Beyond the Real Number Line
The real number system cannot solve certain simple equations, like x² = -1. To solve this, mathematicians defined the imaginary unit, i.
i = √-1
This means i² = -1.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers.
a is called the real part.
b is called the imaginary part.
The set of complex numbers includes all real numbers (where b=0) and all imaginary numbers (where a=0).
Arithmetic with Complex Numbers
Addition/Subtraction: Add or subtract the real and imaginary parts separately.
(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication: Use the distributive property (like FOIL), and remember that i² = -1.
(a + bi)(c + di) = ac + adi + bci + bdi²
= ac + (ad + bc)i + bd(-1)
= (ac - bd) + (ad + bc)i
The Complex Conjugate and Division
The complex conjugate of a complex number a + bi is a - bi.
The product of a complex number and its conjugate is always a real number: (a + bi)(a - bi) = a² + b².
Division: To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. This makes the denominator a real number, allowing for simplification.
Example: (3 + 2i) / (1 - i)
Multiply top and bottom by (1 + i): [(3 + 2i)(1 + i)] / [(1 - i)(1 + i)]
Numerator: (3 - 2) + (3 + 2)i = 1 + 5i
Denominator: 1² + 1² = 2
Result: (1 + 5i) / 2 = 1/2 + 5/2 i
The Complex Plane
Complex numbers can be graphed on a two-dimensional plane called the complex plane.
The horizontal axis is the real axis.
The vertical axis is the imaginary axis.
The number a + bi is plotted at the point (a, b).