Complete Grade 11 Mathematics study guide in English, aligned to the CAPS curriculum. 14 chapters with explanations, worked examples, exercises with solutions, and original practice papers with memos.

Chapters

Chapter 1. Getting Started: Number Systems, Rounding and Algebraic Foundations FREE

Imagine you are pricing a delivery run for a small bakery in Soweto. The petrol costs R23,79 per litre, the van holds 45 litres, and you want to split the cost between 7 customers. Almost every number you meet here behaves differently: some are whole, some are fractions, one of them (the answer after dividing by 7) never stops repeating. Before you can do any Grade 11 algebra, trigonometry or financial mathematics, you must be confident about which kind of number you are working with, how to round it sensibly, and how to manipulate algebraic expressions without making careless errors. This chapter rebuilds those foundations.

1.1 The real number system

Every number you use in Grade 11 lives inside the set of real numbers, written \(\mathbb{R}\). Inside \(\mathbb{R}\) there are smaller, nested sets:

Natural numbers \(\mathbb{N} = \{1, 2, 3, \dots\}\) - the counting numbers.
Whole numbers \(\mathbb{N}_0 = \{0, 1, 2, 3, \dots\}\) - the naturals together with zero.
Integers \(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\) - whole numbers and their negatives.
Rational numbers \(\mathbb{Q}\) - any number that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). Their decimal form either terminates (for example \(0{,}75\)) or recurs (for example \(0{,}333\dots\)).
Irrational numbers \(\mathbb{Q}'\) - real numbers that cannot be written as such a fraction. Their decimals go on forever with no repeating block, for example \(\sqrt{2} = 1{,}41421\dots\) and \(\pi = 3{,}14159\dots\)

A useful test: a surd like \(\sqrt{n}\) is rational only when \(n\) is a perfect square. So \(\sqrt{49} = 7\) is rational, but \(\sqrt{50}\) is irrational. Take care: \(\frac{22}{7}\) is rational (it is a fraction) even though it is a famous approximation of the irrational \(\pi\).

Example: Classify \(-3\), \(\frac{5}{8}\), \(\sqrt{16}\), \(\sqrt{7}\) and \(0{,}272727\dots\) Solution: \(-3\) is an integer (and rational and real). \(\frac{5}{8}\) is rational. \(\sqrt{16} = 4\) is a natural number (and rational). \(\sqrt{7}\) is irrational. \(0{,}272727\dots\) recurs, so it is rational.

1.2 Recurring decimals as fractions

Because every recurring decimal is rational, we can always write it as a fraction. The trick is to multiply by a power of 10 so the repeating tail lines up, then subtract.

Example: Write \(0{,}4\dot{5}\) (that is \(0{,}4555\dots\)) as a common fraction. Let \(x = 0{,}4555\dots\) Then \(10x = 4{,}555\dots\) and \(100x = 45{,}55\dots\) Subtract: \(100x - 10x = 45{,}55\dots - 4{,}55\dots\), so \(90x = 41\) and \(x = \frac{41}{90}\).

1.3 Rounding, decimal places and significant figures

In real problems we round to keep answers practical. To round to a given number of decimal places, look at the next digit: if it is 5 or more, round up; otherwise leave the last kept digit unchanged. For money we usually round to two decimal places (the nearest cent).

Significant figures count from the first non-zero digit. For example \(0{,}004067\) to three significant figures is \(0{,}00407\). When you estimate, round each number to one significant figure first.

Example: A spaza shop owner buys 13 crates of cooldrink at R87,40 each. Estimate the total, then round the exact total to the nearest rand. Estimate: \(13 \approx 10\) and \(87{,}40 \approx 90\), so about \(R900\). Exact: \(13 \times 87{,}40 = R1\,136{,}20\), which rounds to \(R1\,136\). The estimate confirms the answer is sensible.

1.4 Algebraic foundations: simplifying expressions

Algebra is generalised arithmetic. A term is a product of numbers and variables (for example \(-5x^2y\)); like terms have identical variable parts and can be added. Always respect the order of operations (brackets, exponents, multiplication and division, then addition and subtraction).

Two skills you must own before Chapter 3:

Multiplying out (distribution): \(2x(3x - 4) = 6x^2 - 8x\), and \((x + 5)(x - 2) = x^2 + 3x - 10\).
Factorising - the reverse process. Take out a common factor first: \(6x^2 - 8x = 2x(3x - 4)\). Recognise a difference of two squares: \(x^2 - 9 = (x - 3)(x + 3)\). Factorise a simple trinomial: \(x^2 + 3x - 10 = (x + 5)(x - 2)\).

Example: Simplify \(3(2a - 5) - (a - 4)\). Distribute: \(6a - 15 - a + 4\). Collect like terms: \((6a - a) + (-15 + 4) = 5a - 11\).

1.5 Algebraic fractions

To add or subtract algebraic fractions, find a common denominator, exactly as with ordinary fractions. To simplify, factorise the numerator and denominator first, then cancel common factors. You may only cancel factors, never individual terms.

Example: Simplify \(\dfrac{x^2 - 9}{x^2 + 4x + 3}\). Factorise top and bottom: \(\dfrac{(x-3)(x+3)}{(x+1)(x+3)}\). Cancel the common factor \((x+3)\): \(\dfrac{x - 3}{x + 1}\), provided \(x \neq -3\) and \(x \neq -1\).

Master these foundations and the rest of Grade 11 - exponents, equations, functions and finance - becomes far less intimidating, because every one of those topics leans on confident number sense and clean algebra.