Complex numbers arise naturally when solving quadratic equations. We know that the solutions to are . What are the solutions to ?
is often used to denote the imaginary unit, which satisfies the equation . 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.
We say that for a complex number , it has real part , denoted as , and imaginary part , denoted as . Take note that the imaginary part is a real number.
Let's see how the usual arithmetic operations work:
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:
The conjugate of the complex number is . Notice that the conjugate of the conjugate is the identity, i.e. .
We now 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:
The norm of the complex number is .The absolute value of the complex number is the positive square root of the norm, and is given by .
With this, we have the following:
4) Division: If is non-zero, then .
is known as the Rectangular Form of the complex number. In an upcoming post, we will study the Polar Form of complex numbers.
1. If and are real numbers such that , then and .
This should be obvious to you, but let's show it. If , then . Squaring both sides, we get . By non-negativity of squares, we have , which implies that equality must hold throughout. Thus, , which gives .
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 ?
Solution 1: Since , . We seek the value of so will use the division operation, to obtain that .
Solution 2: Using the language of norm and conjugates, we can express this directly as:.
3. Verify that conjugation distributes over multiplication and division. Specifically, show that and .
Hence 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.