1.2 Positional Number Systems

In this system, numbers are expressed as a sequence of digits, with each digit's position carrying a specific weight. The overall value of the number is determined by adding up these weighted digits. Stated more simply, it's the total of each digit multiplied by its corresponding place value.

A number D in the form d_nd_n−1…d₁d₀.d₋₁d₋₂…d_−n has the value

where:

d = digit

r = base or radix

i = digit position

rⁱ = weight of the digit

p = number of digits to the left of the radix point

n = number of digits to the right of the radix point

Part 1.2.1 Decimal Number System

We use decimal numbers in day to day life. This is the first number system we learn way back to our grade school. They are based on the number 10. They are represented by 0,1,2,3,4,5,6,7,8 and 9.

Decimal number representation.

Back in elementary school, we learned about place value. We understood that numbers are made up of digits, and each digit has a specific value based on its position.

Example No. 1

For Example, the Number 1995.

We represent this number as:

Position	Digit	Multiplier
Ones	5	x 10^0	5
Tens	9	x 10^1	90
Hundreds	9	x 10^2	900
Thousands	1	x 10^3	1000

After adding all the numbers, we get 1995.

Example No. 2

For Example, the Number 3.141.

We represent this number as:

Position	Digit	Multiplier
Thousandths	1	x 10^-3	.001
Hundredths	4	x 10^-2	.04
Tenths	1	x 10^-1	.1
Ones	3	x 10^0	3

After adding all the numbers, we get 3.141.

Part 1.2.2 Binary Number System

Binary numbers, the foundation of computer systems, use only two digits: 0 and 1. This simple system allows computers to process information efficiently.

Note: Binary numbers, denoted by the subscript 2.

Binary number representation.

Unlike decimal numbers that use 10 as base, binary numbers use 2 as base.

Example No. 3

For Example, the Number 11100011₂

We represent this binary number as:

Index	Digit	Multiplier
0	1	x 2^0	1
1	1	x 2^1	2
2	0	x 2^2	0
3	0	x 2^3	0
4	0	x 2^4	0
5	1	x 2^5	32
6	1	x 2^6	64
7	1	x 2^7	128

By adding the numbers, we get 99.

Example No. 4

For Example, the Number 11111111₂

We represent this binary number as:

Index	Digit	Multiplier
0	1	x 2^0	1
1	1	x 2^1	2
2	1	x 2^2	4
3	1	x 2^3	8
4	1	x 2^4	16
5	1	x 2^5	32
6	1	x 2^6	64
7	1	x 2^7	128

By adding the numbers, we get 255.

Example No. 5

For Example, the Number 1111.1111₂

We represent this binary number as:

Index	Digit	Multiplier
-4	1	x 2^-4	0.0625
-3	1	x 2^-3	0.125
-2	1	x 2^-2	0.25
-1	1	x 2^-1	.5
0	1	x 2^0	1
1	1	x 2^1	32
2	1	x 2^2	64
3	1	x 2^3	128

By adding the numbers, we get 15.9375.

Part 1.2.3 Octal Number System

Octal numbers are a number system based on eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. They are often used in computer science and digital electronics as a more compact representation of binary numbers.

Note: Octal numbers, denoted by the subscript 8.

Octal number representation.

Octal numbers use the base 8.

Example No. 6

For Example, the number 1567₈

We represent this octal number as:

Index	Digit	Multiplier
0	7	x 8^0	7
1	6	x 8^1	48
2	5	x 8^2	320
3	1	x 8^3	512

By adding the numbers, we get 887.

Part 1.2.4 Hexadecimal Number System

This number is more compact if compared to binary and octal numbers. Its use 0,1,2,3,4,5,6,7,8,9, A, B, C, D, E and F.

Note: Hexadecimal numbers, denoted by the subscript 16.

Hexadecimal Number Representation.

Hexadecimal number used the base 16.

Example No. 7

For Example, the number 15AC₁₆

We represent this Hexadecimal number as:

Index	Digit	Multiplier
0	C = 12	x 16^0	12
1	A = 10	x 16^1	160
2	5	x 16^2	1280
3	1	x 16^3	4096

By adding the numbers, we get 5548.