CSC 105 MT1-Deck 4: Binary Numbers & Number Systems

How do computers represent information?

Computers ultimately represent and process information using binary digits (0s and 1s).

Text, images, videos and numbers are encoded into binary.

Natural Numbers

Non-negative whole numbers: 0, 1, 2, 3, 4, etc.

Negative Numbers

Numbers less than zero. Examples: -1, -5 and -1.2.

Negative numbers can be integers or non-integers.

Integers

Whole numbers and their negatives, including zero.

Examples: -3, -1, 0, 2 and 7. Integers do not contain fractional parts.

Rational Numbers

Numbers that can be expressed as a fraction of two integers, where the denominator is not zero.

Examples: 3/4, -5, 2.4 and 0.125.

Number System

A method of representing numbers using a particular base and set of digits.

Examples: Decimal (base 10), binary (base 2) and hexadecimal (base 16).

Base (Radix)

The number of unique digits available in a number system.

Base 10 uses 0–9; base 2 uses 0–1; base 16 uses 0–9 and A–F.

Decimal Number System

A base-10 number system using the digits 0–9.

Each position represents a power of 10. Example: 943 = 9×100 + 4×10 + 3×1.

Binary Number System

A base-2 number system using only 0 and 1.

Each position represents a power of 2. Example: 1110₂ = 14₁₀.

Hexadecimal Number System

A base-16 number system using digits 0–9 and letters A–F.
The letters represent values 10–15.

Example: 1A₁₆ = 26₁₀.Hexadecimal Digit Values

Positional Notation

A system where a digit's value depends on its position and the number system's base.

Moving left increases the power of the base. The rightmost position starts at base⁰.

Power of the Base

Each digit's place value is calculated using the base raised to a power.

Decimal uses powers of 10, binary uses powers of 2 and hexadecimal uses powers of 16.

How do you convert binary to decimal?

Multiply each binary digit by its corresponding power of 2, starting with 2⁰ on the right, then add the results.

Example: 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11₁₀.

Binary-to-Decimal Conversion Example

Convert 1110₂ to decimal. From right to left, the place values are 1, 2, 4 and 8,

Calculate 1×8 + 1×4 + 1×2 + 0×1 = 14₁₀.

How do you convert hexadecimal to decimal?

Multiply each hexadecimal digit by its corresponding power of 16, starting with 16⁰ on the right, then add the results.

Remember that A–F represent 10–15.

Hexadecimal-to-Decimal Conversion Example

Convert 9E3A₁₆ to decimal.

Calculate 9×16³ + 14×16² + 3×16¹ + 10×16⁰ = 36864 + 3584 + 48 + 10 = 40506₁₀.

Binary Addition Rules

Binary addition uses only 0 and 1.

Rules: 0+0=0; 0+1=1; 1+0=1; 1+1=10₂ (write 0 and carry 1).

If three 1s are added, the result is 11₂ (write 1 and carry 1).

Carry (Binary Addition)

A value transferred to the next column on the left when a binary addition produces a result of 2 or more.

Example: 1+1=10₂, so write 0 and carry 1.

Binary Addition Example

Add 10011₂ + 10001₂.

Working right to left and carrying when necessary gives 100100₂. Decimal check: 19 + 17 = 36.

Binary Subtraction Rules

Binary subtraction uses these rules: 0-0=0; 1-0=1; 1-1=0; 0-1 requires borrowing from the next available 1 to the left.

Borrow (Binary Subtraction)

When subtracting 1 from 0, borrow from a higher binary position. Borrowing 1 from the next column represents 2 in the current column, allowing 10₂-1₂=1₂.

Binary Subtraction Example

Subtract 10011₂ - 10001₂. Working from right to left gives 00010₂, or simply 10₂. Decimal check: 19 - 17 = 2.

Augend, Addend, Sum, Minuend, Subtrahend & Difference

Binary addition: Augend = first number being added; addend = number added; sum = result.

Binary subtraction: Minuend = number being subtracted from; subtrahend = number being subtracted; difference = result.