# Binary Number System

A Binary Number is made up of only **0**s and **1**s.

110100 |

Example of a Binary Number |

There is no 2, 3, 4, 5, 6, 7, 8 or 9 in Binary!

A "**bit**" is a single **b**inary dig**it**. The number above has 6 bits.

Binary numbers have many uses in mathematics and beyond.

In fact the digital world uses binary digits.

## How do we Count using Binary?

It is just like counting in decimal except we reach 10 much sooner.

Binary | ||

0 | We start at 0 | |

1 | Then 1 | |

??? | But then there is no symbol for 2 ... what do we do? |

Well how do we count in Decimal? | |||

0 | Start at 0 | ||

... | Count 1,2,3,4,5,6,7,8, and then... | ||

9 | This is the last digit in Decimal | ||

10 | So we start back at 0 again, but add 1 on the left |

The same thing is done in binary ...

Binary | |||

0 | Start at 0 | ||

• | 1 | Then 1 | |

•• | 10 | Now start back at 0 again, but add 1 on the left | |

••• | 11 | 1 more | |

•••• | ??? | But NOW what ... ? |

What happens in Decimal? | |||

99 | When we run out of digits, we ... | ||

100 | ... start back at 0 again, but add 1 on the left |

And that is what we do in binary ...

Binary | |||

0 | Start at 0 | ||

• | 1 | Then 1 | |

•• | 10 | Start back at 0 again, but add 1 on the left | |

••• | 11 | ||

•••• | 100 | start back at 0 again, and add one to the number on the left... ... but that number is already at 1 so it also goes back to 0 ... ... and 1 is added to the next position on the left | |

••••• | 101 | ||

•••••• | 110 | ||

••••••• | 111 | ||

•••••••• | 1000 | Start back at 0 again (for all 3 digits), add 1 on the left | |

••••••••• | 1001 | And so on! |

See how it is done in this little demonstration (press play button):

## Decimal vs Binary

Here are some equivalent values:

Decimal: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Binary: | 0 | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |

### Symmetry

Binary numbers also have a beautiful and elegant pattern:

Here are some larger values:

Decimal: | 20 | 25 | 30 | 40 | 50 | 100 | 200 | 500 |
---|---|---|---|---|---|---|---|---|

Binary: | 10100 | 11001 | 11110 | 101000 | 110010 | 1100100 | 11001000 | 111110100 |

"Binary is as easy as 1, 10, 11."

Now see how to use Binary to count past 1,000 on your fingers:

## Position

In the Decimal System there are Ones, Tens, Hundreds, etc

In **Binary** there are Ones, Twos, Fours, etc, like this:

This is 1×8 + 1×4 + 0×2 + 1 + 1×(1/2) + 0×(1/4) + 1×(1/8)

= **13.625 in Decimal**

Numbers can be placed to the left or right of the point, to show values greater than one and less than one.

10.1 | |

The number to the left of the point is a whole number (such as 10) | |

As we move further left, every number place gets 2 times bigger. | |

The first digit on the right means halves (1/2). | |

As we move further right, every number place gets 2 times smaller (half as big). |

### Example: 10.1

- The "10" means 2 in decimal,
- The ".1" means half,
- So "10.1" in binary is 2.5 in decimal

You can do conversions at Binary to Decimal to Hexadecimal Converter.

## Words

The word **binary** comes from "Bi-" meaning two. We see "bi-" in words such as "bicycle" (two wheels) or "binocular" (two eyes).

When you say a binary number, pronounce each digit (example, the binary number "101" is spoken as "one zero one", or sometimes "one-oh-one"). This way people don't get confused with the decimal number. |

A single binary digit (like "0" or "1") is called a "bit".

For example **11010** is five bits long.

The word** bit** is made up from the words "**b**inary dig**it**"

## How to Show that a Number is Binary

To show that a number is a *binary* number, follow it with a little 2 like this: **101 _{2}**

This way people won't think it is the decimal number "101" (one hundred and one).

## Examples

### Example: What is 1111_{2} in Decimal?

_{2}

- The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8)
- The next "1" is in the "2×2" position, so that means 1×2×2 (=4)
- The next "1" is in the "2" position, so that means 1×2 (=2)
- The last "1" is in the ones position, so that means 1
- Answer: 1111 = 8+4+2+1 = 15 in Decimal

### Example: What is 1001_{2} in Decimal?

_{2}

- The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8)
- The "0" is in the "2×2" position, so that means 0×2×2 (=0)
- The next "0" is in the "2" position, so that means 0×2 (=0)
- The last "1" is in the ones position, so that means 1
- Answer: 1001 = 8+0+0+1 = 9 in Decimal

### Example: What is 1.1_{2} in Decimal?

_{2}

- The "1" on the left side iaas in the ones position, so that means 1.
- The 1 on the right side is in the "halves" position, so that means 1×(1/2)
- So, 1.1 is "1 and 1 half" = 1.5 in Decimal

### Example: What is 10.11_{2} in Decimal?

_{2}

- The "1" is in the "2" position, so that means 1×2 (=2)
- The "0" is in the ones position, so that means 0
- The "1" on the right of the point is in the "halves" position, so that means 1×(1/2)
- The last "1" on the right side is in the "quarters" position, so that means 1×(1/4)
- So, 10.11 is 2+0+1/2+1/4 = 2.75 in Decimal

"There are 10 kinds of people in the world,

those who understand binary numbers, and those who don't."