Complex Numbers in Cartesian Form and the Argand Diagram
The Imaginary Unit
You have probably come across equations such as $x^2 + 4 = 0$, and written "no real solutions". Well, some mathematicians back in the 16th century essentially said "nah" to that, and came up with a new imaginary unit, defined as:
$$i^2 = -1$$
These, despite being called "imaginary", are actual mathematical objects you can do operations on.
Understanding Complex Numbers in Cartesian Form
Complex numbers are expressions of the form $a + bi$, where:
- $a$ is the real part
- $b$ is the imaginary part
The real and imaginary parts can be any real number, including zero. When $b = 0$, we have a real number, and when $a = 0$, we have a purely imaginary number.
Basic Operations with Complex Numbers
- Addition and Subtraction works exactly as you'd expect.
$$(a + bi) ± (c + di) = (a ± c) + (b ± d)i$$ - Multiplication works similarly to multiplying two binomials. However, remember to convert any terms containing $i^2$ back to real (but negative) terms.
$$(a + bi)(c + di) = ac + adi + bci + bdi^2=(ac - bd) + (ad + bc)i$$
Let's multiply $(2 + 3i)(1 - 2i)$:
- First term: $2(1) - 2(3i) = 2 - 6i$
- Second term: $3i(1) - 3i(2i) = 3i + 6$
- Combining: $(2 - 6i) + (3i + 6) = 8 - 3i$
Complex conjugates
For any complex number $z = a + bi$, its conjugate is defined as $z* = a - bi$.
Adding or multiplying a complex number with its complex conjugate always yields a real number.
- Addition: $z + z* = a + bi + a - bi = 2a$
- Multiplication: $zz* = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2$
The conjugate of a complex number $z$ can also be represented as $\bar{z}$.
The Argand Diagram
The Argand diagram (also known as the complex plane) is a geometric representation of complex numbers where:
- The horizontal axis represents real numbers
- The vertical axis represents imaginary numbers
- Each point $(x, y)$ represents the complex number $x + yi$
Think of the Argand diagram as an extension of the real number line into two dimensions, where we can visualize complex numbers as points or vectors. As seen on the diagram above, we have three points $A,B,C$. The $x$-axis represents the real component while the $y$-axis represents the imaginary component.
Point $A$ has coordinate $(1,1)$ hence is the complex number $1+i$.
Point $B$ has coordinate $(-1,-1.5)$ hence is the complex number $-1-1.5i$
Point $C$ has coordinate $(-1,0)$ hence is the complex number $-1+0i$ or in other words, has no complex part and therefore is the real number $-1$.
Properties on the Argand Diagram
- Modulus
- The modulus of a complex number $z = a + bi$ is its distance from the origin
- Calculated using: $|z| = \sqrt{a^2 + b^2}$
- Complex Conjugates
- For $z = a + bi$, its conjugate $z* = a - bi$ is its reflection across the real axis.
Geometric Interpretations
- Addition
- Represented by vector addition
- Follow the parallelogram rule or tip-to-tail method
- Multiplication by $i$
- Rotates a complex number by 90° counterclockwise
- Multiplying by $i$ twice rotates by 180°
- Multiplication by -1
- Rotates a complex number by 180°
- Alternatively, reflects it across the origin
- Multiplication by a scalar $k$
- Increases the modulus of a complex number by a factor of $k$
If $z = 2 + i$:
- $iz = i(2 + i) = 2i + i^2 = -1 + 2i$ (90° rotation)
- $i^2z = i(i(2 + i)) = -2 - i$ (180° rotation)
When working with complex numbers geometrically, always sketch the Argand diagram to visualize transformations and relationships.
Applications in Problem Solving
Complex numbers in Cartesian form are particularly useful for:
- Solving quadratic equations with negative discriminants
- Representing rotations and scaling in 2D geometry
- Analyzing electrical circuits in engineering
The Cartesian form is just one way to represent complex numbers. In later topics, we'll explore polar form, which is often more convenient for multiplication and division operations.
