Binary Number System and Its Representation

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.

For 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 signaled. And we say that the signal state corresponds to 1.

If the current is not flowing, it is not signaled. The not signal state corresponds to 0. So, 1 and 0, in binary, translate to a signal or non-signal in an electrical device, and we can have multiple wires or multiple inputs to represent multiple ones and zeros.

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

As we see, 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 a number expressed in the base-2 number system 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 that only the two 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 ) × $2^{5}$ ] + [ ( 0 ) × $2^{4}$ ] + [ ( 0 ) × $2^{3}$ ] + [ ( 1 ) × $2^{2}$ ] + [ ( 0 ) × $2^{1}$ ] + [ ( 1 ) × $2^{0}$]

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

$100101_{2}$ = $37_{10}$

---

Below is a widget that shows output of these concepts in HTML. In the next lessons, we will cover output more in-depth.

Decimal to binary representation

Below is the 32-bit binary representation.

5 -> 00000000 00000000 00000000 00000101

Below is another widget that shows output in HTML. In the next lessons, we will cover this topic in-depth.

---

Representing decimals & ASCII in binary

A computer only understands byte-code that is made of 0’s and 1’s. We have to represent every decimal character as binary digits so a computer can understand the instructions we give.

Decimal numbers in binary (8-bit representation)

Each software programming language uses its own 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