The Math Daily News

Imaginary Numbers & Solving Systems with 3 Variables

Imaginary Numbers!? Don't worry, it's simple!

Imaginary numbers at a glance seem extremely complicated, but when broken down, they are extremely simple. As you can see in the picture below, "i" simply equals the square root of -1 (outlined in orange). "i^2", therefore, equals the square root of -1 multiplied by the square root of -1. As we know from our basic laws of square roots, when you take the product of two square roots with the same radicand, it simply equals that radicand. Therefore, "i^2", equals -1. "i^3" can be broken down into the product of i^2 and i. We just determined that i^2 equals -1, making "i^3" equal -1*i, or -i. Finally, "i^4" can be broken down into i^2 multiplied by i^2. We know now that i^2 equals -1, so "i^4" equals -1*-1, or 1.
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Example 1: Simplifying Basic Expressions

In Example 1, below, we are asked to simplify "i^70". Using the main concept learned above, i^70 can be broken down into factors of i that are more basic. In the first step, i^70 is broken down into i^68 and i^2. Because i^68 is a factor of i^4, i^68 equals 1, and i^2, as we learned, equals -1. Therefore, i^70 equals 1*-1, or -1.
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Example 2: Adding and Subtracting

In the picture below, there are two examples: one adding, and one subtracting. In the first example, the like terms are simply combined and put into a+bi form. In the second example, where there is subtraction, the same process happens. However, when simplifying the expression, it is VERY important to remember to distribute the negative to both the 3 AND -6i, as shown in the work below. The final answers after the like terms were combined are outlined in red.
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Example 3: Multiplying

In this example, where we are multiplying two complex expressions, we begin by foiling the expression while treating the i's as if they were a variable. After foiling, you substitute -1 in for i^2, as we determined them equal to each other in the first diagram. Once this is done, you can combine like terms until you reach a simplified expression in a +bi format. In this problem, the answer is 10-10i.
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Example 4: Simplifying Using Complex Conjugates

Complex Conjugates are two complex numbers in the form a+bi and a-bi. The product of complex conjugates is ALWAYS a real number, so they can be used to eliminate imaginary numbers from the denominator when simplifying an expression that has a complex number in the denominator.

In the work shown below, the numerator and denominator of the expression we are told to simplify is multiplied by the complex conjugate of the original denominator. In step 2, the product of those two expressions is found. In step 3, -1 is substituted for i^2, and as you can see, when step 4 simplified the expression, there is now a real number in the denominator. Finally, in step 5, everything is divided by 3, and the expressions is put into a+bi form, yielding a final answer of (4/15) + (2/15i).

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Recap: Solving Systems of Equations with 3 Variables


1) Eliminate one variable by using two pairs of equations.

2) Solve the system of equations to the two variables that weren't eliminated.

3) Substitute the two known variables into one of the original equations in order to find the third variable that was eliminated in step one.

4) Check your work! Substitute your 3 variables in to one of the original equations to make sure it works.

Mistakes in video:

  • 2:27- I say 26, where it should say, as it is written on the whiteboard, -26
  • 4:57 & 5:06- I say "write it in ordered pair form, whereas using the correct terminology, I should say "written as an ordered triple".

SUMMARY Of Concepts Learned

CONCEPT: Imaginary Numbers

-Simplify basic expressions

-Add and subtract


-Simplify using complex conjugates

CONCEPT: Solving Systems With 3 Variables

-Video of example

-Steps written out

Real Life Application of Imaginary Numbers

Imaginary Numbers are used everyday by electrical engineers that deal with the voltage, current, and impedance of a circuit.

Real life application of systems of equations with 3 variables

The ordered triple for a system of equations with 3 variables often represents points on a graph. That ordered triple is then directly involved in graphing planes in a three-dimensional space. Architects, surveyors, and cartographers use coordinate graphing in 3D everyday at work.

Upcoming Assesments

Unit 4 Quiz

Monday, Dec. 7th, 10:45am

Mrs. Govoni's Classroom, Hopkinton High School


Thursday, Dec. 10th, 9:30am

Mrs. Govoni's Classroom, Hopkinton High School