Summary (Grade 10 NSC Matric Mathematics): Revision Notes

Summary

Number systems

Understanding different types of numbers forms the foundation of algebraic expressions. Numbers can be classified into several important categories.

Natural numbers (N) are the counting numbers we use in everyday life: {1, 2, 3, 4, ...}. These numbers start from 1 and continue infinitely.

Whole numbers (N₀) include all natural numbers plus zero: {0, 1, 2, 3, 4, ...}. The subscript ₀ reminds us that zero is included in this set.

Integers (Z) extend further to include all positive and negative whole numbers: {..., -3, -2, -1, 0, 1, 2, 3, ...}. The symbol Z comes from the German word "Zahlen" meaning numbers.

This Venn diagram shows how these number sets relate to each other, with natural numbers contained within whole numbers, which are contained within integers.

Rational and irrational numbers

Rational numbers are any numbers that can be expressed as a fraction $\frac{a}{b}$ where both $a$ and $b$ are integers and $b \neq 0$. This large category includes:

• All fractions where both the numerator and denominator are integers
• All integers (since any integer $n$ can be written as $\frac{n}{1}$)
• All decimal numbers that terminate (like 0.25 = $\frac{1}{4}$)
• All decimal numbers that recur or repeat (like 0.333... = $\frac{1}{3}$)

Irrational numbers are numbers that cannot be written as a fraction with integer numerator and denominator. These numbers have decimal representations that neither terminate nor repeat in a pattern.

Surds are a special type of irrational number. When the $n^{th}$ root of a number cannot be simplified to give a rational result, we call it a surd. For example, $\sqrt{2}$ and $\sqrt{5}$ are surds because they cannot be expressed as exact fractions.

An important property of surds is that if $a$ and $b$ are positive whole numbers with $a \< b$, then $\sqrt{a} \< \sqrt{b}$.

Binomials and polynomial expressions

A binomial is any algebraic expression that contains exactly two terms. Examples include $(x + 3)$, $(2a - b)$, and $(m^2 + 5n)$.

When we multiply two identical binomials, we get the square of the binomial. This creates important patterns that are useful for both expanding and factorising expressions.

For more complex expressions, when we multiply a binomial by a trinomial (three terms), we use the distributive property:

$(A + B)(C + D + E) = A(C + D + E) + B(C + D + E)$

Factorisation techniques

Factorisation is the reverse process of expanding brackets. Instead of multiplying out expressions, we break them down into their component factors.

The most basic method is taking out a common factor. This involves identifying the largest factor that divides all terms in the expression and factoring it out.

For more complex expressions, we often use grouping to factorise polynomials. This technique involves rearranging and grouping terms strategically to reveal common factors.

Quadratic factorisation requires us to find the two binomials that, when multiplied together, produce the original quadratic expression.

Two important special cases are the sum and difference of cubes:

Sum of two cubes: $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

Difference of two cubes: $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

Simplifying algebraic fractions

Working with algebraic fractions requires understanding several key principles. We can simplify fractions by using the factorisation methods we have learned to break down complex expressions.

When adding or subtracting algebraic fractions, the denominators of all fractions must be the same. If they are different, we need to find a common denominator first, just as we do with numerical fractions.

The same principles of fraction operations that apply to numbers also apply to algebraic expressions, but we must be more careful with the algebraic manipulation required.