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Chapter Summary

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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.