Arithmetic and Logical Operations

Data

Represents fundamental information such as numbers, characters, and other values.

Data are particular values that conform to specific types and structures

Examples of Simple Value Data (Primitives)

• Integers
• Floating-point numbers
• Characters
• Booleans

Transmission

Data is transmitted between components of a computer system electronically

Voltage

Electric potential difference: energy per unit charge.

Current

The rate of electric charge flow.

Binary

Most computers today are binary digital machines that electronically process data in the form of binary digits (bits)

Digital vs. Analog

Digital has a finite number of values, while analog has an infinite number of values.

Binary

Binary is a base-2 number system that uses two digits, 0 and 1. Therefore, it can be represented in 2 states.

Example

A low state can be represented as a voltage range of 0-0.8V, while a high state can be represented as a voltage range of 2-5V.

Bit

The basic unit of information in computing and digital communications, representing a single binary digit.

Grouping Bits

Bits are grouped into larger units to represent more complex data.

• Byte: 8 bits (representing 256 values)
• 10 bits: 1,024 values
• 12 bits: 4,096 values
• $n$ bits: $2^n$ values

Tip

To calculate the number of bits needed to represent $n$ values, use $b=\lceil {\log }_{2}n\rceil$ bits, for $n\ge 1$

Encoding

The process of converting one representation of data into another representation.

Weighted Positional Notation

Given a base $b$ and a set of $n$ digits, the value of a number is calculated as:

Base Systems

• Base-2: Binary
• Base-8: Octal
• Base-10: Decimal (most common)
• Base-16: Hexadecimal
• Base-64: Used in encoding

Most Significant Digit (MSD)

The digit that represents the largest value in a number.

Least Significant Digit (LSD)

The digit that represents the smallest value in a number.

Unsigned Integers

Unsigned integers are nonnegative integers that are represented in binary form.

Signed Integers

A data type to represent both positive and negative numbers.

Sign-Magnitude

The most significant bit (MSB) represents the sign of the number.

• 0: Positive
• 1: Negative

However, this method has two representations for zero.

One’s Complement

The negative of a number is represented by flipping all the bits.

However, this method has two representations for zero.

Two’s Complement

The negative of a number is represented by flipping all the bits and adding 1.

This method has only one representation for zero.

What is the 2's complement of the following (taking the 2's complement)

• $01000_2\to 10111_2+1=11000_2$
• $10100_2\to 01011_2+1=01100_2$
• $00011_2\to 11100_2+1=11101_2$
• $11111_2\to 00000_2+1=00001_2$

Two’s complement is the most common method for representing signed integers, as it is preferred for arithmetic operations.

Related: lecture-02, lecture-03, lecture-04, lecture-05

Source

Original retained in wisconsin-legacy.