The binary number system: Practice Questions

What is the maximum decimal value that can be represented using an 8-bit unsigned binary integer?

An 8-bit register stores the unsigned binary integer \(10110001_2\). If the unsigned binary integer \(01101111_2\) is added to it, what will be the 8-bit result and the status of the overflow flag?

A floating point system uses an 8-bit mantissa and a 4-bit exponent, both in two's complement. The binary point in the mantissa is after the first bit.
Normalize the following un-normalized value: Mantissa: \(00011000\) Exponent: \(0101\).

What is the decimal equivalent of the 8-bit two's complement binary integer \(11110101_2\)?

The decimal value 8.0 is intended to be stored, but due to a system error, it is represented as 7.75. Calculate the relative error of this representation.

Calculate the maximum unsigned integer value that can be represented using \( 10 \) bits, and provide the formula used for this calculation.

Determine the range of decimal integers (both minimum and maximum) that can be represented using an 8-bit signed integer in two's complement format.

A computer system uses 8-bit two's complement binary to represent signed integers.

(a) Convert the decimal value \(-43_{10}\) into 8-bit two's complement binary. Show your working.
(b) Perform the following binary arithmetic operation: \(00110101_2 + 11010101_2\). Show your working and state the final result in decimal.
(c) Identify whether an overflow error has occurred in part (b) and justify your answer.

A 12-bit floating point system is used where 8 bits are allocated for the mantissa and 4 bits for the exponent. Both parts are represented in two's complement.

(a) Represent the decimal value \(-9.5_{10}\) in this floating point format. You must ensure the mantissa is normalised.
(b) Calculate the relative error when the value \(0.1_{10}\) is represented in a system that can only store it as the binary fraction \(0.00011001_2\). Show your working.
(c) Describe the trade-off between the number of bits allocated to the mantissa and the number of bits allocated to the exponent in terms of precision and range.