Show the IEEE 754 singleprecision representation of 1234567

Show the IEEE 754 single-precision representation of 123.4567. Display the 32-bits of the representation. Round mantissa to nearest even. Show work.

Solution

For a 32 bit float representation: the 32nd bit will be used to represent the sign. And the bits from 31st bit to 24th bit i.e., a total of 8 bits will represent the exponent, and the remaining 23 bits represent the mantissa.

As the given number is: 123.4567, its a positive number. So, the leading bit is 0.

And the integral part is: 123₁₀ = 01111011₂ 133 - 127 = 6. Therefore, 10000101.

The fractional part is: .4567₁₀ = .01110₂

.4567 x 2 = 0.9134

.9134 x 2 = 1.8268

.8268 x 2 = 1.6536

.6536 x 2 = 1.3072

.3072 x 2 = 0.6144

.6144 x 2 = 1.2288

.2288 x 2 = 0.4576

.4576 x 2 = 0.9152

.9152 x 2 = 1.8304

.8304 x 2 = 1.6608

.6608 x 2 = 1.3216

.3216 x 2 = 0.6432

.6432 x 2 = 1.2864

.2864 x 2 = 0.5728

So, the binary representation is:  01111011.01110 = 1.1110110111010011 x 2⁶.

Therefore, the answer is: 100001011110110111010011