# 2.1: Basic Notions

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The set of complex numbers is obtained algebraically by adjoining the number $$i$$ to the set $$\mathbb{R}$$ of real numbers, where $$i$$ is defined by the property that $$i^2=−1$$. We will take a geometric approach and define a complex number to be an ordered pair $$(x,y)$$ of real numbers. We let $$\mathbb{C}$$ denote the set of all complex numbers,

$\mathbb{C} =\{(x,y) | x,y∈ \mathbb{R}\}.$

Given the complex number $$z=(x,y)$$, $$x$$ is called the real part of $$z$$, denoted Re$$(z)$$; and $$y$$ is called the imaginary part of $$z$$, denoted Im$$(z)$$. The set of real numbers is a subset of $$\mathbb{C}$$ under the identification $$x↔(x,0)$$, for any real number $$x$$.

Addition in $$\mathbb{C}$$ is componentwise,

$(x,y)+(s,t)=(x+s,y+t),$

and if $$k$$ is a real number, we define scalar multiplication by

$k⋅(x,y)=(kx,ky).$

Within this framework, $$i=(0,1)$$, meaning that any complex number $$(x,y)$$ can be expressed as $$x+yi$$ as suggested here:

$\begin{array} (x,y) &= (x,0)+(0,y) \\ &= x(1,0)+y(0,1) \\ &= x + yi \end{array}$

The expression $$x+yi$$ is called the Cartesian form of the complex number. This form can be helpful when doing arithmetic of complex numbers, but it can also be a bit gangly. We often let a single letter such as $$z$$ or $$w$$ represent a complex number. So, $$z=x+yi$$ means that the complex number we're calling $$z$$ corresponds to the point $$(x,y)$$ in the plane.

It is sometimes helpful to view a complex number as a vector, and complex addition corresponds to vector addition in the plane. The same holds for scalar multiplication. For instance, in Figure $$\PageIndex{1}$$ we have represented $$z=2+i$$, $$w=−1+1.5i$$, as well as $$z+w=1+2.5i$$, as vectors from the origin to these points in $$\mathbb{C}$$. The complex number $$z−w$$ can be represented by the vector from $$w$$ to $$z$$ in the plane.

We define complex multiplication using the fact that $$i^2=−1$$.

$\begin{array} (x+yi)⋅(s+ti) &= xs+ysi+xti+yti^2 \\ &= (xs−yt)+(ys+xt)i \end{array}$

The modulus of $$z=x+yi$$, denoted $$|z|$$, is given by

$|z|= \sqrt{x^2+y^2}.$

Note that $$|z|$$ gives the Euclidean distance of $$z$$ to the point $$(0,0)$$.

The conjugate of $$z=x+yi$$, denoted $$\overline{z}$$, is

$\overline{z} =x−yi$

In the exercises, the reader is asked to prove various useful properties of the modulus and conjugate.

##### Example $$\PageIndex{1}$$: Arithmetic of Complex Numbers

Suppose $$z=3−4i$$ and $$w=2+7i$$.

Then $$z+w=5+3i$$, and

$$$\begin{array} \cdot z \cdot w &= (3−4i)(2+7i) \\ &= 6+28−8i+21i \\ &= 34+13i \end{array}$$$

A few other computations:

$\begin{array} 4z &= 12−16i \\ |z| &= \sqrt{3^2 + (-4)^2} = 5 \\ \overline{zw} &= 34-13i \end{array}$

## Exercises

##### Exercise $$\PageIndex{1}$$

In each case, determine $$z+w$$, $$sz$$, $$|z|$$, and $$z⋅w$$.

1. $$z=5+2i$$, $$s=−4$$, $$w=−1+2i$$
2. $$z=3i$$, $$s= \dfrac{1}{2}$$, $$w=−3+2i$$
3. $$z=1+i$$, $$s=0.6$$, $$w=1−i$$

a. $$z+w=4+4i$$, $$sz=−20−8i$$, $$|z|=\sqrt{29}$$, and $$zw=−9+8i$$.

##### Exercise $$\PageIndex{2}$$

Show that $$z⋅ \overline{z} =|z|^2$$, where $$\overline{z}$$ is the conjugate of $$z$$.

Hint

Let $$z=a+bi$$, and show that the two sides of the equation agree.

##### Exercise $$\PageIndex{3}$$

Suppose $$z=x+yi$$ and $$w=s+ti$$ are two complex numbers. Prove the following properties of the conjugate and the modulus.

1. $$|w⋅z|=|w|⋅|z|$$.
2. $$\overline{zw}= \overline{z} \cdot \overline{w}$$
3. $$\overline{z+w}= \overline{z} + \overline{w}$$
4. $$z+\overline{z}= 2 \text{ Re} (z).$$ (Hence, $$z+ \overline{z}$$ is a real number.)
5. $$z- \overline{z}= 2 \text{ Im} (z)i.$$
6. $$|z|=|\overline{z}|.$$
##### Exercise $$\PageIndex{4}$$

A Pythagorean triple consists of three integers $$(a,b,c)$$ such that $$a^2+b^2=c^2$$. We can use complex numbers to generate Pythagorean triples. Suppose $$z=x+yi$$ where $$x$$ and $$y$$ are positive integers. Let

$$a= \text{ Re}(z^2) \;\;\;\;\; b= \text{ Im}(z^2) \;\;\;\;\; c=z \overline{z}.$$

1. Prove that $$a^2+b^2=c^2$$.
2. Find the complex number $$z=x+yi$$ that generates the famous triple $$(3,4,5)$$.
3. Find the complex number that generates the triple $$(5,12,13)$$.
4. Find five other Pythagorean triples, generated using complex numbers of the form $$z=x+yi$$, where $$x$$ and $$y$$ are positive integers with no common divisors.

This page titled 2.1: Basic Notions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael P. Hitchman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.