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1.5 Summary (EMBFD)

  • The number system:

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

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

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

    • \(\mathbb{Q}\): rational numbers are numbers which can be written as \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\ne 0\), or as a terminating or recurring decimal number.

    • \(\mathbb{Q}'\): irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers. Irrational numbers also include decimal numbers that neither terminate nor recur.

    • \(\mathbb{R}\): real numbers include all rational and irrational numbers.

    • \(\mathbb{R}'\): non-real numbers or imaginary numbers are numbers that are not real.

  • Definitions:

    • \({a}^{n}=a\times a\times a\times \cdots \times a \left(n \text{ times}\right) \left(a\in \mathbb{R},n\in \mathbb{N}\right)\)

    • \({a}^{0}=1\) (\(a \ne 0\) because \(0^0\) is undefined)

    • \({a}^{-n}=\frac{1}{{a}^{n}}\) (\(a \ne 0\) because \(\dfrac{1}{0}\) is undefined)

  • Laws of exponents:

    • \({a}^{m} \times {a}^{n}={a}^{m+n}\)
    • \(\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}\)
    • \({\left(ab\right)}^{n}={a}^{n}{b}^{n}\)
    • \({\left(\frac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}\)
    • \({\left({a}^{m}\right)}^{n}={a}^{mn}\)
    where \(a > 0\), \(b > 0\) and \(m, n \in \mathbb{Z}\).
  • Rational exponents and surds:

    • If \(r^n = a\), then \(r = \sqrt[n]{a} \quad (n \geq 2)\)
    • \(a^{\frac{1}{n}} = \sqrt[n]{a}\)
    • \(a^{-\frac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \sqrt[n]{\dfrac{1}{a}}\)
    • \(a^{\frac{m}{n}} = (a^{m})^{\frac{1}{n}} = \sqrt[n]{a^m}\)
    where \(a > 0\), \(r > 0\) and \(m,n \in \mathbb{Z}\), \(n \ne 0\).
  • Simplification of surds:

    • \(\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}\)
    • \(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
    • \(\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}\)