A computer or any digital system works in a binary manner. The main number systems used in digital hardware are as follows.

**DECIMAL NUMBER SYSTEM**: The decimal number system (base10) number system has ten as it base. It uses various symbols called digit for ten distinct value (0,1,3,4,5,6,7,8 and 9) to represent numbers. It requires 10 different types of electronic pulse.

The decimal system is a position number system. It has position for unit, tens, hundred e.t.c The position of each digit conveys the multiplier( a power of ten) to be used with the digit- each position has value to ten time of a position to its right. For example:

275 = 2×100 + 7×10 + 5×1

2x 10^{2} + 7 + 10^{1} + 5 x 10^{0}

**BINARY NUMBER**

The binary number (base 2) number system represents values using symbols typically 0 and 1. In other words, the binary number system is a position number system with a power of two (2). Owing to its relatively straightforward implementation in electronic circuitry, the binary is used internally by virtually in all modern computers.

The numerals 0 and 1 have the same meaning in the decimal system, but a different interpretation is placed on the position occupied by a digit.

In the binary number system, the individual digit represent the coefficient of power 2 rather than 10 as in the decimal number system. For example, the decimal number system 19 is written in the binary representation as 10011

1^{4}0^{3}0^{2}1^{1}1^{0}_{2} = 1 x 2^{4} +0 x 2^{3} + 0 x 2^{2} + 1 x 2^{1} + 1 x 2^{0}

= 16 + 0 + 0 + 2 + 1.

Let’s look at base-two, or binary, numbers. How would you write, for instance, 12_{10} (“twelve, base ten”) as a binary number? You would have to convert to base-two columns, the analogue of base-ten columns. In base ten, you have columns or “places” for 10^{0} = 1, 10^{1} = 10, 10^{2} = 100, 10^{3} = 1000, and so forth. Similarly in base two, you have columns or “places” for 2^{0} = 1, 2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16, and so forth.

The first column in base-two math is the units column. But only “0” or “1” can go in the units column. When you get to “two”, you find that there is no single solitary digit that stands for “two” in base-two math. Instead, you put a “1” in the twos column and a “0” in the units column, indicating “1 two and 0 ones”. The base-ten “two” (2_{10}) is written in binary as 10_{2}.

A “three” in base two is actually “1 two and 1 one”, so it is written as 11_{2}. “Four” is actually two-times-two, so we zero out the twos column and the units column, and put a “1” in the fours column; 4_{10} is written in binary form as 100_{2}. Here is a listing of the first few numbers:

decimal (base 10) | binary (base 2) | |

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 | 0 ones 1 one 1 two and zero ones 1 two and 1 one 1 four, 0 twos, and 0 ones 1 four, 0 twos, and 1 one 1 four, 1 two, and 0 ones 1 four, 1 two, and 1 one 1 eight, 0 fours, 0 twos, and 0 ones 1 eight, 0 fours, 0 twos, and 1 one 1 eight, 0 fours, 1 two, and 0 ones 1 eight, 0 fours, 1 two, and 1 one 1 eight, 1 four, 0 twos, and 0 ones 1 eight, 1 four, 0 twos, and 1 one 1 eight, 1 four, 1 two, and 0 ones 1 eight, 1 four, 1 two, and 1 one 1 sixteen, 0 eights, 0 fours, 0 twos, and 0 ones |

Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two.

**Convert 101100101**_{2}to the corresponding base-ten number.

I will list the digits in order, and count them off from the RIGHT, starting with zero:

digits: | 1 0 1 1 0 0 1 0 1 |

numbering: | 8 7 6 5 4 3 2 1 0 |

The first row above (labelled “digits”) contains the digits from the binary number; the second row (labelled ” numbering”) contains the power of 2 (the base) corresponding to each digits. I will use this listing to convert each digit to the power of two that it represents:

1×2^{8} + 0×2^{7} + 1×2^{6} + 1×2^{5} + 0×2^{4} + 0×2^{3} + 1×2^{2} + 0×2^{1} + 1×2^{0}

= 1×256 + 0×128 + 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1

= 256 + 64 + 32 + 4 + 1

= 357

DECIMAL | BINARY |

9 | 1001 |

10 | 1010 |

11 | 1011 |

12 | 1100 |

13 | 1101 |

14 | 1110 |

15 | 1111 |

100 | 0100100 |

512 | 1000000000 |

= 19

DECIMAL | BINARY |

0 | 0000 |

1 | 0001 |

2 | 0010 |

3 | 0011 |

4 | 0100 |

5 | 0101 |

6 | 0110 |

7 | 0111 |

8 | 1000 |

- OCTAL NUMBER SYSTEM

The octal number system is a base 8 number system, and uses the digits from 0 to 7. Programs often display in an octal format because it can be translate relatively in binary format, each digit in the octal number system represents a power of base 8. For example the binary representation for decimal 74 is 1001010, which group into 1001010, so the octal representation is 112

1128 = 1 x 8^{2} + 1 x 8^{1} + 2 x 8^{0}

= (1 x 64) + (1 x 8) + (2 x 1)

= 64 + 8 + 2

= 74.

So, the decimal equivalent of octal number 1128 is 7410. Since there are only 8 digit (0-8) in the octal number system, 3 bits are sufficient to represent an octal number in a binary digits.

OCTAL | BINARY |

0 | 000 |

1 | 001 |

2 | 101 |

3 | 011 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

With this table, it is easy to translate octal and binary system for example

65_{8} = 110 101_{2}

17_{8} = 001 111_{2}

**HEXADECIMAL NUMBER SYSTEM**

In the hexadecimal number system is a number with a base of 16, usually written using symbols 0-9 and A-F. for example, the decimal number 79 whose binary representation is 01001111 can be written as 4F in hexadecimal ( 4 = 0100, F = 1111 ) for example 1FF16 = 1 x 132 + F x 161 + F x 160

= 1 x 256 + 15 x 16 + 16 x 1

= 511.

Thus, the decimal equivalent of hexadecimal number 1FF16 is 51110. Since there are only 16 digits in the hexadecimal number system, 4 bits are sufficient to represent any hexadecimal number in binary.

The current decimal number system was first introduced to the computing world in 1963 by international business machine (IBM). An early version that used the digit 0-9 and u-2 was introduced in 1956, in the Bendix G-15 computer

The tale given below displays the binary and decimal equivalent of some hexadecimal numbers

HEXADECIMAL | BINARY | DECIMAL |

0 | 0000 | 0 |

1 | 0001 | 1 |

2 | 0010 | 2 |

3 | 0011 | 3 |

4 | 0100 | 4 |

5 | 0101 | 5 |

6 | 0110 | 6 |

7 | 0111 | 7 |

8 | 1000 | 8 |

9 | 1001 | 9 |

A | 1010 | 10 |

B | 1011 | 11 |

C | 1100 | 12 |

D | 1101 | 13 |

E | 1110 | 14 |

F | 1111 | 15 |

The hexadecimal number 4B3A translates the following binary number.

A B 3 A

0100 1011 0011 1010

**ASSESSMENT**

The main number systems used in digital hardware are___

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