# Exponents, Scientific Notation

In Science sometimes we have do deal with very large numbers, or very small numbers. Scientific notation solves the problem of writing out these numbers.

## Exponents

Large multiples of 10 can be represented as:

number | multiples of 10 | exponent form |
---|---|---|

100 | 10 x 10 | 10^{2} |

1,000 | 10 x 10 x 10 | 10^{3} |

10,000 | 10 x 10 x 10 x 10 | 10^{4} |

100,000 | 10 x 10 x 10 x 10 x 10 | 10^{5} |

And so a large number like 100,000,000,000,000,000,000,000 can be written as
10^{23} because it is a 1 with 23 zeros following it.

Conversely for small powers of 10:

number | fraction | exponent form |
---|---|---|

0.1 | 1/10 | 10^{-1} |

0.01 | 1/100 | 10^{-2} |

0.001 | 1/1,000 | 10^{-3} |

0.000,1 | 1/10,000 | 10^{-4} |

Therefore a small number like 0.000,000,000,000,000,034 could be written as
3.4 x 10^{-17} because you'd have to move the decimal 17 times to the
right in order to get 3.4 .

Notice that the *negative *exponent does *not *mean a
negative number. It means a small number less than 1.

The combined table, with 1 and 10 included:

number | exponent form |
---|---|

1,000 | 10^{3} |

100 | 10^{2} |

10 | 10^{1} |

1 | 10^{0} |

0.1 | 10^{-1} |

0.01 | 10^{-2} |

0.001 | 10^{-3} |

etc.. |

## Scientific notation

There are 602,000,000,000,000,000,000,000 atoms in 12 grams of carbon.

602,000,000,000,000,000,000,000 could be written as 6.02 x 100,000,000,000,000,000,000,000

Using what we learned above, this can be written as 6.02 x 10^{23}.

Alternatively, one atom of carbon weighs 0.000,000,000,000,000,000,000,0199
grams or 1.99 x 10^{-23} grams.

## Multiplication and Division

Notice that if you multiply exponents they add, and if you divide they subtract:

100,000 x 100 = 10,000,000 or 10^{5}
x 10^{2} = 10^{7} (notice that 5 +
2 = 7)

100,000 / 100 = 100 or 10^{5} / 10^{2
}= 10^{3} (notice that 5 - 2 = 3)

100 / 100,000 = 0.001 or 10^{2} / 10^{5
}= 10^{-3} (notice that 2 - 5 = -3 and also that
10^{-3 }is not a negative number)