15 points Determine the binary form of the following numbers

(15 points) Determine the binary form of the following numbers:
5.78
87.2
0.005

Solution

as given we need to find the bnary form of the given numbers.

how to conevrt integer part :

Divide the integer part by note the quotient and remainder again divide the quotient by 2 and not the new quotient and remainder.Repeat the process till we have 0 as quotient and write the remainder from bottom to top.

which is the binary equivalent of given integer part.

how to convert fraction part :

multiply farction part with 2 note the product and integer part and again multiply the fraction of the previous product with 2 and note the new product and integer part.

Repeat the prcess till the pattern repeats ot upto to 8 to 10 bits and write the integer part from top to bottom which is the binary equivalent of given fraction

1)

5.78

in integer part we have 5 and in fraction part we have 0.78

first we convert integer part to binary form

quotient remainder

5/2 2 1

2/2 1 0

1/2 0 1

now write the remainder from bottom to top

we can say that 5₁₀ = 101₂

now we will convert fraction 0.78 to binary

product integer part

0.78*2 1.56 1

0.56*2 1.12 1

0.12*2 0.24 0

0.24*2 0.48 0

0.48*2 0.96 0

0.96*2 1.92 1

0.92*2 1.84 1

0.84*2 1.68 1

0.68*2 1.36 1

write the integer part from top to bottom we have 0.78₁₀ = 0.110001111

hence we can say that 5.78₁₀ = 101.110001111₂

2)

we have 87.2

in integer part we have 87 and in fraction part we have 0.2

first we convert integer part to binary form

quotient remainder

87/2 43 1

43/2 21 1

21/2 10 1

10/2 5 0

5/2 2 1

2/2 1 0

1/2 0 1

now write the remainder from bottom to top

we can say that 87₁₀ = 1010111₂

now we will convert fraction 0.2 to binary

product integer part

0.2*2 0.4 0

0.4*2 0.8 0

0.8*2 1.6 1

0.6*2 1.2 1

pattern repeats now so,

write the integer part from top to bottom we have 0.2₁₀ = 0.0011₂

hence we can say that 87.2₁₀ = 1010111.0011₂

3)

we have a number 0.005

in integer part we have 0 and in fraction part we have 0.005

so binary equivalent of integer part is 0

we will find the binary equivalent of fraction part as :

product integer part

0.005*2 0.01 0

0.01*2 0.02 0

0.02*2 0.04 0

0.04*2 0.08 0

0.08*2 0.16 0

0.16*2 0.32 0

0.32*2 0.64 0

0.64*2 1.28 1

0.28*2 0.56 0

0.56*2 1.12 1

0.12*2 0.24 0

0.24*2 0.48 0

0.48*2 0.96 0

write the integer part from top to bottom we have 0.005₁₀ = 0.0000000101000₂

so we can say that 0.005₁₀ =  0.0000000101000₂