Question

a. Convert the following decimal numbers to 32-bit floating point number. 1. 25.875 2.-7.0625 b. What decimal value is represented by the following 32-bit floating point number? 1. C10B000016 2. 46AB000016

please answer with steps.

Answer 1

a.

1. 25.875

--convert the integer part 25 to binary by dividing it with 2 until the quotient becomes 0.

Division=quotient remainder

25÷2= 12 1

12÷2= 6 0

6÷2=3 0

3÷2=1 1

1÷2=0 1

-- take the remainder values from down to up.

11001

---convert the fraction part .875 to binary by multiplying with 2 and take integer part of result until the fraction part becomes 0.

Multiplication = integer part fraction part

0.875 × 2 = 1 0.75

0.75 × 2 =1 0.5

0.5 × 2 =1 0

Take integer part value = 111

Total integer+ fraction value = 11001.111

Move decimal point to left for 4 places= 1.1001111×2^4 here mantissa is 1001111.

E=127+4 = 131(10000011)

Floating point representation:

Sign E Mantissa

0 10000011 1001 111 1111 1111 1111 1111

2. -7.0625

Follow the same steps as above.

Integer part 7

Division = quotient remainder

7÷2 = 3 1

3÷2 = 1 1

1÷2 = 0 1

7 value is 111

-- now fraction part .0625

Multiplication = integerpart fraction part

0.625×2 = 0 0.125

0.125×2 = 0 0.25

0.25×2 = 0 0.5

0.5×2 = 1 0

0.625 value is : 0001

Total value : 111.0001

Decimal point move to left for 2 places: 1.110001×2^2

E=127+2=129 (10000001)

Floating point representation:

S E Mantissa

1 10000001 1100 0100 0000 0000 0000 000

b.

1.given : C 1 0 B 0 0 0 0

- convert to binary

1100 0001 0000 1011 0000 0000 0000 0000

S E Mantissa

1st bit ---- S

Next 8 bits -----E

Next 23 bits -----mantissa

E= 10000010 =130

Exponential bias = 130-127 = 3

Mantissa part 000 1011 0000 ...........

2^-4 + 2^-6 + 2^-7 = 0.0859375

Decimal value will be obtained by the following formula :

(-1)^S × (1+Mantissa ) × 2^E

(-1)^1 × (1+0.0859375)×2^3

=-8.6875

2. Given : 4 6 A B 0 0 0 0

Binary value :

0100 0110 1010 1011 0000 0000 0000 0000

S E Mantissa

1st bit ---S

Next 8 bits ---E

Next 23 bits -- Mantissa

E= 10001101 = 141

Exponential bias : 141-127=14

Mantissa part : 010 1011 00000 .......

2^-2 +2^-4 + 2^-6 +2^-7 = 0.3359375

Binary value will be obtained from the formula :

(-1)^S × (1+Mantissa )×2^E

(-1)^0 × (1+0.3359375) × 2^ 14

=21888