Complex Number Arithmetic: A First Look
Imagine you're trying to solve x2+1=0. You know that no real number squared gives −1. The square of any real number is either zero or positive. So this equation has no real solution. But what if we invent a number whose square is −1? That's exactly what mathematicians did — and that invention is the imaginary unit i, defined by:
A complex number is any number of the form a+bi, where a and b are real numbers. Here a is called the real part, and b is called the imaginary part. For example, 3+4i has real part 3 and imaginary part 4.
The name "imaginary" is unfortunate — these numbers are just as real (in the mathematical sense) as the numbers you already know. They're simply a different kind of number.
Why Bother?
Complex numbers let you solve equations that real numbers can't. Every polynomial equation — no matter how complicated — has a solution in the complex numbers. This is the Fundamental Theorem of Algebra, and it's one of the most important results in mathematics.
Arithmetic Operations
The rules are straightforward: treat i like a variable, but remember that i2=−1.
Addition and Subtraction
Add (or subtract) real parts with real parts, imaginary parts with imaginary parts.
(a+bi)+(c+di)=(a+c)+(b+d)i
(a+bi)−(c+di)=(a−c)+(b−d)i
Example: (2+3i)+(4−5i)=(2+4)+(3−5)i=6−2i
Multiplication
Multiply like binomials, then replace i2 with −1.
(a+bi)(c+di)=ac+adi+bci+bdi2=(ac−bd)+(ad+bc)i
Example: (2+3i)(4−5i)=2(4)+2(−5i)+3i(4)+3i(−5i)
=8−10i+12i−15i2
=8+2i−15(−1)
=8+2i+15=23+2i
The most common mistake: forgetting that i2=−1, not 1. Always check your final step.
Division
Division is trickier. The key idea: multiply numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of a+bi is a−bi. When you multiply a complex number by its conjugate, you get a real number:
(a+bi)(a−bi)=a2−(bi)2=a2−b2i2=a2+b2
So to divide:
c+dia+bi=(c+di)(c−di)(a+bi)(c−di)=c2+d2(a+bi)(c−di)
Example: 4−5i2+3i=(4−5i)(4+5i)(2+3i)(4+5i)=16+258+10i+12i+15i2=418+22i−15=41−7+22i=−417+4122i
To divide quickly: multiply top and bottom by the conjugate of the denominator, then simplify. The denominator always becomes c2+d2, a positive real number.
The Big Picture …