Summary of Fundamental Mathematics Concepts

## Introduction This guide covers Grade 8 Mathematics topics: integers and whole numbers, algebraic expressions and equations, fractions, and exponents. Each section breaks concepts into small steps, gives clear rules, worked examples, and real-world uses to help you build confidence and skills. ## Integers & Whole Numbers > **Definition:** An *integer* is a whole number that can be positive, negative, or zero. A *whole number* is an integer that is zero or positive. ### Number System and Classification - Natural numbers: $1$, $2$, $3$, ... - Whole numbers: $0$, $1$, $2$, ... - Integers: $\ldots$, $-2$, $-1$, $0$, $1$, $2$, $\ldots$ | Subset | Example elements | |---|---:| | Natural | $1$, $2$, $3$ | | Whole | $0$, $1$, $2$ | | Integers | $-3$, $-2$, $-1$, $0$, $1$ | ### Properties of Operations (on integers) - **Commutative:** $a+b=b+a$, $ab=ba$. - **Associative:** $(a+b)+c=a+(b+c)$, $(ab)c=a(bc)$. - **Distributive:** $a(b+c)=ab+ac$. - **Identity elements:** Additive identity $0$ since $a+0=a$. Multiplicative identity $1$ since $a\cdot1=a$. ### Rules for Operations - Subtraction: "Keep, Change, Change". Example: $5-3=5+(-3)$ and $5-(-3)=5+3$. - Addition: same sign add and keep sign; different sign subtract absolute values and take sign of larger absolute value. - Multiplication/Division sign rules: positive times positive is positive; negative times negative is positive; negative times positive is negative. Fun fact: Negative numbers were once controversial and not widely accepted in mathematics until several centuries into its history. ### Order of Operations (PEDMAS/BODMAS) 1. Parentheses/Brackets 2. Exponents/Orders 3. Division and Multiplication (left to right) 4. Addition and Subtraction (left to right) Example: $$5 + (2 \times -3)$$ Evaluate inside brackets: $$2 \times -3 = -6$$ Then add: $$5 + (-6) = -1$$ ### Simplifying Expressions (integers) 1. Apply distributive property when needed. 2. Combine like terms (same variable and exponent). Example: $$2(x - 3) + 4x$$ Distribute: $$2x - 6 + 4x$$ Combine like terms: $$(2 + 4)x - 6$$ $$6x - 6$$ ## Algebra Essentials > **Definition:** An *algebraic expression* uses variables, constants, and operations. An *equation* states that two expressions are equal. ### Algebraic Language - Variable: symbol such as $x$, $y$. - Term: a single product of numbers and variables, e.g. $3x^2$. - Coefficient: numerical factor of a term, e.g. $7$ in $7x$. - Constant: a fixed number, e.g. $5$ in $3x+5$. ### Like Terms and Distributive Law - Like terms: same variable parts, e.g. $3x$ and $-5x$. - Distributive law: $a(b+c)=ab+ac$. Example: $3(x+4)=3x+12$. ### Solving Linear Equations (one variable) Goal: isolate the variable on one side. Steps: 1. Undo addition/subtraction. 2. Undo multiplication/division. 3. Check your solution by substitution. Example: $$3x-5=10$$ Add $5$ to both sides: $$3x=15$$ Divide both sides by $3$: $$x=5$$ Practice problems (set up and solve): - $$-3-(x+2)=x+8$$ - $$4+\frac{x-2}{4}=-2$$ - $$3^{x-1}=81$$ Fun fact: Exponential equations such as $3^{x-1}=81$ connect algebra to rapid growth models like population growth and compound interest. ## Fractions & Exponents > **Definition:** A *fraction* $\frac{a}{b}$ represents division of $a$ by $b$. An *exponent* $x^n$ means multiply $x$ by itself $n$ times. ### Fraction Types | Type | Example | |---|---:| | Proper fraction | $\frac{3}{4}$ | | Improper fraction | $\frac{5}{4}$ | | Mixed number | $1\,\frac{1}{4}$ | ### Fraction Operations - Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. - Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. - Addition/Subtraction: use common denominator: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$. Example: $$\frac{1}{2} \times \frac{3}{4} + \frac{1}{3}$$ Multiply first: $$\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$$ Common denominator with $\frac{1}{3}$ is $24$: $$\frac{3}{8} = \frac{9}{24}, \quad \frac{1}{3}