## Teach Any Computer Science Class

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View the Lessons →### Decimal Numbering System

Decimal is a base 10 numbering system which is made up of 10 numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is the most commonly used numbering system. The reason for this is convenience. We have 10 fingers that we use for counting, so it is easier to count with a base 10 numbering system, thus decimal is widely used.

### Computers and Binary

Binary is the main language used in computers for the following reasons:

- It is simple. Since there are only two possible values, 0 and 1, it is easier to store and manipulate numbers. It is also less expensive to use binary than other numbering systems.
- It is easy to identify OFF and ON states. The distinction is clear between the two states, which makes it reliable.
- It is efficient in controlling logic circuits. It requires the least amount of circuitry, which in turn requires the least cost, space and energy consumption.

### Conversion from Decimal to Binary

Conversion can be done by dividing the decimal number by 2 repeatedly until the final result is 0.

For example, the decimal number 357 is converted to a binary number as follows:

Division | Result | Remainder |

357 / 2 | 178 | 1 |

178 / 2 | 89 | 0 |

89 / 2 | 44 | 1 |

44 / 2 | 22 | 0 |

22 / 2 | 11 | 0 |

11 / 2 | 5 | 1 |

5 / 2 | 2 | 1 |

2 / 2 | 1 | 0 |

1 / 2 | 0 | 1 |

The binary number is taken from the remainder starting from the last to the first, or in the illustration above, from bottom to top, which is 101100101.

So decimal number 357 is 101100101 in the binary number system.

### Conversion from Binary to Decimal

Conversion can be done by plotting each binary digit value in each column corresponding to its decimal digit value. Each column is the number 2 raised to an exponent. The exponent increases by one from right to left. To get the total value, you add the value of those columns tagged as ON or equivalent to 1.

For example, the binary number 101100101 is converted to a decimal number as follows:

Exponent | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 |

Value | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

ON / OFF | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |

Total value = (256+64+32+4+1) = 357.

So the binary number 101100101 is 357 in the decimal number system.

### Advantages of the Binary System

Here are some advantages of using the binary system:

- It is easy to implement.
- The logic is easy to understand.
- It is the lowest possible base and allows for higher numbering systems to be easily encoded.

### Disadvantages of the Binary System

Here are some disadvantages of using the binary system:

- Numbers are expressed in longer form, thus taking up more space.
- There are rounding issues since fractional values don’t follow a base 2 numbering system.
- People find it difficult to read, write and manipulate binary.