Numbers and Arithmetic

Numbers are fundamental objects in mathematics used to count, measure, and label. They form the basis of many fields of study and are crucial in everyday life. Over time, various types of numbers have been developed to solve different kinds of problems.

Numbers

Types of Numbers

• Natural Numbers: 1, 2, 3, 4, ... (used for counting).
• Whole Numbers: 0, 1, 2, 3, ... (natural numbers plus zero).
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (positive and negative whole numbers).
• Rational Numbers: Numbers that can be expressed as a fraction of two integers.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction (like √2, π).
• Real Numbers: All rational and irrational numbers.

Representation of Numbers

• Decimals: Numbers are usually represented in decimal (base 10) form. Reference: Decimals.
• Fractions: Rational numbers can also be represented by fractions. Reference: Fractions.
• Percentages: Percentages are a way of expressing a number as a fraction of 100. For example, 50% is equivalent to 50/100 or 0.5. Reference: Percentages.
• Standard Index Form: This is a variation on the decimal form, especially useful for very large or very small numbers. Reference: Standard Index Form.
• Ratios: A ratio is a comparison of two quantities by division. For example, the ratio 3:2 means that for every 3 units of the first quantity, there are 2 units of the second quantity. Reference: Ratios.
• Other Bases: Numbers can be represented in bases other than 10 (decimal). Typically computers esentially work in base 2 or binary representation. Reference: Change of Base, Binary Representation of Numbers.

Arithmetic

Arithmetic is the branch of mathematics that deals with basic operations on numbers.

Basic Operations

The operations that form the foundation of most mathematical concepts include:

• Addition - Combining two numbers to get their total.
• Subtraction - Finding the difference between two numbers.
• Multiplication - Repeated addition of a number.
• Division - Splitting a number into equal parts.

Integer Arithmetic

Integer arithmetic involves operations such as addition, subtraction, multiplication, and division performed on integers. Beyond basic operations, integer arithmetic includes important concepts like finding the Highest Common Factor (HCF), the Lowest Common Multiple (LCM), distinguishing prime and composite numbers, and understanding modular arithmetic.

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

The Highest Common Factor (HCF) of two or more integers is the largest positive integer that divides each of the numbers without a remainder. It is also known as the Greatest Common Divisor (GCD). Example: HCF of 18 and 24 is 6.

The Lowest Common Multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of each of the numbers. Example: LCM of 4 and 5 is 20.

Reference: Lowest Common Multiple : Highest Common Factor.

Prime and Composite Numbers

Prime Numbers are numbers greater than 1 that have exactly two distinct positive divisors: 1 and themselves. Examples: 2, 3, 5, 7, 11, 13, 17...

Composite Numbers are numbers greater than 1 that have more than two positive divisors. Examples: 4, 6, 8, 9, 12, 15...

Reference: Prime and Composite Numbers.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value called the modulus. Example: In modulo 5 arithmetic, 7 ≡ 2 (mod 5), because when 7 is divided by 5, the remainder is 2.

Reference: Modular Arithmetic.

Powers, Roots and Logarithms

Powers, roots and logarithms are fundamental operations in mathematics that extend the concepts of multiplication and division. They are used widely in algebra, calculus, physics, engineering, and computer science.

Powers and Roots

Powers (Indices or Exponents)

A power represents repeated multiplication of the same number. For a number a raised to the power n, it is written as aⁿ, meaning:

aⁿ = a × a × ... × a (n times)

Examples:

• 2³ = 2 × 2 × 2 = 8
• 5² = 5 × 5 = 25

Roots

A root is the inverse operation of raising a number to a power. The square root of a number a is a number that, when multiplied by itself, gives a. It is denoted as √a.

Examples:

• √9 = 3, because 3² = 9
• ∛8 = 2, because 2³ = 8

There are square roots, cube roots, and higher-order roots depending on the degree of the root.

Reference: Powers and Roots.

Logarithms

A logarithm answers the question: "To what exponent must a base number be raised to produce a given number?"

The logarithm of a number a with base b is written as log_b(a) and satisfies: b^x = a is equivalent to log_b(a) = x

Examples:

• log₂(8) = 3, because 2³ = 8
• log₁₀(1000) = 3, because 10³= 1000

Common types of logarithms include:

• Common logarithm (log): Base 10
• Natural logarithm (ln): Base e (where e is 2.718 (3 d.p.))

Reference: Logarithms.

Rules of Arithmetic (BODMAS)

Reference: BODMAS: Rules of Arithmetic.