Besides the decimal system, one of the most familiar number systems is the binary numeration system; this is mainly because of its use in computers (for more about computers, see “Math in Computing”). In a binary numeration system, only 1 and 0 are used, or a base 2 system. Converting between binary and decimal systems is fairly simple; just remember that each digit in the binary number represents a power of two.

The first column in the base two math is the *units column,* then the twos, fours, eights, etc. columns, all of which can only be filled with 0s or 1s. Since there is no single digit that stands for “2” in base 2, when you get to what stands for 2, you put a 1 in the 2’s column and a 0 in the units column, creating one 2 and no 1s. Thus, the base ten “two” (2_{10} or just 2 in decimal form) is written in the binary as 10_{2}; a 3 (3_{10}, or just 3 in decimal form) in base 2 is actually “one 2 and one 1,” or 11_{2}. The number 4 is actually 2 × 2, so you eliminate the 2’s and unit columns and put a 1 in the 4s column. Thus, 4_{10} (or just 4 in decimal form) is written in binary form as 100_{2}. To see how computers “translate” decimal to binary numbers, here are the first ten conversions:

**Decimal** |
**Binary** |
**Explanation** |

0 |
0 |
no 1s |

1 |
1 |
one 1 |

2 |
10 |
one 2 and no 1s |

3 |
11 |
one 2 and one 1 |

4 |
100 |
one 4, no 2s, and no 1s |

5 |
101 |
one 4, no 2s, and one 1 |

6 |
110 |
one 4, one 2, and no 1s |

7 |
111 |
one 4, one 2, and one 1 |

8 |
1000 |
one 8, no 4s, no 2s, and no 1s |

9 |
1001 |
one 8, no 4s, no 2s, and one 1 |

10 |
1010 |
one 8, no 4s, one 2, and no 1s |