1.9 Chapter summary
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1.9 Chapter summary (EMAR)


\(\mathbb{N}\): 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 \(\left(ax+b\right)\left(axb\right)\)

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: \[{x}^{3}{y}^{3}=\left(xy\right)\left({x}^{2}+xy+{y}^{2}\right)\]

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.
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