Standard Form

The Sun is about 150\,000\,000 km away. A single virus is roughly 0.000\,000\,02 m across. Try writing either of those down in a hurry and you will lose count of the zeros — and one wrong zero makes the number ten times too big or too small. Scientists deal with the very large and the very small all day long, so they need a way to write these numbers that is short, safe, and easy to compare.

That way is standard form (also called scientific notation). The trick is to split every number into two honest pieces: a tidy number between 1 and 10, times a power of ten that does all the zero-counting for you:

150\,000\,000 = 1.5 \times 10^{8} \qquad 0.000\,000\,02 = 2 \times 10^{-8}

Same numbers — but now you can read them at a glance, and the power of ten tells you the scale instantly. No more miscounted zeros.

The format

Every number in standard form looks like this:

a \times 10^{n}

Here a is a number with exactly one non-zero digit before the decimal point — that is, 1 \le a < 10 — and n is a whole number that can be positive, negative or zero. The power n simply counts how many places the decimal point has moved.

Big numbers (bigger than 10) get a positive power.
Small numbers (between 0 and 1) get a negative power.
A number already between 1 and 10 has power 0, since 10^{0} = 1.

Worked example 1 — a big number

Write 4\,500\,000 in standard form.

Put the decimal point right after the first non-zero digit: 4.5. That is our a. Now count how many places the point had to slide left to get there — from the end of 4\,500\,000. back to 4.5 is six places:

4\,500\,000 = 4.5 \times 10^{6}

Sliding the point left made the number smaller, so we multiply by a positive power of ten to build it back up. Big number, positive power — it checks out.

Worked example 2 — a small number

Write 0.000\,34 in standard form.

Again put the point after the first non-zero digit: 3.4. Count how many places the point slides right to get there — from 0.000\,34 to 3.4 is four places:

0.000\,34 = 3.4 \times 10^{-4}

Sliding the point right made the number bigger, so we multiply by a negative power to shrink it back down. Small number, negative power.

To turn 3.4 \times 10^{-4} back into an ordinary number, just do what the power says: move the decimal point 4 places left (negative = left, towards small), filling in zeros: 3.4 \to 0.000\,34. A positive power moves the point right: 4.5 \times 10^{6} \to 4\,500\,000. The sign of the power is just a direction sign for the decimal point.

Worked example 3 — multiplying in standard form

Here is where standard form really earns its keep. To multiply two numbers written this way, multiply the fronts (the a's) and add the powers:

(2 \times 10^{3}) \times (3 \times 10^{4}) = (2 \times 3) \times 10^{3+4} = 6 \times 10^{7}

The powers add because of the index laws: 10^{3} \times 10^{4} = 10^{7}. No zeros were harmed. Sometimes the fronts multiply to 10 or more, and then you must tidy up — for example (5 \times 10^{2}) \times (4 \times 10^{3}) = 20 \times 10^{5}, but 20 is not between 1 and 10, so rewrite it as 2 \times 10^{1} and roll the extra ten into the power:

20 \times 10^{5} = 2 \times 10^{6}

A number in standard form is written a \times 10^{n}, where:

a has exactly one non-zero digit before the decimal point, so 1 \le a < 10;
the power n is positive for large numbers and negative for small ones, counting how far the decimal point moves;
to multiply two such numbers, multiply the a's and add the powers — (2 \times 10^{3}) \times (3 \times 10^{4}) = 6 \times 10^{7}.

Two classic slip-ups snare almost everybody:

The front number must be between 1 and 10. Writing 45 \times 10^{5} is not proper standard form — the 45 is too big. Fix it: 45 = 4.5 \times 10^{1}, so 45 \times 10^{5} = 4.5 \times 10^{6}. Likewise 0.7 \times 10^{3} is wrong because 0.7 is below 1.
A negative power means a small number, not a negative one. 10^{-3} is 0.001 — a tiny positive number — not -1000. The minus sign lives on the power, and it only tells the decimal point which way to move. The number 3.4 \times 10^{-4} is a small positive number, 0.000\,34.

Standard form lets you write — and instantly compare — things whose sizes are almost impossible to picture side by side. Astronomers clock the distance across a galaxy at around 10^{21} km; physicists measure an atom at about 10^{-10} m. You could never line those numbers up written out in full, but as powers of ten the gap is obvious at a glance: the powers differ by about 31, so one is roughly 10^{31} times the other.

It is also exactly how calculators and computers show huge or tiny results. When your calculator displays 1.5\text{E}8, the "E" is short for "times ten to the power", so it means 1.5 \times 10^{8} = 150\,000\,000. That "E notation" is the everyday shorthand of science, engineering and every spreadsheet on Earth.