Imaginary (Complex) Numbers

This section covers:

Introduction to Imaginary Numbers

Think of imaginary numbers as numbers that are typically used in mathematical computations to get to/from “real” numbers (because they are more easily used in advanced computations), but really don’t exist in life as we know it.  Yet they are real in the sense that they do exist and can be explained quite easily in terms of math – think  ofInfinity(infinity) – does that really exist?.  And we’ve all had imaginary friends, right?image700

Actually, imaginary numbers are used quite frequently in engineering and physics, such as an alternating current in electrical engineering, which is usually represented by a complex number.  (Don’t worry; I don’t know what an alternating current is, either.)

A complex number consists of a “real” part and an “imaginary” part, and typically  looks like a plus bi, where “a” is the real part, and “b” is the imaginary part, following by “i”, to indicate the “imaginary” unit.  For now, let’s just learn about the imaginary part.   (Note that all real numbers, even without the imaginary part, are considered to be complex, as we saw in Types of Numbers).

Let’s look at the following graphs and notice that the parabolas never touch the x axis, so there aren’t any x-intercepts, although the “roots”, “solutions”, and “values” are “complex” or “imaginary”.

As we learned earlier, since these graphs have no roots, they have a discriminantDiscriminant that is less than 0.  The graphs will never touch the x axis, yet we can still find imaginary roots, and the roots will have “i”s in them, as we see later.   But we can’t find these roots with a graphing calculator!

Quadratics with Imaginary Solutions

Working with “i”,  the Square Root of -1 Large

Before working problems that have imaginary solutions, we need to learn about the value of a special “number” called “i”.   i is simply Square Root of -1, which can’t exist in our “real” system, since we can never take two “real” numbers multiplied together to get  –1.  Since i equals Square Root of -1, then it follows that:

i Squared = -1

So, Square Root of -16  Similarly, Square Root of -3 (if you put the i at the end, make sure it is clearly outside of the square root sign in this case).

And here’s something really cool: when we multiply i’s together, we notice a pattern:

Patterns of i

Note that there’s a repeating pattern when raising “i” to an exponent. Notice that every fourth exponent number repeats, so Repeating i's, and so on.  Because of this, we can easily compute “i” raised to any exponent by dividing that exponent by four, and examining the remainder, as shown in the examples:

I Pattern Examples

Again, when dealing with complex numbers, expressions contain a real part and an imaginary part.  Together they form a complex number that typically looks like a + bi, where “a” is the real part, and “b” is the imaginary part, following by “i”, to indicate the “imaginary” unit.  (Later, in Pre-Calculus, we’ll see how these can be graphed on a coordinate system, where the “x” is the real part and the “y” is the imaginary part.)

For example, “4 + 5i” indicates the number Unsimplified Imaginary Expression, and we cannot mix the real parts with the imaginary parts when adding or subtracting, so that the “i’s” are treated somewhat like variables (like radicals were thought as variables, back in the Exponents and Radicals in Algebra section).

So when we perform operations on i, we pretty much treat it like a variable, except when we’re multiplying the “i’s” together – and then we can simplify.   Note that for good “math grammar” we want our final answer to be in the form a + bi.  Here are some examples:

Complex Number Examples

You can also put complex expressions in the graphing calculator:

Complex Numbers in Graphing Calculator

(Note that the complex conjugate that we used to simplify a denominator with an imaginary number in it is similar to the radical conjugate we learned about here in the Introduction to Quadratics section.)

Quadratic Formula with Complex Solutions

Now let’s solve a quadratic equation that has complex (imaginary) solutions.

Let’s take the equation x squared minus 2x plus 2.  We know that since the discriminant Discriminant for a, b, and c in ax squared +bx +c is negative ( –4), there are no real solutions to the equation.

Let’s use the quadratic equation to find this solution, and one that’s a little more complicated:

Using Quadratic Formula to get Complex Solutions

Completing the Square with Complex Solutions

Let’s try completing the square with a quadratic with complex solutions:

Completing the Square with Complex Solutions

Yeah!  We got the same answers as when we solved with the Quadratic Equation!

We learned earlier here in the Introduction to Quadratics section that when we have an irrational value for a root, the conjugate is also a root.  (For example, if 3 + square root of 17 is a root, then 3 - square root of 17 is also a root).

Similarly, if we have a complex root, the complex conjugate is also a root; this is called the Complex Conjugate Root Theorem, or Complex Conjugate Zeros Theorem.  For example, if  3 + i is a root, then 3 – i is also a root.  Interesting!

Learn these rules and practice, practice, practice!


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On to Compositions of Functions, Even and Odd, and Increasing and Decreasing – you are ready!

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