Standard Form

Some numbers are awkwardly big, and some are awkwardly small. Writing 3\,400\,000 or 0.000\,52 in full means counting zeros, and it is easy to get one wrong. Standard form (also called scientific notation) writes any such number compactly as a single digit-and-decimal multiplied by a power of ten:

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 (it may be negative). The power n counts how many places the decimal point moves.

Big numbers have a positive power. To turn 3\,400\,000 into standard form, slide the point left until just one digit is in front of it — that is six places:

3\,400\,000 = 3.4 \times 10^{6}

Small numbers have a negative power. To turn 0.000\,52 into standard form, slide the point right until one non-zero digit is in front of it — that is four places:

0.000\,52 = 5.2 \times 10^{-4}

The two rules — keep a between 1 and 10, and count the decimal moves into the power — let you write and multiply enormous or tiny numbers without ever writing a string of zeros.

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}.