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 ${10}^{5}$ :
$0.000076=7.6\times {10}^{5}$
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