# 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, \(z^2-2z+5=0\):

- What is \(z\)?
- 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:

- \(x= 3 + 5i\) and \(y = 2 - 3i\)
- \(x= 5 + 1i\) and \(y = 3 + i\)
- \(x = y = 5 + 5i\)
- \(x = y = -3 + 2i\)

### 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