Binary Representation - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Binary Representation

How a Decimal Number is Represented

2 1 0 1 2

− −

= × + × + × + × + ×

Convert Base 10 Integer to binary

representation

Table 1 Converting a base-10 integer to binary representation. Quotient Remainder 11/2 5 5/2 2 2/2 1 1/2 0

1 = a 0

1 = a 1

0 = a 2

1 = a 3

Hence

2

10 3 2 1 0 2 ( 1011 )

( 11 ) ( )

=

= a a a a

Start

Input (N) (^10)

i = 0

Divide N by 2 to get quotient Q & remainder R

ai = R

Is Q = 0?

n = i (N) 10 = (an.. .a 0 ) (^2)

STOP

Integer N to be converted to binary format

i=i+1,N=Q

No

Yes

Start

Input (F) (^10)

Multiply F by 2 to get number before decimal, S and after decimal, T

ai = R

Is T =0?

n = i (F) 10 = (a-1.. .a-n) (^2)

STOP

Fraction F to be converted to binary format

No

Yes

i= i−1,F= T

i =− 1

Decimal Number to Binary

( ) ( )

( 11 ) 10 = ( 1011 ) 2

( 0. 1875 ) 10 = ( 0. 0011 ) 2

and

we have ( 11. 1875 ) 10 = ( 1011. 0011 ) 2

Since

Another Way to Look at Conversion

Convert (^11.^1875 ) 10 to base 2

3 2 1 0

3 1 0

3 1

3 10

1011

1 2 0 2 1 2 1 2

2 2 2

2 2 1

11 2 3

=

= × + × + × + ×

= + +

= + +

= +

1 2 3 4

3 4

3 10

= × + × + × + ×

− − − −

− −

−