Numbers

MATH1051

1.1 Number Systems

Irrational numbers are real numbers which cannot be represented as a ratio of integers. For example, $\sqrt{2},\sqrt{3},\sqrt{5},\pi =3.1415926\dots ,e=2.71828\dots$ are all irrational.

Note:

Proving Rationality or Irrationality of a given number can be quite subtle.

1.2 Real Number line and Ordering of $\mathbb{R}$

The real number system can be visualised by imagining each real number as a point on a line, with the positive direction being to the right, and an arbitrary origin being chosen to represent 0.

The real numbers are ordered, i.e. given any 2 real numbers $a$ and $b$ there holds precisely one of the following: $a>b,a<b$ or $a=b$. This means we can use the symbols ${}^{\prime }{<}^{\prime }{,}^{\prime }{>}^{\prime }{,}^{\prime }{\le }^{\prime }{,}^{\prime }{\ge }^{\prime }$ to write statements such as $1\le 2$ and $3>\sqrt{3}$. geometrically, $a<b$ means that $a$ lies to the left of $b$ on the real number line.

Note:

$a\le b$ means that either $a<b$ or $a=b$.

1.3 Definition: Intervals

An interval is a set of real numbers that can be thought of as a segment of the real number line. For $a<b$, the open interval from $a$ to $b$ is given by

$$(a,b)=\{x\in \mathbb{R}\mid a<x<b\}$$

There are also infinite intervals such as

$$\begin{array}{rl}(-\infty ,a]& =\{x\in \mathbb{R}\mid x\le a\}\\ (-\infty ,a)& =\{x\in \mathbb{R}\mid x<a\}\\ [a,\infty )& =\{x\in \mathbb{R}\mid a\le x\}\\ (a,\infty )& =\{x\in \mathbb{R}\mid a<x\}\\ \mathbb{R}& =(-\infty ,\infty )\end{array}$$

Note that $\pm \infty$ can never be included in an interval.

1.4 Absolute Value

We define

$$|x|=\{\begin{array}{ll}x,& \text{if }x\ge 0\\ -x,& \text{if }x<0\end{array}$$

1.4.3 Convention for ​

For $a>0$, $\sqrt{a}$ always denotes the positive solution of $x^2=a$. Thus $\sqrt{4}=2$ and so on. This means that we can now solve $x^2=a$. For $a>0$, solutions to $x^2=a$ are $x=\pm \sqrt{a}$.

1.5 Complex Numbers

Complex numbers were introduced in the 16th century to obtain roots of polynomial equations. A complex number is of the form

$$z=x+iy$$

where $x,y\in \mathbb{R}$ and $i$ is a symbol satisfying $i^2=-1$. The quantity $x$ is called the real part of $z$ and $y$ is called the imaginary part of $z$.

The set of all complex numbers is denoted $\mathbb{C}$.

1.5.1 Example

The real part of $3-2i$ is $3$ and the imaginary part is $-2$ (not $-2i$).

Complex numbers can be added and multiplied by replacing $i^2$ everywhere with $-1$. For example, $(2i{)}^2=4i^2=-4$.

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If $z=a+bi$ is a complex number, the number $a-bi$ is called the complex conjugate of $z$. The complex conjugate of $3+2i$ is $\overline{3+2i}=3-2i$.

1.6 Polar Form

Real numbers are often represented on the real line. A complex number $z=x+iy$ may be represented by a point in the complex plane, where the horizontal axis is the real axis and the vertical axis is the imaginary axis.

We can also specify $z$ by giving the length $r$ and the angle $\theta$. The quantity $r$ is called the modulus of $z$, denoted $|z|$. It measures the distance of $z$ from the origin. The angle $\theta$ is called the argument of $z$.

1.7 Euler’s Formula

Euler’s formula states for any real number $\theta :$

$$\cos \theta +i\sin \theta =e^{i\theta }$$

Thus every complex number $z=x+iy$ can be represented in polar form

$$z=r{e}^{i\theta }$$