Define complex numbers, discuss the algebraic operations with complex numbers, introduce the complex plane, and explain the polar form of complex numbers.
Introduction to Complex Numbers
Key ideas, definitions, operations, and visual tools
| Symbol | Meaning |
|---|---|
| ( \mathbb{R} ) | Set of real numbers |
| ( i ) | Imaginary unit, defined by ( i^2 = -1 ) |
| ( \mathbb{C} ) | Set of all numbers of the form ( a + bi ) with ( a,b \in \mathbb{R} ) |
A complex number is written ( z = a + bi ).
If ( b = 0 ) the number is purely real; if ( a = 0 ) it is purely imaginary.
A point ((a,b)) corresponds to the complex number (a+bi).
[ z = r(\cos\theta + i\sin\theta) = re^{i\theta} ]
| Operation | Algebraic Form | Polar Form |
|---|---|---|
| Addition | ((a+bi)+(c+di) = (a+c) + (b+d)i) | Add vectors tip‑to‑tail in the plane |
| Subtraction | ((a+bi)-(c+di) = (a-c) + (b-d)i) | Same as addition with the opposite vector |
| Multiplication | ((a+bi)(c+di) = (ac-bd) + (ad+bc)i) | ( r_1r_2,e^{i(\theta_1+\theta_2)}) |
| Division | (\displaystyle\frac{a+bi}{c+di}= \frac{(ac+bd)+(bc-ad)i}{c^{2}+d^{2}}) | (\displaystyle\frac{r_1}{r_2},e^{i(\theta_1-\theta_2)}) |
| Conjugate | (\overline{z}=a-bi) | (\overline{z}=re^{-i\theta}) |
| Modulus | ( | z |
| Argument | (\arg(z)=\theta) (principal value (-\pi<\theta\le\pi)) | Same |
Why the conjugate helps:
[
z\overline{z}= (a+bi)(a-bi)=a^{2}+b^{2}=|z|^{2}
]