This post is targeted at a Level 3 student.
After having learned about rational numbers and irrational numbers, which form the set of real numbers, the next extension that students see will be the complex numbers. This arises naturally when we’re looking to solve a quadratic equation. We know that the solutions to are . What are the solutions to ?
is often used to denote the imaginary unit, which satisfies the equation . If so, and will be the roots to the equation . With this symbol, we can extend the real numbers to obtain the set of complex numbers, which are of the form , where and are real numbers.
Let’s see how the usual arithmetic operations work:
1) Addition: .
2) Subtraction: .
3) Multiplication: .
Division becomes slightly tricky, because we only know how to divide by a real number. Rewriting as isn’t very helpful. If only we could make the denominator a real number … To do so, we introduce the idea of a conjugate:
We have , using our nifty multiplication formula. This gives us a real non-negative value. Now, we introduce the idea of a Norm and absolute value:
With this, we have the following:
4) Division: If is non-zero, then .
In an upcoming post, we will study the Polar Form of complex numbers.
1. If and are real numbers such that , then and .
Corollary: This allows us to compare coefficients of real and imaginary parts. In particular, if , then we must have .
2. With real numbers, we are familiar with the concept of reciprocals. For example, 2 and are reciprocals of each other because . What is the reciprocal of a non-zero complex number ?
3. Verify that conjugation distributes over multiplication and division. Specifically, show that and .
Let then, whileHence they are the same.As for the division, let’s use the language of norm and conjugates that we’ve learned. You can also do this using as above.
You should easily verify that conjugation distributes over addition and subtraction.
1. Determine the 4 roots of the equation , and find their real and imaginary parts.
2. Verify that the norm distributes over multiplication and division. Specifically, show that and that . Give examples to show that the norm does NOT distribute over addition and subtraction.
3. If , what is ?
Hint: There is no need to determine the exact values of and .
4. Determine the square root of .
Note: We can show that the square root of is equal to . Currently, our only way to show this is through brute force multiplication. We will be learning how to approach this problem otherwise.