# Number Systems

### number systems introduction

### Decimal number systems:

For example we have a decimal number 8243 which is represented as 8*10^3+2*10^2+4*10^1+3*10^0.

For overall calculation we represent a number in decimal system with constants is,

A3A2A1A0.A-1A-2

representation: A3*10^3+A2*10^2+A1*10^1+A0*10^0.A-1*10^-1+A-2*10^-2

like above we can represent any number in decimal system.

Ex: 123.62

representation in decimal system: (1*10^2+2*10^1+3*10^0)

**.(**6*10^-1+2*10^-2)
the decimal number system is said to be of base, radix 10. Because it uses 10 digits and the coefficients are multiplied by powers of 10.

### Binary system:

the binary system is a different number system.the coefficients of the binary numbers system have two possible values : 0 and 1. for example, the binary number 110.11 is expressed in terms of base or radix 2.

(110.11)2= (1*2^2+1*2^1+0*2^0).(1*2^-1+1*2^-2)

=(4+2+0).(1/2+1/4)

=(6).(0.5+0.25)

=(6.75)10

*** In general, a number expressed in base r system has coefficients multiplied by powers of r ***

### Octal number system:

The radix or base 8 contains the system is known as octal number system.

for example (112)8=(1*8^2+1*8^1+2*8^0)

=(72)10

### Hexadecimal number system:

tTe radix or base is 16 for the hexadecimal system.

A-10

B-11

C-12

D-13

E-14

F-15

.

.

.

like this we gave the notation in hexadecimal system.

An example of a hexadecimal number is,

(B65F)16 = (B*16^3+6*16^2+5*16^1+F*16^0)

=(46687)10

This way we have the notations of number systems.

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