# 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

$391×{10}^{18}$

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:

$3.91×{10}^{20}$

The other big number at the top of the page was $8.64×{10}^{19}$ . 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 ${10}^{-5}$ :

$0.000076=7.6×{10}^{-5}$