Basic Arithmetic Classrooms

1.1 Introduction to Basic Arithmetic

Basic arithmetic operations are the foundation of all mathematics. Mastering these concepts is crucial for everyday life and for tackling more complex mathematical problems. This section will review addition, subtraction, multiplication, division, HCF, LCM, and the order of operations.

Each example is designed to be followed step-by-step. Click "Show Next Step" to reveal the next part of the solution and explanation, just like a teacher would guide you.

1.2 Addition & Subtraction Review

Addition combines quantities, while subtraction finds the difference between them. Understanding these operations with various types of numbers (positive, negative, decimals, fractions) is key.

1.3 Multiplication & Division Review

Multiplication is repeated addition, and division is the process of splitting a quantity into equal parts. These operations have specific rules, especially concerning positive and negative numbers.

1.4 HCF & LCM

The Highest Common Factor (HCF) and Lowest Common Multiple (LCM) are fundamental concepts for working with fractions and understanding number properties.

1.5 Order of Operations & Brackets

To ensure consistency in mathematical calculations, a specific order of operations (often remembered as PEMDAS/BODMAS) must be followed. Brackets are used to override this natural order.

1.6 Power, Roots & Indices

Powers (also known as exponents or indices) and roots are essential mathematical operations. A power indicates how many times a base number is multiplied by itself, while a root is the inverse operation, finding the base that was multiplied.

• Power: $a^n$ means 'a' multiplied by itself 'n' times. For example, $2^3 = 2 \times 2 \times 2 = 8$.
• Root: $\sqrt[n]{a}$ means the number that, when multiplied by itself 'n' times, equals 'a'. For example, $\sqrt[3]{8} = 2$. The square root $\sqrt{a}$ is when $n=2$.
• Indices (laws of exponents): Rules that simplify expressions involving powers, such as $a^m \times a^n = a^{m+n}$.

1.7 Ratio & Proportion

Ratio is a way of comparing two or more quantities of the same kind. It shows how much of one quantity there is compared to another. Ratios can be written as $a:b$ or $\frac{a}{b}$.

Proportion is a statement that two ratios are equal. If $a:b = c:d$, then $a, b, c, d$ are in proportion. This means $\frac{a}{b} = \frac{c}{d}$.

1.8 Simple Linear Equations

A linear equation is an equation where the highest power of the variable (usually 'x') is 1. When plotted on a graph, it forms a straight line. Solving a linear equation means finding the value of the variable that makes the equation true.

The goal is to isolate the variable on one side of the equation. We do this by performing inverse operations to both sides of the equation, maintaining balance.

1.9 Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. Its standard form is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are coefficients and $a \neq 0$. These equations can have two, one, or zero real solutions.

Common methods for solving quadratic equations include factoring, completing the square, and using the quadratic formula.