# Binary Numbers

Whether it be from math class or computer science class, eventually we will need to learn about number systems, and the mathematics that is involved.

## Binary numbers

, as with decimal, octal, and hexadecimal numbers, are organized into columns. To learn binary math, we first need to understand how number systems operate. Let's take a look at the decimal system first, since it is simple and easier to think about. We can consider the number "1234" as,Thousands | Hundreds | Tens | Ones |

1 | 2 | 3 | 4 |

Which means,

1234 = 1x1000 + 2x100 + 3x10 + 4x1

1000 | = 10^3 = 10x10x10 |

100 | = 10^2 = 10x10 |

10 | = 10^1 = 10 |

1 | = 10^0 (any number to the exponent zero is one, except for zero) |

The table above can be represented as,

Thousands | Hundreds | Tens | Ones |

10^3 | 10^2 | 10^1 | 10^0 |

1 | 2 | 3 | 4 |

such that,

1234 = 1x1000 + 2x100 + 3x10 + 4x1

= 1x10^3 + 2x10^2 + 3x10^1 + 4x10^0

The decimal system, as with decimal math, operates in "base 10" (*dec* being the Latin prefix for ten) using the digits 0-9 to represent numbers, whereas the binary system, as well as its math, operates in "base 2" (*bi* being the Latin prefix for two) using the digits 0-1 to represent numbers. The base is also known as the radix. In other words, the table above can be represented as,

Thousands | Hundreds | Tens | Ones | |

Decimal | 10^3 | 10^2 | 10^1 | 10^0 |

Binary | 2^3 | 2^2 | 2^1 | 2^0 |

In base 10, we put the digits 0-9 in columns 10^0, 10^1, 10^2, and so on. To put a number that is greater than 9 into 10^n, we must add to 10^(n+1). For example, adding 10 to column 10^0 requires us to add 1 to the column 10^1.

In base 2, we put the digits 0-1 in columns 2^0, 2^1, 2^3, and so on. To put a number that is greater than 1 into 2^n, we must add to 2^(n+1). For example, adding 3 to column 2^0 requires us to add 1 to the column 2^1.

Here are some decimal numbers represented in binary.

Decimal | Binary |

1 | 1 |

2 | 10 |

3 | 11 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

8 | 1000 |

9 | 1001 |

10 | 1010 |

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Binary Numbers | Add & Subtract | Multiply & Divide | Exercises | Conversion