Introducing Complex Numbers
Imaginary Unit Number and Complex plane
Define the imaginary unit number (Euler's notation) as \(i^2 = -1\). Then a complex number is an expression of the form \(z=x+iy\) where
- \(\text{Re}(z):=x\) is the real part of \(z\)
- \(\text{Im}(z):=y\) is the imaginary part of \(z\).
If \(y=0\), then \(z\) is real, if \(x=0\), then \(z\) is pure imaginary.
Therefore, we can define the set of complex numbers as
Geometrically, each complex number can be placed on a 2-D coordinate system \((x, y)\), such system is defined as a complex plane.
Note that \(z\) can also be represented by the polar coordinate
so that the complex number \(z\) can also reside in the polar coordinate
And we can define the modulus of \(z\) as its absolute value
and the argument of \(z\) is defined by its angle when \(z\neq 0\)
Note that \(\theta\) is periodic with period \(\pi\), which means it's multivalued.
Geometry of Arithmetic
Using the complex plane, we can better understand the "addition" and "multiplication". Let \(z, w\) be two complex variables. Then, addition is the addition of their vectors in the complex plane.
Multiple \(z\) by \(i\) will rotate \(z\) counterclockwise by 90 degrees.
Multiple \(z\) by \(c\in\mathbb R\) simply scales the vector.
Multiple \(z=a+bi, w=c+di\in\mathbb C\) gives
Geometrically, it scales \(z\) by \(|w|\) and rotates from \(w\) counterclockwise for \(\theta\).
Polar Exponential
Define the polar exponential by
So that \(z\) can be written in the polar exponential form as
Note that this notation is so convenient as we can prove exponential properties from trigonometric identities.
Theorem
Example complex number to polar exponential form (since \(\theta\) is periodic, we will only take values in \(0<\theta\leq 2\pi\))
Example polar exponential form to \(x+yi\)
Example Define \(\cos z = \frac{e^{iz} + e^{-iz}}2, e^z = e^{x}e^{iy}\), evaluate \(\cos(c+i\frac{\pi}4)\) where \(c\in\mathbb R\)
Powers of Complex Numbers
Consider an equation of the form for \(z, w\in\mathbb C, n\in\mathbb N^+\)
so that
Since the period is \(2\pi\), this will yield \(n\) unique roots.
Example
Arithmetic Operations
Equivalence \(z_1 = z_2\) IFF their real and imaginary parts are respectively equal, i.e.
Also note that given the definition as \(\mathbb C = \{a+ib:a,b\in\mathbb R\}\), Addition, subtraction, multiplication, division follows the same rules as real numbers. and the commutative, associative, distributive laws of addition and multiplication hold.
Complex Conjugate
For \(z = x+iy =re^{i\theta}\), define its complex conjugate as \(\bar z = x - iy = re^{-i\theta}\). Consequently, we can have the following results,
and the division can be better viewed as
Results of Elementary Functions
Let \(z=a+ib, w =c+id\in\mathbb C\),
Theorem \(\overline{z+w} = \bar z + \bar w\)
proof \(z+w=(a+c)+i(b+d), \overline{z+w}= (a+c) -i(b+d) = (a-ib)+(c-id) = \bar z + \bar w\)
Theorem \(|z-w| \leq |z|+|w|\)
proof \(|z-w|^2 = (a-c)^2 + (b-d)^2 \leq a^2+b^2 + c^2 + d^2 = |z|^2 + |w|^2\),
since the absolute values are all real numbers, it follows triangle inequality that \(|z|^2 + |w|^2 \leq (|z|+|w|)^2\)
Given \(|z-w| \geq 0, |z|+|w|\geq 0, |z-w|\leq |z|+|w|\)
Theorem \(z-\bar z = 2i Im(z)\)
proof \(z-\bar z = a+ib-(a-ib) = 2ib\)
Theorem \(Re(z)\leq |z|\)
proof \(a \leq \sqrt{a^2+b^2}\)
Theorem \(|wz| = |w||z|\)
proof We will equivalently prove \(|wz|^2 = |w|^2|z|^2\)
\(|wz|^2 = (ac-bd)^2 + (ad+bc)^2 = \cdots = (a^2+b^2)(c^2+d^2)\)
Triangular Inequality
proof First note that
Then,
Using the results above,
So that \(|z+w|^2 \leq |z|^2 + 2|z||w| + |w|^2 = (|z| + |w|)^2\), and take a square root, we have the right inequality.
Note that the reverse triangular inequality is a corollary of triangular inequality, which
Theorem \(|w\bar z + \bar w z| \leq 2|wz|\)
proof First note that \(|z| = |\bar z|\) as \(a^2+b^2 = a^2 + (-b)^2\), then by trig-inequality and the theorem above,
\(|w\bar z + \bar w z| \leq |w\bar z| + |\bar w z| = 2|w||z| = 2|wz|\)