Number System Base

Topic 1 : Number System

Number System Base

The number system is a basic symbol to represent a set of quantities.

Consider what the number 12345.321 means:

12345.321 = (1 x 10000) + (2 x 1000) + (3 x 100) + (4 x 10) + 5 +

 (3 x 0.1) + (2 x 0.01) + (1 x 0.001)

The example above using a system based on decimal digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, to represent numbers and refer to the system as the decimal system.

The decimal system is said to have a base of 10. Each digit in the number is multiplied by 10 raised to a power corresponding to that digit’s position:

12345.321 = (1 x 10⁴) + (2 x 10³) + (3 x 10²) + (4 x 10¹) + (5 x 10⁰) +

 (3 x 10^-1) + (2 x 10^-2) + (1 x 10^-3)

The decimal system is one of type of number system and there are many types of number system.

Here we only focus on the decimal, hexadecimal and binary number.

System	Base	Possible Digits
Binary	2	0 1
Decimal	10	0 1 2 3 4 5 6 7 8 9
Hexadecimal	16	0 1 2 3 4 5 6 7 8 9 A B C D E F

What is Decimal number?

ü Base of 10.

ü The number consists from 0 to 9.

üIn general, for the decimal representation of 

   the value of X is

   

 What is Binary number?

ü Base of 2.

ü The number consists 0 and 1 only.

ü In general, for the binary representation of 

the value of Y is

What is Hexadecimal number?

ü Base of 16.

ü The composer number start from 0 to F

Decimal (base 10)	Binary (base 2)	Hexadecimal (base 16)
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
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F

Number System Conversion

1.   Convert decimal number 120.125 to binary number.

Consider integer part:

120 = 1111000₂

Consider fraction part:

0.125 = 0.001₂

Therefore, 120.125 = 1111000.001₂

   2 .  Convert binary number 1111011.01 to decimal number.

1111011.01 = (1 x 2⁰) + (1 x 2¹) + (0 x 2²) + (1 x 2³) + (1 x 2⁴) +

                      (1 x 2⁵) + (1 x 2⁶) + (0 x 2^-1) + (1 x 2^-2)

                   = 123.25

   3.   Convert decimal number 100.5 to hexadecimal.

                         

Consider integer part:

100 = 64₁₆

Consider fraction part:

0.5 = 0.8₁₆

Therefore, 100.5 = 64.8₁₆

   4.   Convert hexadecimal number 1234.9₁₆ to binary number.

1 = 0001₂

2 = 0010₂

3 = 0011₂

4 = 0100₂

9 = 1001₂

Therefore, 1234.9₁₆ = 1001000110100.1001₂