### Counting in Base 10: (Decimal)

The usual system of counting is called the decimal or denary system.

“dec “means ten which means our usual decimal system is in **Base 10**.

In Base 10 we have digits for the numbers 0 – 9. We do not have a single digit for ten.

We saw in the previous lesson that the Romans had a single character for 10 which is X.

In Base 10, we write 10 for the number ten, but this stands for **1** ten and **0** ones. This is two digits; we have no single solitary digit that stands for “ten”.

We count in Base 10 like this:

0 | Start at 0 | ||

• | 1 | Then 1 | |

•• | 2 | Then 2 | |

⋮ | |||

••••••••• | 9 | Up to 9 | |

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

•••••••••• • | 11 | ||

•••••••••• •• | 12 | ||

⋮ | |||

•••••••••• ••••••••• | 19 | ||

•••••••••• •••••••••• | 20 | Start back at 0 again, but add 1 on the left | |

•••••••••• •••••••••• • | 21 | And so on! |

This system enables us to write small or large numbers using a combination of digits.

Numbers in base 10 are usually written without the subscript ten or 10

431 _{ten} is simply written as 431.

Apart from base 10, numbers are counted in other bases such as base two, base five, base seven, base eight, etc.

For example, 64_{eight }means 64 in base 8.

### Counting in Groups of Twos: (Binary)

**Binary numbers** are numbers written using only 0s and 1s.

Numbers 2, 3, 4, 5, 6, 7, 8, and 9 do not exist in binary.

A major difference between decimal numbers are binary numbers is that decimal numbers are in base 10 and binary numbers are in base 2.

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).

As discussed in base 10 there are ten numbers: we count 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

In base 2 we have two numbers: we count 0, 1

How do we Count in Base 2?

In the base 2 number system, we also count in groups of 2

When counting we go to a new group each time we reach a number that is a power of 2.

We, therefore, count as follows:

\(\scriptsize \circledast\) | 1 unit (1) |

\(\scriptsize \color{green}{\circledast \, \circledast}\) | 1 group of 2 and 0 units (10) |

\(\scriptsize \color{green}{\circledast \, \circledast} \, \color{black}{\circledast}\) | 1 group of 2 and 1 unit (11) |

If we add another object, we have 2 groups of two, which is 4. We continue counting as follows:

\(\scriptsize \color{blue}{\circledast \, \circledast \, \circledast \, \circledast}\) | 1 group of four, 0 groups of two, and 0 units (100) |

\(\scriptsize \color{blue}{\circledast \, \circledast \, \circledast \, \circledast}\, \color{black}{\circledast}\) | 1 group of four, 0 groups of two, and 1 unit (101) |

\(\scriptsize \color{blue}{\circledast \, \circledast \, \circledast \, \circledast} \, \color{green}{\circledast \, \circledast}\) | 1 group of four, 1 group of two, and 0 units (110) |

\(\scriptsize \color{blue}{\circledast \, \circledast \, \circledast \, \circledast} \, \color{green}{\circledast \, \circledast} \, \color{black}{\circledast}\) | 1 group of four, 1 group of two, and 1 unit (111) |

If we add another object, we have 3 groups of two, which is 8. We continue counting as follows:

\(\scriptsize \color{red}{\circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast}\) | 1 group of eight, 0 groups of four, 0 groups of two, and 0 unit (1000) |

\(\scriptsize \color{red}{\circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast} \, \color{black}{\circledast}\) | 1 group of eight, 0 groups of four, 0 groups of two, and 1 unit (1001) |

\(\scriptsize \color{red}{\circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast}\, \color{green}{\circledast\, \circledast}\) | 1 group of eight, 0 groups of four, 1 group of two, and 0 units (1010) |

\(\scriptsize \color{red}{\circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast}\, \color{green}{\circledast\, \circledast}\, \color{black}{\circledast}\) | 1 group of eight, 0 group of four, 1 group of two, and 1 unit (1011) |

\(\scriptsize \color{red}{\circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast \, \circledast}\, \color{blue}{\circledast\, \circledast\, \circledast\, \circledast}\) | 1 group of eight, 1 group of four, 0 groups of two, and 0 units (1100) |

### Decimal vs. Binary

Binary | Decimal |

0 | 0 |

1 | 1 |

10 | 2 |

11 | 3 |

100 | 4 |

101 | 5 |

110 | 6 |

111 | 7 |

1000 | 8 |

1001 | 9 |

1010 | 10 |

1011 | 11 |

1100 | 12 |

10 as a decimal number is 1010 as a binary number

5 as a decimal number is 101 as a binary number

The four most commonly used number systems are;

**1. **Decimal Number System**2.** Binary Number System**3.** Octal Number System**4.** Hexadecimal Number System

Number System | Base | Digits Used |

Decimal Number System | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (Total 10 digits) |

Binary Number System | 2 | 0, 1 (Total 2 digits) |

Octal Number System | 8 | 0, 1, 2, 3, 4, 5, 6, 7 (Total 8 digits) |

Hexadecimal Number System | 16 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F (Total 16 digits) |

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