# Worksheet 0: Introduction to Complex Numbers

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## Complex Numbers

A complex number is a number that can be expressed in the form $$a + bi$$, where $$a$$ and $$b$$ are real numbers, and $$i$$ is the imaginary unit (which satisfies the equation $$i^2 = −1$$). In this expression, $$a$$ is called the real part of the complex number, and $$b$$ is called the imaginary part. If $${\displaystyle z=a+bi}$$, then we write $${\displaystyle \operatorname {Re} (z)=a,}$$ and $${\displaystyle \operatorname {Im} (z)=b}$$.

### Q1

Use the quadratic formula to find roots for $$z^2-2z+5=0$$. (Recall that the roots for a quadratic equation, $$ax^2+bx+c=0$$, can be found from quadratic formula:

$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}.$

For the quadratic equation, $$z^2-2z+5=0$$:

1. What is $$z$$?
2. What are $${Re} (z)$$ and $${Im} (z)$$?

## Complex Conjugates

The complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude, but the complex value is opposite in sign. For example, the complex conjugate of $$3 + 4i$$ is $$3 − 4i$$. In general, the complex conjugate of a complex number is simply that number with the sign of the imaginary part reversed, that is, if $$z=x+iy$$, then the complex conjugate is $$z^*=x-iy$$.

### Q2

What is the value of $$xy^*$$ for the following complex numbers:

1. $$x= 3 + 5i$$ and $$y = 2 - 3i$$
2. $$x= 5 + 1i$$ and $$y = 3 + i$$
3. $$x = y = 5 + 5i$$
4. $$x = y = -3 + 2i$$

### Q3

Which sums in Q2 are real? Can you identify the pattern for when the sum of two complex numbers will be real?

## Plotting Complex Numbers

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number $$a + bi$$ can be identified with the point $$(a, b)$$ in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone. We often conceptualize complex numbers in a graphical manner, on a complex plane. The real part is the $$x$$-axis, and the imaginary part is the $$y$$-axis.

### Q4

What is the length of the vector shown in the figure? This can be computed using the Pythagorean theorem.

## Absolute Value

The absolute value or modulus $$|x|$$ of a real number $$x$$ is the non-negative value of $$x$$ without regard to its sign (i.e., $$|x| = x$$ for a positive $$x$$, $$|x| = −x$$ for a negative $$x$$ when $$-x$$ is positive, and $$|0| = 0$$). the definition given above for the real absolute value cannot be directly generalized for a complex number. However the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalized. The absolute value of a complex number is defined as its distance in the complex plane from the origin. That product of any complex number $$z$$ and its complex conjugate $$z^*$$ is always a non-negative real number.

### Q5

When is $$z^*z = z^2$$?

## Phase Angle

We refer to the angle, $$\theta$$, shown in the figure above, as the phase angle. We can find it from the legs of the triangle, that is $$\theta =\tan{\frac{y}{x}}$$.

### Q6

For $$z=2+2i$$, what is the length of this vector and what is its phase angle?

## Euler's Formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x:

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

This is

From this. Then $$x=r\cos{\theta}$$ and $$y=r\sin{\theta}$$.

### Q7

The property involving the product of exponentials

$e^{a}e^{b}=e^{a+b}$

also applies to complex arguments.

What is the product of $$e^{4+3i}$$ and $$e^{5+2i}$$?

### Q8

What is the product of $$e^{z}$$ and $$e^{z^*}$$ where $$z$$ is any complex number?

## Relationship to Trigonometry

Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:

{\begin{align}\cos x&=\operatorname {Re} \left(e^{ix}\right)={\frac {e^{ix}+e^{-ix}}{2}}\\\sin x&=\operatorname {Im} \left(e^{ix}\right)={\frac {e^{ix}-e^{-ix}}{2i}}\end{align}}

### Q9

Expand $$y= \cos x \sin x$$ into a function of complex exponentials