 equals 4 times 10 to the 3
Divide the \(A\) values and subtract the powers: \(4 \times 10^3\)

Adding and subtracting

Powers of 10 are the same

Add or subtract the \(A\) values; keep the power unchanged. Adjust if needed.

Powers of 10 are different

Rewrite one number so both have the same power of 10, then add or subtract.

Calculate \((4.2 \times 10^5) + (1.8 \times 10^5)\).

Worked solution: 4.2 times 10 to the 5 plus 1.8 times 10 to the 5 equals 6.0 times 10 to the 5
Same power of 10 — add the \(A\) values: \(6.0 \times 10^5\)

Paper 2 (non-calculator)

After multiplying, \(A\) may leave the range. Example: \(5 \times 10^3 \times 4 \times 10^2 = 20 \times 10^5\). Rewrite as \(2.0 \times 10^6\).

Converting back to ordinary numbers

Positive \(n\): move the decimal point \(n\) places to the right, filling with zeros if needed.

Negative \(n\): move the decimal point \(|n|\) places to the left.

Write \(4.56 \times 10^4\) as an ordinary number.

Worked solution: 4.56 times 10 to the 4 equals 45600
Move the decimal 4 places right: \(4.56 \times 10^4 = 45\,600\)

Try this

Write \(7.3 \times 10^{-2}\) as an ordinary number and calculate \((5 \times 10^3) \times (4 \times 10^2)\).

Show answer
Answer

  1. Negative \(n\) — move the decimal 2 places left

    \[ 7.3 \times 10^{-2} = 0.073 \]

  2. Multiply the \(A\) values and add the powers

    \[ (5 \times 10^{3}) \times (4 \times 10^{2}) = 20 \times 10^{5} \]

  3. \(20\) is not between \(1\) and \(10\) — rewrite

    \[ 20 \times 10^{5} = 2.0 \times 10^{6} \]

  4. Final answers

    \[ 0.073,\quad 2.0 \times 10^{6} \]

Exam Traps

  • \(12 \times 10^4\) is not standard form because \(12 \not< 10\). Rewrite as \(1.2 \times 10^5\).
  • Do not add the powers when adding the numbers — first make the powers of 10 the same.

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