Complex Numbers
Understanding the fundamentals of complex numbers and their properties
Why Do We Need Complex Numbers?
Throughout the history of mathematics, the number system has been continuously extended to solve new problems:
- Natural numbers (\(\mathbb{N}\)) for counting
- Integer numbers (\(\mathbb{Z}\)) to include negatives
- Rational numbers (\(\mathbb{Q}\)) for fractions
- Real numbers (\(\mathbb{R}\)) for irrational values
- Complex numbers (\(\mathbb{C}\)) to solve equations like \(x^2 = -1\)
Even with real numbers, certain fundamental equations cannot be solved. For example:
\(\displaystyle x^2 + 1 = 0 \quad \text{or} \quad x^2 = -1\)
This equation has no solution in the real numbers because the square of any real number is always non-negative. To solve this problem, mathematicians extended the number system to include complex numbers.
Complex numbers were initially considered "imaginary" because they seemed unreal. However, they are fundamental to modern mathematics, physics, and engineering.
The Imaginary Unit \(i\)
To solve the equation \(x^2 = -1\), mathematicians introduced a new number called the imaginary unit, denoted by the letter \(i\).
The imaginary unit \(i\) is defined as a number that, when squared, equals \(-1\):
\(\displaystyle i^2 = -1\) or equivalently \(\displaystyle i = \sqrt{-1}\)
The imaginary unit \(i\) is not a real number because no real number, when squared, produces a negative result. However, using the ordinary calculation rules (principle of permanence), we can perform operations with \(i\).
Powers of the Imaginary Unit
\(i^1\)
\(\displaystyle i^1 = i\)\(i^2\)
\(\displaystyle i^2 = -1\)(Definition)
\(i^3\)
\(\displaystyle i^3 = i^2 \cdot i = (-1) \cdot i = -i\)\(i^4\)
\(\displaystyle i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1\)\(i^5\)
\(\displaystyle i^5 = i^4 \cdot i = 1 \cdot i = i\)(Pattern repeats)
Pattern
Powers of \(i\) repeat with period 4:\(i, -1, -i, 1, i, -1, ...\)
Example: Calculate \(i^{10}\)
Since powers of \(i\) repeat every 4 terms:
\(\displaystyle i^{10} = i^{8+2} = (i^4)^2 \cdot i^2 = 1^2 \cdot (-1) = -1\)
Or: \(\displaystyle 10 = 4 \times 2 + 2\), so \(\displaystyle i^{10} = i^2 = -1\)
Definition of Complex Numbers
A complex number is formed by combining a real number with an imaginary number.
A complex number \(z\) is defined as:
\(\displaystyle z = a + bi\)
where:
- \(a\) is a real number (the real part)
- \(b\) is a real number (the coefficient of the imaginary part)
- \(i\) is the imaginary unit with \(i^2 = -1\)
Examples of Complex Numbers
Complex Number Examples
- \(\displaystyle 3 + 4i\) — Real part: 3, Imaginary coefficient: 4
- \(\displaystyle -1 + 3i\) — Real part: -1, Imaginary coefficient: 3
- \(\displaystyle 2 - 3i\) — Real part: 2, Imaginary coefficient: -3
- \(\displaystyle 5\) — Real part: 5, Imaginary coefficient: 0 (purely real)
- \(\displaystyle 2i\) — Real part: 0, Imaginary coefficient: 2 (purely imaginary)
- \(\displaystyle 0\) — Real part: 0, Imaginary coefficient: 0 (zero)
Alternative Notation: Pair Form
Complex numbers can also be written as ordered pairs of real numbers:
\(\displaystyle z = (a, b) \quad \text{or} \quad z = (\text{Re}(z), \text{Im}(z))\)
Pair Form Examples
- \(2 + 3i \leftrightarrow (2, 3)\)
- \(5 + 0i \leftrightarrow (5, 0)\) — Real number
- \(0 + 2i \leftrightarrow (0, 2)\) — Purely imaginary
- \(-1 - 4i \leftrightarrow (-1, -4)\)
Real and Imaginary Parts
For any complex number, we can identify two key components using specific notation.
For a complex number \(z = a + bi\):
Real Part
\(\displaystyle \text{Re}(z) = a\)The real-valued component
Imaginary Part
\(\displaystyle \text{Im}(z) = b\)The real coefficient of \(i\)
For \(z = 2 - 5i\):
- \(\text{Re}(z) = 2\) (the real part)
- \(\text{Im}(z) = -5\) (the imaginary part is just the coefficient)
- NOT \(\text{Im}(z) = -5i\) — The imaginary part is a real number!
Example: Identifying Parts
For the complex number \(z = 3 - 7i\):
- Real part: \(\text{Re}(z) = 3\)
- Imaginary part: \(\text{Im}(z) = -7\)
- Imaginary unit multiple: \(-7i\)
Complex Numbers as an Extension of Real Numbers
Complex numbers form a superset of real numbers. Every real number can be written as a complex number with an imaginary part of zero.
Number System Hierarchy:
\(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\)
Natural Numbers ⊂ Integers ⊂ Rationals ⊂ Reals ⊂ Complex Numbers
This relationship means:
- Every real number \(a\) can be written as \(a + 0i\)
- Real numbers are complex numbers with \(\text{Im}(z) = 0\)
- The set of complex numbers includes all real numbers plus imaginary numbers
Why We Cannot Order Complex Numbers
Because complex numbers consist of two independent components (real and imaginary), we cannot order them with relations like "greater than" or "less than" as we do with real numbers.
We cannot say \(3 + 2i > 2 + 3i\) or \(2i < 5i\)
These comparisons are undefined for complex numbers!
Geometric Representation: The Complex Plane
Because a complex number has two components, it cannot be represented on a one-dimensional number line. Instead, we use a two-dimensional coordinate system called the complex plane or Argand diagram.
In the complex plane:
- Horizontal axis (x-axis): Represents the real part \(\text{Re}(z)\)
- Vertical axis (y-axis): Represents the imaginary part \(\text{Im}(z)\)
Complex Plane Representation:
The complex number \(z = a + bi\) is represented as point \((a, b)\)
Example: \(z = 3 + 4i\) is plotted at coordinates (3, 4)
This geometric representation allows us to:
- Visualize complex numbers as points in a plane
- Understand operations on complex numbers geometrically
- Calculate distances and angles involving complex numbers
- Study the behavior of complex-valued functions
The complex plane is also called the "Argand diagram" after Swiss mathematician Jean-Robert Argand, who developed this geometric representation.
Summary of Complex Number Basics
| Concept | Definition | Notation |
|---|---|---|
| Complex Number | Combination of real and imaginary parts | \(z = a + bi\) or \((a, b)\) |
| Imaginary Unit | Number with property \(i^2 = -1\) | \(i\) |
| Real Part | The real component \(a\) | \(\text{Re}(z) = a\) |
| Imaginary Part | The coefficient \(b\) of \(i\) (a real number) | \(\text{Im}(z) = b\) |
| Purely Real | \(\text{Im}(z) = 0\) | \(a + 0i = a\) |
| Purely Imaginary | \(\text{Re}(z) = 0\) | \(0 + bi = bi\) |
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