Physics loves numbers that are ridiculous. The width of an atom is about 0.0000000001 metres. Light races along at roughly 300000000 metres every second. Write those out in full and you spend your life counting zeros — and one slip turns a right answer into nonsense.

Scientists get around this with two neat tools. Metric prefixes (like kilo, milli, mega) rename a unit so a huge or tiny quantity comes out as a friendly, ordinary-sized number. And standard form (also called scientific notation) writes any number as a single digit-or-so times a power of ten, so 300000000 becomes the compact 3\times10^{8}. Both are really the same idea wearing different clothes: counting in powers of ten.

Prefixes: renaming a unit by powers of ten

A prefix is a little word stuck on the front of a unit that multiplies it by a power of ten. You already meet one every day: kilo means "a thousand", so a kilometre is 1000 metres and a kilogram is 1000 grams. The same handful of prefixes works for every unit — metres, grams, seconds, watts, amperes — so learning them once unlocks the lot.

Here is the ladder of prefixes you need for GCSE, from the giant tera down to the tiny pico:

• tera (T) — \times10^{12} — a million million
• giga (G) — \times10^{9} — a thousand million (a "billion")
• mega (M) — \times10^{6} — a million
• kilo (k) — \times10^{3} — a thousand
• (the plain unit) — \times10^{0}=\times1
• milli (m) — \times10^{-3} — a thousandth
• micro (µ) — \times10^{-6} — a millionth
• nano (n) — \times10^{-9} — a thousand-millionth
• pico (p) — \times10^{-12} — a million-millionth

Notice the pattern: each step up the ladder multiplies by 1000 (adds 3 to the power of ten), and each step down divides by 1000. So 1 megawatt is a thousand kilowatts, which is a million watts. The prefixes are just rungs three powers of ten apart.

Converting from one prefix to another

The golden rule: to swap a prefix for the plain unit, multiply the number by the prefix's power of ten. To go the other way, divide. Let's work some out.

Example 1 — kilo to the plain unit. Write 3\ \text{km} in metres.

Kilo means \times10^{3}=\times1000, so we multiply:

Example 2 — milli to the plain unit. Write 5\ \text{mA} (5 milliamps) in amperes.

Milli means \times10^{-3}, which is the same as dividing by 1000:

Example 3 — the plain unit back to a prefix. Write 4500\ \text{g} in kilograms.

Going into kilograms means we divide by 1000 (the reverse of Example 1):

A quick sense-check every time: a bigger prefix should give a smaller number for the same quantity (fewer, chunkier units), and a smaller prefix gives a bigger number. Two thousand metres is only 2 km but a whopping 2\,000\,000 mm.

See it: the same length, every prefix

Here is a single fixed length — 4500 metres — and nothing about the actual distance ever changes. Slide up and down the prefix ladder and watch the number stretch and shrink as the unit is renamed. Only near kilometres does it become a comfortable, human-sized number (4.5 km); pick a tiny prefix and the same length balloons into a monster with a dozen zeros. Notice, too, that the standard-form version at the bottom stays exactly the same whatever prefix you choose — because standard form describes the quantity, not the unit.

Standard form: taming any number

Prefixes are handy, but they only come in steps of a thousand — and sometimes you just want the raw number under control. Standard form writes any number as

The digit part A is always at least 1 but below 10 (so exactly one non-zero digit sits before the decimal point), and the power n records how many places the decimal point has moved.

Example 4 — a big number. Write the speed of light, 300000000\ \text{m/s}, in standard form.

Put the decimal point after the first non-zero digit — 3 — then count how far it has to travel to get back to the end: 8 places to the right. A big number needs a positive power:

Example 5 — a tiny number. Write 0.0000004 in standard form.

The first non-zero digit is 4. Slide the decimal point right until it sits just after that 4 — that's 7 places. A number smaller than 1 needs a negative power:

So the sign of the power is a size label: positive means big, negative means small. And 10^{0}=1, so an ordinary number like 7.2 is just 7.2\times10^{0}.

What standard form means

A number is in standard form when it is written as A\times10^{n} with:

• 1\le A<10 — one non-zero digit before the decimal point;
• n a whole number (it may be negative or zero);
• n>0 for numbers bigger than 10, and n<0 for numbers below 1.

Multiplying and dividing powers of ten

Standard form pays off the moment you calculate with monster numbers, because powers of ten obey two beautifully simple rules:

To multiply, add the powers; to divide, subtract them. That's it — no counting zeros.

Example 6 — a multiplication. How far does light travel in 2\times10^{2} seconds?

Multiply the speed by the time — handle the digit parts and the powers of ten separately:

Example 7 — a division. Work out \dfrac{8\times10^{9}}{4\times10^{6}}.

One warning: after multiplying the digit parts you might land outside the 1\le A<10 range — for instance (5\times10^{3})\times(4\times10^{3})=20\times10^{6}. That's correct, but not yet tidy standard form; nudge the point back one place to get 2\times10^{7}.

Why physics lives at these scales

These aren't school-exercise numbers — they're where the real furniture of the universe sits. An atom is about 10^{-10}\ \text{m} across; a virus a few hundred nanometres; a human cell tens of micrometres. Look outward and light crosses space at 3\times10^{8}\ \text{m/s}, the Sun is 1.5\times10^{11}\ \text{m} away, and the observable universe stretches to some 10^{27}\ \text{m}. Without prefixes and standard form you simply could not write physics down, let alone calculate with it.