Revision Revision Notes for Grade 11 NSC Matric Mathematics

Revision

The number system

Understanding the different types of numbers is essential for working with exponents and surds. Numbers are organised into a hierarchical system where each type has specific properties.

The number system consists of several categories, each containing the previous categories within it. Understanding these key definitions will provide the foundation for all mathematical work with different number types.

Natural numbers (ℕ)

These are the counting numbers we use in everyday life.

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Definition: ℕ = {1; 2; 3; ...}

Natural numbers are the most basic numbers we encounter - they represent counting and ordering in the real world.

Whole numbers (ℕ₀)

Natural numbers with zero included.

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Definition: ℕ₀ = {0; 1; 2; 3; ...}

The addition of zero to the natural numbers creates the set of whole numbers, which is useful for many mathematical operations.

Integers (ℤ)

All positive and negative whole numbers, including zero.

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Definition: ℤ = {...; -3; -2; -1; 0; 1; 2; 3; ...}

Integers extend the whole numbers to include negative values, allowing us to represent concepts like debt, temperature below zero, and direction.

Rational numbers (ℚ)

Numbers that can be expressed as a fraction where both the numerator and denominator are integers, and the denominator is not zero. These include terminating and recurring decimal numbers.

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Definition: Any number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ .

Rational numbers include all fractions, decimals that terminate, and decimals that repeat in a pattern.

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Examples of Rational Numbers:

$-\frac{7}{5}$ (proper fraction)
$-2,25$ (terminating decimal)
$0$ (can be written as $\frac{0}{1}$ )
$\sqrt{9}$ (equals 3, which is $\frac{3}{1}$ )
$0,8$ (terminating decimal)
$\frac{23}{1}$ (integer expressed as fraction)

Irrational numbers (ℚ')

Numbers that cannot be written as a fraction with integer numerator and denominator. These are decimal numbers that neither terminate nor have a repeating pattern.

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Examples of Irrational Numbers:

$\sqrt{3}$ (non-perfect square root)
$\sqrt[3]{2}$ (non-perfect cube root)
$\pi$ (mathematical constant)
$\frac{1+\sqrt{5}}{2}$ (golden ratio)
$1,27548...$ (non-repeating, non-terminating decimal)

Real numbers (ℝ)

The combination of all rational and irrational numbers. This set includes virtually all numbers we work with in everyday mathematics.

Non-real numbers (ℝ')

Numbers that involve the square root of negative numbers, which are not real.

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Examples of Non-real Numbers:

$\sqrt{-25}$ (square root of negative number)
$\sqrt[3]{-1}$ (cube root of negative number)
$-\sqrt{-\frac{1}{16}}$ (negative square root of negative fraction)

Laws of exponents

Exponential notation is a shorthand way to show that a number or variable is multiplied by itself a specific number of times.

Understanding exponential notation

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Definition: $a^n = a \times a \times a \times ... \times a$ (n times), where $a \in \mathbb{R}$ and $n \in \mathbb{N}$

In the expression $a^n$ :

$a$ is called the base
$n$ is called the exponent, index, or power

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Examples of Exponential Notation:

$2 \times 2 \times 2 \times 2 = 2^4$
$0,71 \times 0,71 \times 0,71 = (0,71)^3$
$(501)^2 = 501 \times 501$
$k^6 = k \times k \times k \times k \times k \times k$

Special vocabulary

Understanding the language of exponents helps with mathematical communication:

For $x^2$ , we say " $x$ is squared"
For $y^3$ , we say " $y$ is cubed"
For higher powers like $k^6$ , we say " $k$ is raised to the sixth power"

Special exponent rules

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Rule 1: $a^0 = 1$ (where $a \neq 0$ because $0^0$ is undefined)

Rule 2: $a^{-n} = \frac{1}{a^n}$ (where $a \neq 0$ because $\frac{1}{0}$ is undefined)

These rules are fundamental and apply to all exponential expressions. Remember that zero cannot be a base when dealing with negative exponents or the zero power rule.

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Examples of Special Exponent Rules:

$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
$(-36)^0 x = (1)x = x$
$\frac{7p^{-1}}{q^3 t^{-2}} = \frac{7t^2}{pq^3}$

The five laws of exponents

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The Five Fundamental Laws

These laws apply when $a > 0$ , $b > 0$ and $m, n \in \mathbb{Z}$ :

Law 1: $a^m \times a^n = a^{m+n}$ (Multiplication law)

Law 2: $\frac{a^m}{a^n} = a^{m-n}$ (Division law)

Law 3: $(ab)^n = a^n b^n$ (Product law)

Law 4: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (Quotient law)

Law 5: $(a^m)^n = a^{mn}$ (Power law)

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Worked Example 1: Simplifying expressions

Question: Simplify the following:

$5(m^{2t})^p \times 2(m^{3p})^t$
$\frac{8k^3 x^2}{(xk)^2}$
$\frac{2^2 \times 3 \times 7^1}{(7 \times 2)^4}$
$3(3^b)^a$

Solution:

$5(m^{2t})^p \times 2(m^{3p})^t = 10m^{2pt+3pt} = 10m^{5pt}$
$\frac{8k^3 x^2}{(xk)^2} = \frac{8k^3 x^2}{x^2 k^2} = 8k^{(3-2)} x^{(2-2)} = 8k^1 x^0 = 8k$
$\frac{2^2 \times 3 \times 7^1}{(7 \times 2)^4} = \frac{2^2 \times 3 \times 7^1}{7^4 \times 2^4} = 2^{(2-4)} \times 3 \times 7^{(1-4)} = 2^{-2} \times 3 = \frac{3}{4}$
$3(3^b)^a = 3 \times 3^{ab} = 3^{ab+1}$

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Worked Example 2: Factorising with exponents

Question: Simplify: $\frac{3^m - 3^{m+1}}{4 \times 3^m - 3^m}$

Solution:

Step 1: Simplify to a form that can be factorised

$\frac{3^m - 3^{m+1}}{4 \times 3^m - 3^m} = \frac{3^m - (3^m \times 3)}{4 \times 3^m - 3^m}$

Step 2: Take out a common factor

$= \frac{3^m(1 - 3)}{3^m(4 - 1)}$

Step 3: Cancel the common factor and simplify

$= \frac{1 - 3}{4 - 1} = \frac{-2}{3}$

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Key Points to Remember:

Number system hierarchy: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real, with irrational numbers also being real numbers
Special exponent rules: $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ (where $a \neq 0$ )
Five laws of exponents: When multiplying same bases, add exponents; when dividing, subtract exponents; when raising a power to a power, multiply exponents
Always write final answers with positive exponents when possible
Look for common factors when simplifying complex expressions with exponents

Revision (Grade 11 NSC Matric Mathematics): Revision Notes

Revision

The number system

Natural numbers (ℕ)

Whole numbers (ℕ₀)

Integers (ℤ)

Rational numbers (ℚ)

Irrational numbers (ℚ')

Real numbers (ℝ)

Non-real numbers (ℝ')

Laws of exponents

Understanding exponential notation

Special vocabulary

Special exponent rules

The five laws of exponents

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