Summary and Main Ideas

Summary

• 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 $$\cfrac{a}{b}$$ 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:

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

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

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

• Laws of exponents:

• $${a}^{m} \times {a}^{n}={a}^{m+n}$$
• $$\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}$$
• $${(ab)}^{n}={a}^{n}{b}^{n}$$
• $${\left(\cfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}$$
• $${({a}^{m})}^{n}={a}^{mn}$$

where $$a > 0$$, $$b > 0$$ and $$m, n \in \mathbb{Z}$$.

• Rational exponents and surds:

• If $$r^n = a$$, then $$r = \sqrt[n]{a} \quad (n \geq 2)$$
• $$a^{\frac{1}{n}} = \sqrt[n]{a}$$
• $$a^{-\cfrac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \sqrt[n]{\dfrac{1}{a}}$$
• $$a^{\frac{m}{n}} = (a^{m})^{\frac{1}{n}} = \sqrt[n]{a^m}$$

where $$a > 0$$, $$r > 0$$ and $$m,n \in \mathbb{Z}$$, $$n \ne 0$$.

• Simplification of surds:

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