Home Practice
For learners and parents For teachers and schools
Textbooks
Full catalogue
Leaderboards
Learners Leaderboard Classes/Grades Leaderboard Schools Leaderboard
Pricing Support
Help centre Contact us
Log in

We think you are located in United States. Is this correct?

1.9 Chapter summary

Test yourself now

High marks in science are the key to your success and future plans. Test yourself and learn more on Siyavula Practice.

Sign up and test yourself

1.9 Chapter summary (EMAR)

Presentation: 2DR6

    • \(\mathbb{N}\): natural numbers are \(\left\{1; 2; 3; \ldots\right\}\)

    • \(\mathbb{N}_0\): whole numbers are \(\left\{0; 1; 2; 3; \ldots\right\}\)

    • \(\mathbb{Z}\): integers are \(\left\{\ldots; -3; -2; -1; 0; 1; 2; 3; \ldots\right\}\)

  • A rational number is any number that can be written as \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\ne 0\).

  • The following are rational numbers:

    • Fractions with both numerator and denominator as integers

    • Integers

    • Decimal numbers that terminate

    • Decimal numbers that recur (repeat)

  • Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers.

  • If the \(n^{\text{th}}\) root of a number cannot be simplified to a rational number, it is called a surd.

  • If \(a\) and \(b\) are positive whole numbers, and \(a<b\), then \(\sqrt[n]{a}<\sqrt[n]{b}\).

  • A binomial is an expression with two terms.

  • The product of two identical binomials is known as the square of the binomial.

  • We get the difference of two squares when we multiply \(\left(ax+b\right)\left(ax-b\right)\)

  • Factorising is the opposite process of expanding the brackets.

  • The product of a binomial and a trinomial is:

    \[\left(A+B\right)\left(C+D+E\right)=A\left(C+D+E\right)+B\left(C+D+E\right)\]
  • Taking out a common factor is the basic factorisation method.

  • We often need to use grouping to factorise polynomials.

  • To factorise a quadratic we find the two binomials that were multiplied together to give the quadratic.

  • The sum of two cubes can be factorised as: \[{x}^{3}+{y}^{3}=\left(x+y\right)\left({x}^{2}-xy+{y}^{2}\right)\]

  • The difference of two cubes can be factorised as: \[{x}^{3}-{y}^{3}=\left(x-y\right)\left({x}^{2}+xy+{y}^{2}\right)\]

  • We can simplify fractions by incorporating the methods we have learnt to factorise expressions.

  • Only factors can be cancelled out in fractions, never terms.

  • To add or subtract fractions, the denominators of all the fractions must be the same.