Question

Here is a number represented in the IEEE standard single precision format

11010101010101000000000000000000
What is the decimal number represented by this string in the IEEE standard single
precision floating point format?

Answer 1

The IEEE standard single precision floating point number 11010101010101000000000000000000 is converted to a decimal by identifying the sign, exponent, and mantissa. After converting the exponent and mantissa to decimal and considering the sign bit, the number is represented as -1.625 x 2^43.

Step-by-step explanation:

The IEEE standard single precision floating point format number 11010101010101000000000000000000 represents a decimal number. To convert this, we need to break it down into three parts:

• The sign bit: '1', which indicates a negative number.
• The exponent: '10101010', an 8-bit value which is 170 in decimal. The actual exponent is calculated by subtracting 127 (the bias for single precision) from this value, resulting in 43.
• The mantissa (or significand): '10101000000000000000000', which represents the significant digits of the number.

For IEEE 754 single precision, the leading bit of the mantissa (which is always 1 for normalized numbers) is not stored and is represented implicitly, so the actual mantissa we use in the calculation is '1.10101000000000000000000' (in binary).

Now we translate the exponent and mantissa to decimal form. The mantissa in decimal form is 1 + 0.5 (2-1) + 0.125 (2-3) which sums to 1.625. Then, we use the exponent to scale this value, resulting in 1.625 x 243. The sign bit tells us the number is negative, so the final value is -1.625 x 243, which we can also express in scientific notation if needed for clarity.