# Binary Number System and Its Representation

In mathematics and digital electronics, a binary number is expressed in the base-2 number system, using only two symbols: "0" and "1". Another number system that became famous after the decimal is the binary number system, which has only two digits, 0 and 1.

## Table of Contents

In mathematics and digital electronics, a binary number is expressed in the base-2 number system, using only two symbols: "0" and "1".

## What is the binary number system?

Another number system that became famous after the decimal is the binary number system, which has only two digits, 0 and 1.

If a number system has ** n** digits, we say that the base of the number system is n. So the binary number system can also be called the base-2 number system.

### Why does a computer understand binary?

The simplest explanation would be that a computer is an electrical device, and all electrical devices understand electrical signals, which have only two states.

### Example

If we have an input wire to this machine, there are only two possible states for this wire: either the current is flowing through this wire, or it is not flowing through this wire. If the current is flowing, we say that the state of this wire is signalled. And we say that the signal state corresponds to ** 1**.

If the current is not flowing, it is not signalled. The not signal state corresponds to ** 0**. So,

**and**

**1****, in binary, translate to a signal or non-signal in an electrical device, and we can have multiple wires or inputs to represent multiple ones and zeros.**

**0**### Powers of 2

Power of two | Binary | Decimal Value |
---|---|---|

2^0 | 0001 | 1 |

2^1 | 0010 | 2 |

2^2 | 0100 | 4 |

2^3 | 1000 | 8 |

2^4 | 0001 0000 | 16 |

2^5 | 0010 0000 | 32 |

2^6 | 0100 0000 | 64 |

2^7 | 1000 0000 | 128 |

2^8 | 0001 0000 0000 | 256 |

2^9 | 0010 0000 0000 | 512 |

2^10 | 0100 0000 0000 | 1,024 |

The powers of 2 are increasing, so the bits go from right to left based on the decimal value given as input. All other left bits will be `0`

.

**For example:**

`125`

can be represented as `01111101`

in the computer binary system. Anything in computer language gets converted into a binary number system.

## What do binary numbers represent?

In mathematics and digital electronics:

- A binary number is expressed in the base-2 or binary number system.
- It uses only two symbols: typically “0” (zero) and “1” (one).

The base-2 number system is a positional notation with a radix of 2. Each digit is referred to as a bit.

### Binary counting

Binary counting follows the same procedure, except only the symbols `0`

and `1`

are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left.

0000,

0001, (rightmost digit starts over, and next digit is incremented)

0010, 0011, (rightmost two digits start over, and next digit is incremented)

0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)

1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111,…

### Binary to decimal conversion

In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 2^{0}, the next representing 2^{1}, then 2^{2}, and so on. The value of a binary number is the sum of the powers of 2 represented by each “1” digit. For example, the binary number 100101 is converted to decimal form as follows:

100101_{2} = [ ( 1 ) x 2^{5} ] + [ ( 0 ) x 2^{4} ] + [ ( 0 ) x 2^{3} ] + [ ( 1 ) x 2^{2} ] + [ ( 0 ) x 2^{1} ] + [ ( 1 ) x 2^{0} ]

100101_{2} = [ ( 1 ) x 32 ] + [ ( 0 ) x 16 ] + [ ( 0 ) x 8 ] + [ ( 1 ) x 4 ] + [ ( 0 ) x 2 ] + [ ( 1 ) x 0 ]

100101_{2} = 37_{10}

### Decimal to binary representation

Below is the 32-bit binary representation.

`5 -> 00000000 00000000 00000000 00000101`

### Representing decimals & ASCII in binary

A computer only understands byte-code made of 0’s and 1’s. We must represent every decimal character as binary digits so a computer can understand our instructions.

#### Decimal numbers in binary (8-bit representation)

Each software programming language uses its pre-defined sizes for primitive data types. So, let’s represent the rightmost 8 bits (1 byte) in binary.

Decimal Number | 8-bit binary representation |
---|---|

0 | 0000 0000 |

1 | 0000 0001 |

2 | 0000 0010 |

3 | 0000 0011 |

4 | 0000 0100 |

5 | 0000 0101 |

6 | 0000 0110 |

7 | 0000 0111 |

8 | 0000 1000 |

9 | 0000 1001 |

10 | 0000 1010 |

#### ASCII - Binary character table

Alphabets in binary (capital letters & lowercase letters)

Letter |
ASCII Code |
Binary |
Letter |
ASCII Code |
Binary |
---|---|---|---|---|---|

a | 097 | 01100001 | A | 065 | 01000001 |

b | 098 | 01100010 | B | 066 | 01000010 |

c | 099 | 01100011 | C | 067 | 01000011 |

d | 100 | 01100100 | D | 068 | 01000100 |

e | 101 | 01100101 | E | 069 | 01000101 |

f | 102 | 01100110 | F | 070 | 01000110 |

g | 103 | 01100111 | G | 071 | 01000111 |

h | 104 | 01101000 | H | 072 | 01001000 |

i | 105 | 01101001 | I | 073 | 01001001 |

j | 106 | 01101010 | J | 074 | 01001010 |

k | 107 | 01101011 | K | 075 | 01001011 |

l | 108 | 01101100 | L | 076 | 01001100 |

m | 109 | 01101101 | M | 077 | 01001101 |

n | 110 | 01101110 | N | 078 | 01001110 |

o | 111 | 01101111 | O | 079 | 01001111 |

p | 112 | 01110000 | P | 080 | 01010000 |

q | 113 | 01110001 | Q | 081 | 01010001 |

r | 114 | 01110010 | R | 082 | 01010010 |

s | 115 | 01110011 | S | 083 | 01010011 |

t | 116 | 01110100 | T | 084 | 01010100 |

u | 117 | 01110101 | U | 085 | 01010101 |

v | 118 | 01110110 | V | 086 | 01010110 |

w | 119 | 01110111 | W | 087 | 01010111 |

x | 120 | 01111000 | X | 088 | 01011000 |

y | 121 | 01111001 | Y | 089 | 01011001 |

z | 122 | 01111010 | Z | 090 | 01011010 |

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