Question

What is the bit pattern (raw binary) of the single precision representation of the decimal number 0.125 following IEEE 754 standard

Answer 1

Answer:

The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) = 0 (a positive number)

Exponent (8 bits) = 0111 1100

Mantissa (23 bits) = 000 0000 0000 0000 0000 0000

Explanation:

1. First, convert to the binary (base 2) the integer part: 0.

Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.

division = quotient + remainder;

0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) = 0(2)

3. Convert to the binary (base 2) the fractional part: 0.125.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.

#) multiplying = integer + fractional part;

1) 0.125 × 2 = 0 + 0.25;

2) 0.25 × 2 = 0 + 0.5;

3) 0.5 × 2 = 1 + 0;

4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:

0.125(10) =

0.001(2)

5. Positive number before normalization:

0.125(10) =

0.001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 3 positions to the right so that only one non zero digit remains to the left of it:

0.125(10) =

0.001(2) =

0.001(2) × 20 =

1(2) × 2-3

7. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign: 0 (a positive number)

Exponent (unadjusted): -3

Mantissa (not normalized):  1

8. Adjust the exponent.

Use the 8 bit excess/bias notation:

Exponent (adjusted) =

Exponent (unadjusted) + 2(8-1) - 1 =

-3 + 2(8-1) - 1 =

(-3 + 127)(10) =

124(10)

9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:

division = quotient + remainder;

124 ÷ 2 = 62 + 0;

62 ÷ 2 = 31 + 0;

31 ÷ 2 = 15 + 1;

15 ÷ 2 = 7 + 1;

7 ÷ 2 = 3 + 1;

3 ÷ 2 = 1 + 1;

1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above:

Exponent (adjusted) =

124(10) =

0111 1100(2)

11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 23 bits, by adding the necessary number of zeros to the right.

Mantissa (normalized) =

1 000 0000 0000 0000 0000 0000 =

000 0000 0000 0000 0000 0000

12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) = 0 (a positive number)

Exponent (8 bits) = 0111 1100

Mantissa (23 bits) = 000 0000 0000 0000 0000 0000