Digital Arithmetic Operations Quiz - Topic 3

Storing integers without a sign, using bits to represent the value, from 0 to 2^N -1.

An 8-bit unsigned integer can store values from 0 to 255.

A 16-bit unsigned integer can store values from 0 to 65,535.

The formula for representing an unsigned integer from decimal to binary is by changing the integer to its binary equivalent, filling with leading zeros until the desired number of bits (N) is reached

An integer value can be stored in a computer's memory through its conversion to binary representation and filling with leading zeros.

Negation is the process of converting a positive number to its negative equivalent or a negative number to its positive equivalent. In the context of signed binary numbers, it's achieved by performing the 2's complement operation.

Negation in the 2's complement system is performed by taking the 2's complement of the binary representation of the number. This involves two steps: 1. Invert all the bits (0s become 1s and 1s become 0s). 2. Add 1 to the result.

For positive numbers, the 2's complement representation is the same as the original binary representation. For example, +9 in binary is 00001001 in the 2's complement system.

For negative numbers, the 2's complement is calculated by inverting all the bits of the binary representation of the magnitude of the number and then adding 1. For example, -9 in binary is 11110111 in the 2's complement system.

The 2's complement representation is used for signed binary numbers because it allows for efficient addition and subtraction of both positive and negative numbers using the same logic unit. This simplifies hardware design and reduces complexity.

To represent +13 in binary form, the true magnitude (13) is represented in 5 bits: 01101. Since it's positive, the sign bit is 0 resulting in 001101.

To represent -9 in binary form, first find the binary representation of the magnitude (9): 01001. Then, invert the bits: 10110. Finally, add 1: 10111. The sign bit for negative numbers is 1, resulting in 110111.

With five bits, you can represent numbers from -16 to +15. This includes 0, which makes a total of 32 possible values.

The first step in floating-point addition/subtraction is to check if either operand is zero. If either operand is zero, the result is simply the other operand.

The second phase of floating-point addition/subtraction involves aligning the significands. This means manipulating the exponents to make them equal. The significand of the smaller exponent is shifted right until the exponents are equal.

The third phase adds or subtracts the aligned significands, taking into account their signs. This can result in a zero result or a significand overflow.

The final phase normalizes the result. This involves shifting the significand left or right and adjusting the exponent to ensure the result is in the correct format.

In some floating-point formats, the most significant bit of the significand is assumed to be 1 and is not explicitly stored. This is called the implicit significand bit. During addition/subtraction, this bit must be made explicit for the ALU calculations.

The process of shifting the significand digits left until the most significant digit (bit or 4 bits for base-16) is non-zero. Each shift decrements the exponent, potentially causing underflow.

The process involves aligning the exponents, adding/subtracting the significands, normalizing the result, and finally rounding.

The process involves multiplying the significands, adding the exponents, normalizing the result, and rounding.

The process involves dividing the significands, subtracting the exponents, normalizing the result, and rounding.

Extra bits used during floating-point operations to improve accuracy by retaining bits that would otherwise be lost during truncation.

The process of approximating a result to the nearest representable value in the floating-point system.

A rounding method that chooses the nearest representable number to the result.

A rounding method that rounds the result up toward positive infinity.

A rounding method that rounds the result down toward negative infinity.

A rounding method that rounds the result toward zero, regardless of its sign.

A condition where the result of an addition is larger than the maximum value that can be stored in the available word size. This can happen when adding two positive numbers or two negative numbers.

A method of representing signed integers where the negative value of a binary number is obtained by inverting all the bits and adding 1. This representation allows for simpler arithmetic operations.

Subtraction is achieved by taking the twos complement of the subtrahend (the number being subtracted) and adding it to the minuend (the number being subtracted from).

The twos complement representation simplifies subtraction because it allows us to use addition instead. By taking the twos complement of the subtrahend, we effectively convert subtraction into an addition operation.

Visualizing twos complement representation on a circle helps understand its properties. The circle represents the range of possible values, with positive numbers on one half and negative numbers on the other. The twos complement of a number is located directly opposite that number in the circle.

The core component for addition and subtraction is a binary adder. This adder takes two numbers as input and produces the sum and an overflow indicator. The operation is performed with the help of registers for storing the numbers and a complementer for handling subtraction.

Multiplication of unsigned binary integers is performed bit by bit, similar to decimal multiplication. Each bit in the multiplier is multiplied by the multiplicand, and the results are added together while shifting them to the left.

The hardware implementation uses a series of steps: multiplying each bit of the multiplier with the multiplicand, storing intermediate results in registers, adding and shifting the results across multiple cycles.

A faster algorithm for twos complement multiplication that uses a sequence of additions and subtractions. It's based on examining pairs of bits in the multiplier to decide whether to add, subtract, or do nothing.

To multiply 7 and 3 using Booth's algorithm, you analyze the multiplier (3) bit by bit. You perform a series of additions or subtractions based on the pairs of bits, then shift the result and the multiplier.

Division involves repeatedly subtracting the divisor from the dividend until it's no longer possible. The count of subtractions represents the quotient, while the remainder is the final difference.

A way of representing numbers with both an integer and a fractional part. It uses a base (usually 2 for binary) and an exponent to represent the magnitude of the number.

Fixed-point notation struggles to represent very large or very small numbers accurately due to the fixed position of the decimal point. It can also lose fractional parts during division.

A condition in floating-point operations where the exponent exceeds the maximum value that can be represented. This can result in an infinite value (positive or negative).

A condition in floating-point operations where the exponent is smaller than the minimum value that can be represented, potentially leading to a very small or zero value.