Consider the following values, using the IEEE 754 single-precision floating-point format. What is the equivalent value as a decimal number?01011100101001000100000000000000Correct answer is '3.69'. Can you explain this answer?

Question

Answer

Given,

The IEEE 754 single-precision floating-point format:

01011100101001000100000000000000

Now, by converting it into decimal number we get,

Sign =0

Exponent =10111001=185

Fraction =01001000100000000000000

= (−1)⁰(1+0.010010001)2¹⁸⁵−¹²⁷

= (1+2⁻²+2⁻⁵+2⁻⁹) ×2⁵⁸

= 3.6986×10¹⁷

= 3.69

So, the equivalent value as a decimal number is 3.69.

Hence, the correct answer is 3.69.

Answer

Converting IEEE 754 single-precision floating-point format to a decimal number

The IEEE 754 single-precision floating-point format is a standard for representing floating-point numbers in computers. It consists of three parts: the sign bit, the exponent, and the fraction.

Binary representation

The given binary representation is 01011100101001000100000000000000.

Sign Bit

The leftmost bit is the sign bit. In this case, it is 0, which means the number is positive.

Exponent

The next 8 bits represent the exponent. In this case, it is 10111001.

To convert the exponent to decimal, we need to subtract the bias value. For single-precision floating-point format, the bias value is 127.

10111001 = 185

185 - 127 = 58

So the exponent is 58.

Fraction

The remaining bits represent the fraction. In this case, it is 01001000100000000000000.

To convert the fraction to decimal, we need to calculate the sum of the powers of 2 corresponding to each bit from left to right. The first bit after the decimal point corresponds to 2^(-1), the second bit corresponds to 2^(-2), and so on.

Sum of the powers of 2 = (0 * 2^(-1)) + (1 * 2^(-2)) + (0 * 2^(-3)) + (0 * 2^(-4)) + (1 * 2^(-5)) + (0 * 2^(-6)) + (0 * 2^(-7)) + (0 * 2^(-8)) + (1 * 2^(-9)) + (0 * 2^(-10)) + (0 * 2^(-11)) + (0 * 2^(-12)) + (0 * 2^(-13)) + (0 * 2^(-14)) + (0 * 2^(-15)) + (0 * 2^(-16)) + (0 * 2^(-17)) + (0 * 2^(-18)) + (0 * 2^(-19)) + (0 * 2^(-20)) + (0 * 2^(-21)) + (0 * 2^(-22)) + (0 * 2^(-23)) + (0 * 2^(-24)) + (0 * 2^(-25)) + (0 * 2^(-26)) + (0 * 2^(-27)) + (0 * 2^(-28)) + (0 * 2^(-29)) + (0 * 2^(-30)) + (0 * 2^(-31))

Sum of the powers of 2 = 0.25 + 0.03125 = 0.28125

Decimal value

To calculate the decimal value, we multiply the fraction by 2 raised to

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