Home

Scientific Notation

Scientific notation is a way to write very large or very small numbers so that they are easier to read and work with. You express a number as the product of a number greater than or equal to $1$ but less than $10$ and an integral power of $10$ .

Which is greater: $391000000000000000000$ or $86400000000000000000$ ?

To tell, you have to count all those zeros. Unless you have really good eyes, it will probably give you a headache.

Scientific notation is a way to express very large or very small numbers more simply, as the product of a number between $1$ and $10$ and a power of $10$ .

Powers of $10$
$10^{- 5} = 0.00001$	$10^{1} = 10$
$10^{- 4} = 0.0001$	$10^{2} = 100$
$10^{- 3} = 0.001$	$10^{3} = 1000$
$10^{- 2} = 0.01$	$10^{4} = 10, 000$
$10^{- 1} = 0.1$	$10^{5} = 100, 000$
$10^{0} = 1$	$10^{6} = 1, 000, 000$

(If you're confused by this table, see the pages on exponents and the properties of exponents .)

For positive powers of $10$ , the exponent is the same as the number of zeros after the $1$ . The negative powers of $10$ show how many places there are to the right of the decimal point.

When a positive number greater than or equal to $10$ is written in scientific notation, the power of $10$ used is positive. When the number is less than $1$ , the power of $10$ used is negative.

If you counted the big numbers at the beginning of the problem, you found that $391000000000000000000$ has $18$ zeros. So you can write it as

$391 \times 10^{18}$

Much easier to read! But, to make scientific notation standard, there is a convention that the first number in the product should be greater than or equal to $1$ , and less than $10$ . So, we divide $391$ by $100$ (or $10^{2}$ ) to get $3.91$ . Then we make up for it by multiplying the second number by $10^{2}$ . So, we end up with the number in scientific notation:

$3.91 \times 10^{20}$

The other big number at the top of the page was $8.64 \times 10^{19}$ . When the numbers are written in scientific notation it's much easier to compare them and do calculations.

The same thing works for small numbers, like $0 .000076$ . First move the decimal point five points to the right to get $7.6$ (which is between $1$ and $10$ ). To compensate, multiply by $10^{- 5}$ :

$0 .000076 = 7.6 \times 10^{- 5}$