Binary number system：練習問題

A 12-bit floating-point system uses 8 bits for the mantissa and 4 bits for the exponent. Both the mantissa and the exponent are stored using two's complement. The mantissa is normalized with the binary point after the first bit (e.g., 0.1... or 1.0...).

What is the decimal representation of the normalized bit pattern 10110000 0011?

Using 8-bit two's complement representation, what is the decimal equivalent of the binary pattern 1111 1100?

Convert the decimal value 0.375 into an unsigned 8-bit fixed-point binary representation with 4 bits for the integer part and 4 bits for the fractional part.

Explain the difference between underflow and overflow in a floating-point system.

A computer scientist is comparing fixed-point and floating-point representations for fractional numbers.

a) Represent the decimal value 6.375 in 8-bit fixed-point binary, where 5 bits are used for the integer part and 3 bits for the fractional part.
b) Identify one advantage of using floating-point over fixed-point for scientific calculations.
c) Explain why the decimal number 0.1 cannot be represented exactly in a binary fractional system.

A normalized floating-point system uses a 10-bit mantissa and a 6-bit exponent, both in two's complement.

a) Normalize the following unnormalized binary number: \(0.001011_2 \times 2^4\).
b) Calculate the decimal value of the following normalized floating-point number:
Mantissa: \(0.1101000000\)
Exponent: \(000011\)
c) What occurs during a floating-point underflow error?