mathsGrade 9-102 min read

Standard form: writing very big and very small numbers

Scientists don't write out the mass of an electron with thirty zeros. Standard form is the shorthand — one rule for the format, and a quick way to multiply and divide enormous numbers in your head.

The distance to the Sun is about $150{,}000{,}000\,\text{km}$ ; an atom is around $0.0000000001\,\text{m}$ across. Writing those out is slow and error-prone, so we use standard form (scientific notation) — a compact way to write any number using a power of ten.

The format

A number in standard form looks like

a \times 10^{n}, \qquad \text{where } 1 \leqslant a < 10.

The key rule is that $a$ must be at least 1 and less than 10 — exactly one non-zero digit before the decimal point. The power $n$ records how far the point moved.

150{,}000{,}000 = 1.5 \times 10^{8}, \qquad 0.0000000001 = 1 \times 10^{-10}

A positive power means a big number (point moves right); a negative power means a small one (point moves left). The power is just a zero-counter.

Converting, both ways

Big number → standard form: put the decimal point after the first digit, then count how many places it moved. $46{,}000 = 4.6 \times 10^{4}$ .
Small number → standard form: same idea, but the power is negative. $0.0032 = 3.2 \times 10^{-3}$ .

Multiplying and dividing become easy

This is where standard form earns its place. Handle the numbers and the powers separately, using the laws of indices on the tens:

(3 \times 10^{4}) \times (2 \times 10^{5}) = (3 \times 2) \times 10^{4+5} = 6 \times 10^{9}

Multiply the front numbers, add the powers. For division, divide the front numbers and subtract the powers. Huge calculations collapse into one easy line.

The trap

After multiplying, your answer might not be in proper standard form. If the front number comes out as $10$ or more, fix it:

(5 \times 10^{3}) \times (4 \times 10^{2}) = 20 \times 10^{5} = 2 \times 10^{6}

Check $a$ is between 1 and 10 at the very end. $20 \times 10^5$ has the right value but the wrong form — bump the extra ten into the power. Examiners take the mark for a front number of 10 or more.

Last revised 2 September 2025.