1.5 Summary
Previous
1.4 Applications of exponentials
|
Next
End of chapter exercises
|
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:
- \({a}^{m} \times {a}^{n}={a}^{m+n}\)
- \(\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}\)
- \({\left(ab\right)}^{n}={a}^{n}{b}^{n}\)
- \({\left(\frac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}\)
- \({\left({a}^{m}\right)}^{n}={a}^{mn}\)
-
Rational exponents and surds:
- If \(r^n = a\), then \(r = \sqrt[n]{a} \quad (n \geq 2)\)
- \(a^{\frac{1}{n}} = \sqrt[n]{a}\)
- \(a^{-\frac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \sqrt[n]{\dfrac{1}{a}}\)
- \(a^{\frac{m}{n}} = (a^{m})^{\frac{1}{n}} = \sqrt[n]{a^m}\)
-
Simplification of surds:
- \(\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}\)
- \(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
- \(\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}\)
Previous
1.4 Applications of exponentials
|
Table of Contents |
Next
End of chapter exercises
|