# Chapter 2 – Mathematics

*Audio version available*

## Pre-Algebra

Pre-algebra is a fundamental branch of mathematics that serves as a bridge between arithmetic and algebra. It covers various topics that form the foundation for more advanced mathematical concepts. In this section, we will explore the key concepts and techniques used in pre-algebra.

**Number Systems**: Pre-algebra introduces students to different number systems, including natural numbers, whole numbers, integers, rational numbers, and real numbers. Understanding the properties and relationships between these number systems is crucial for solving equations and performing operations.**Operations and Expressions**: Pre-algebra involves learning about different operations such as addition, subtraction, multiplication, and division. Students also learn how to simplify expressions by combining like terms and applying order of operations.

Example: Simplify the expression 3x + 2y – 5x + 4y. Solution: Combining like terms, we get -2x + 6y.

**Equations and Inequalities**: Equations are mathematical statements that express the equality of two quantities. Students learn how to solve linear equations with one variable using various methods such as balancing, substitution, or elimination. Inequalities involve comparing two quantities using symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

Example: Solve the equation 2x + 5 = 17. Solution: Subtracting 5 from both sides gives 2x = 12. Dividing by 2 on both sides yields x = 6.

**Factors and Multiples**: Understanding factors and multiples is essential for working with fractions, simplifying expressions, and finding common denominators. Factors are numbers that divide evenly into another number without leaving a remainder, while multiples are obtained by multiplying a number by an integer.

Example: Find the factors of 24. Solution: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

**Fractions and Decimals**: Pre-algebra delves into the concepts of fractions and decimals. Students learn how to perform operations with fractions, convert between fractions and decimals, and solve problems involving ratios and proportions.

Example: Simplify the fraction 16/24. Solution: Dividing both the numerator and denominator by their greatest common divisor (which is 8), we get 2/3.

## Elementary Algebra

Elementary algebra builds upon the foundation of pre-algebra and introduces more advanced concepts and techniques. It focuses on solving equations, manipulating algebraic expressions, and understanding the properties of different algebraic structures.

**Linear Equations**: Elementary algebra extends the study of linear equations to include systems of equations with two or more variables. Students learn methods like substitution, elimination, or graphing to solve these systems.

Example: Solve the system of equations: 2x + y = 5 3x – y = 7 Solution: Adding the two equations eliminates the y variable. We get 5x = 12. Dividing by 5 yields x = 12/5. Substituting this value back into either equation gives y = -1/5.

**Exponents and Radicals**: Exponents represent repeated multiplication, while radicals involve taking roots of numbers. Elementary algebra covers simplifying expressions with exponents, solving exponential equations, and understanding the properties of radicals.

Example: Simplify the expression √(27x^4). Solution: Breaking down the expression into prime factors gives √(3^3 * (x^2)^2). Taking out pairs of factors from under the square root gives x^2√3.

**Quadratic Equations**: Quadratic equations involve variables raised to the power of two. Students learn how to solve quadratic equations using methods like factoring, completing the square, or applying the quadratic formula.

Example: Solve the quadratic equation x^2 – 5x + 6 = 0. Solution: Factoring the equation gives (x – 2)(x – 3) = 0. Setting each factor equal to zero yields x = 2 or x = 3.

**Inequalities and Absolute Value**: Elementary algebra covers solving and graphing linear inequalities, compound inequalities, and absolute value equations. These concepts are crucial for understanding intervals, inequalities on number lines, and regions in coordinate planes.

Example: Solve the inequality 2x – 3 > 5. Solution: Adding 3 to both sides gives 2x > 8. Dividing by 2 on both sides yields x > 4.

**Polynomials**: Polynomials are expressions with one or more terms involving variables raised to non-negative integer powers. Students learn operations with polynomials, including addition, subtraction, multiplication, and division.

Example: Perform the polynomial division of (4x^3 + x^2 – 7x + 10) ÷ (2x – 1). Solution: Using long division or synthetic division, we find that the quotient is 2x^2 + 5x + 3 with a remainder of zero.

## Intermediate Algebra

Intermediate algebra builds upon elementary algebra and covers more advanced topics such as functions, rational expressions, logarithms, and matrices. It serves as a stepping stone towards higher-level mathematics courses.

**Functions**: Intermediate algebra introduces the concept of functions, which relate inputs (independent variables) to outputs (dependent variables). Students learn about domain and range, function notation, composite functions, and inverse functions.

Example: Determine if the relation given by the equation y = 2x + 1 represents a function. Solution: Yes, it represents a function since each input (x) corresponds to exactly one output (y).

**Rational Expressions**: Rational expressions are fractions where the numerator and denominator are polynomials. Intermediate algebra covers simplifying, multiplying, dividing, adding, and subtracting rational expressions.

Example: Simplify the rational expression (3x^2 – 6x) / (6x^3 – 12x^2). Solution: Factoring out common terms from both the numerator and denominator gives (3x(x – 2)) / (6x^2(x – 2)). Canceling out common factors yields 1 / (2x).

**Exponential and Logarithmic Functions**: Intermediate algebra explores exponential functions, logarithmic functions, and their properties. Students learn how to solve exponential and logarithmic equations, graph these functions, and apply them in real-world contexts.

Example: Solve the equation 3^(2x) = 27. Solution: Since 27 is equal to 3^3, we can rewrite the equation as 3^(2x) = 3^3. Equating the exponents gives 2x = 3. Dividing by 2 yields x = 3/2.

**Matrices**: Matrices are rectangular arrays of numbers or variables. Intermediate algebra covers matrix operations such as addition, subtraction, multiplication, and finding inverses. Students also learn about solving systems of linear equations using matrices.

Example: Solve the system of equations using matrices: 2x + y = 5 3x – y = 7 Solution: Representing the coefficients and constants in matrix form, we have: | 2 1 | | x | | 5 | | | | | = | | | 3 -1 | | y | | 7 | Using matrix operations, we find x = 3 and y = -1.

## Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebraic techniques with geometric concepts. It involves studying the relationships between points, lines, curves, and shapes using coordinate systems.

**Cartesian Coordinates**: Coordinate geometry introduces the Cartesian coordinate system, where points are represented by ordered pairs (x, y). Students learn how to plot points on a plane and find distances between them using the distance formula.

Example: Plot the point P(2, -3) on a Cartesian plane. Solution: Locate the point (2, -3) by moving 2 units to the right along the x-axis and 3 units downward along the y-axis.

**Equations of Lines**: Coordinate geometry covers equations of lines in different forms, including slope-intercept form (y = mx + b), point-slope form ((y – y1) = m(x – x1)), and general form (Ax + By = C). Students learn how to determine slopes, intercepts, and parallel or perpendicular lines.

Example: Find the equation of a line passing through the points (-2, 4) and (3, -1). Solution: Using the point-slope form, we have: (y – 4) = (4 – (-1)) / (-2 – 3) Simplifying gives y = -x + 2.

**Conic Sections**: Coordinate geometry explores conic sections such as circles, ellipses, parabolas, and hyperbolas. Students learn how to identify their equations based on geometric properties and translate them into standard forms.

Example: Identify the conic section represented by the equation x^2 + y^2 = 25. Solution: The equation x^2 + y^2 = 25 represents a circle with a radius of 5 units centered at the origin.

**Transformations**: Coordinate geometry covers transformations, including translations, reflections, rotations, and dilations. Students learn how to apply these transformations to points, lines, and shapes on a coordinate plane.

Example: Perform a reflection of the point A(3, -4) across the y-axis. Solution: The reflected point A’ will have the same x-coordinate but the opposite y-coordinate. Therefore, A’ is (-3, -4).

## Plane Geometry

Plane geometry, also known as Euclidean geometry, deals with the properties and relationships of figures in a two-dimensional space. It encompasses concepts such as angles, triangles, polygons, circles, and areas.

**Angles**: Plane geometry introduces different types of angles, including acute, obtuse, right, straight, complementary, and supplementary angles. Students learn about angle measurement, angle bisectors, and angle relationships within polygons.

**Triangles**: Plane geometry covers various properties and classifications of triangles based on their sides (scalene, isosceles, equilateral) and angles (acute, obtuse, right). Students learn about triangle congruence criteria (SSS, SAS, ASA) and the Pythagorean theorem.

**Polygons**: Plane geometry explores polygons, which are closed figures with straight sides. Students learn about regular and irregular polygons, convex and concave polygons, and the properties of interior and exterior angles.

Example: Determine the sum of the interior angles in a hexagon. Solution: The sum of the interior angles in an n-sided polygon is given by (n-2) *180 degrees. For a hexagon, the sum is (6-2) *180 = 720 degrees.

**Circles**: Plane geometry covers the properties of circles, including radius, diameter, chord, secant, tangent, and arc length. Students learn how to calculate the circumference and area of circles and solve problems involving arcs and sectors.

Example: Find the circumference of a circle with a radius of 8 centimeters. Solution: The circumference of a circle is given by the formula C = 2πr. Substituting r = 8 into the formula gives C = 2π(8) = 16π cm.

**Areas**: Plane geometry involves calculating areas of various shapes such as rectangles, triangles, parallelograms, trapezoids, circles, and composite figures. Students learn different formulas for finding area and how to apply them in real-world scenarios.

Example: Find the area of a triangle with base 10 units and height 6 units. Solution: The area of a triangle is given by the formula A = (1/2)bh. Substituting b = 10 and h = 6 into the formula gives A = (1/2)(10)(6) = 30 square units.

## Trigonometry

Trigonometry is the study of relationships between angles and sides of triangles. It has applications in various fields such as physics, engineering, and navigation. Trigonometry involves understanding trigonometric functions, identities, and solving trigonometric equations.

**Trigonometric Functions**: Trigonometry introduces the six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Students learn how to evaluate these functions for given angles and use them to solve problems involving right triangles.

Example: Find the value of sin(45 degrees). Solution: The sine of 45 degrees is equal to the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. For a 45-45-90 triangle, this ratio is 1/√2 or approximately 0.707.

**Trigonometric Identities**: Trigonometry covers various identities that relate trigonometric functions to each other. These identities include reciprocal identities, quotient identities, Pythagorean identities, and sum/difference formulas.

Example: Prove the identity tan^2(x) + 1 = sec^2(x). Solution: Starting with the Pythagorean identity sin^2(x) + cos^2(x) = 1, divide both sides by cos^2(x). This gives tan^2(x) + 1 = sec^2(x).

**Trigonometric Equations**: Trigonometry involves solving equations involving trigonometric functions. Students learn how to solve equations like sin(x) = 0 or cos(2x) = -1 by applying inverse trigonometric functions or using algebraic manipulations.

Example: Solve the equation cos(x) = 1/2. Solution: Taking the inverse cosine (arccos) of both sides gives x = π/3 or x = 5π/3, considering the range of the cosine function.

**Trigonometric Applications**: Trigonometry has numerous applications in real-world scenarios, such as finding distances or heights using angles of elevation or depression, analyzing periodic phenomena, and solving navigation problems.

Example: A ladder is leaning against a wall. If the ladder forms a 60-degree angle with the ground and reaches a height of 10 meters on the wall, find the length of the ladder. Solution: Using the sine function, we have sin(60 degrees) = opposite/hypotenuse. Substituting the known values gives sin(60 degrees) = 10/hypotenuse. Solving for the hypotenuse, we find it to be approximately 11.55 meters.