MPF - Operations of Complex Numbers Lesson

Operations of Complex Numbers

Quadratic Equations and Inequalities are a part of the Polynomials family. Quadratic Equations and Inequalities introduce students to the graphs of quadratics, teach them to find the vertex, intercepts, discriminant, domain and range and interpret the graph in relation to these qualities.

When a real number, equation image indicator LaTeX: a $a$ , is added to an imaginary number, LaTeX: bi $bi$ , it is said to be a complex number. Complex numbers are numbers with both a real and an imaginary part. The imaginary part of a complex number, , consists of two elements: a real number element, b, and an imaginary element, i, defined as LaTeX: i^2=-1 $i^2=-1$ . Imaginary numbers are applied to square roots of negative numbers, allowing them to be simplified in terms of i. The standard form of a complex number is equation image indicator LaTeX: a+bi $a+bi$ . The following are some examples of complex numbers: $LaTeX: 8+4i,\:10-i,\:5i,\:and\:-13i$ $8+4i,\:10-i,\:5i,\:and\:-13i$ . The imaginary number i , where $LaTeX: i=\sqrt[]{-1}$ $i=\sqrt[]{-1}$ , has powers, which can be found from combinations of i and equation image indicator LaTeX: i^2 $i^2$ . They can be determined as shown below.

equation image indicator $LaTeX: i=\sqrt[]{-1}$ $i=\sqrt[]{-1}$

$LaTeX: i^2=\left(\sqrt[]{-1}\right)^1=-1$ $i^2=\left(\sqrt[]{-1}\right)^1=-1$

$LaTeX: i^3=i^2\cdot i=-\sqrt[]{-1}$ $i^3=i^2\cdot i=-\sqrt[]{-1}$

$LaTeX: i^4=i^2\cdot i^2=\left(-1\right)\left(-1\right)=1$ $i^4=i^2\cdot i^2=\left(-1\right)\left(-1\right)=1$

Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers is the same idea as combining like terms. In an expression, the coefficients of i can be summed together just like the coefficients of variables. If an expression has real numbers and square roots of negative numbers, rewrite using i and then combine like terms.

First, we need to review how to simplify radicals that have negatives, using i.

When adding and subtracting complex numbers, you combine like terms.

The real parts are combined and the imaginary parts are combined.

For example: equation image indicator $LaTeX: \left(-3-7i\right)+\left(4+2i\right)=-3+4+-7i+2i=1-5i$ $\left(-3-7i\right)+\left(4+2i\right)=-3+4+-7i+2i=1-5i$

Multiplying and Dividing Complex Numbers

To simplify expressions by multiplying complex numbers, we use exponent rules for equation image indicator $LaTeX: i=\sqrt[]{-1},\:i^2=-1,\:i^3=-\sqrt[]{-1},\:i^4=1$ $i=\sqrt[]{-1},\:i^2=-1,\:i^3=-\sqrt[]{-1},\:i^4=1$ . The first step would be to rewrite the radicals using LaTeX: a+bi $a+bi$ form, multiply them together, and then simplify.

When dividing complex numbers, the first step would be to rewrite the complex numbers in equation image indicator LaTeX: a+bi $a+bi$ form in the numerator and denominator. The second step is to multiply the numerator and denominator by the complex conjugate of the denominator. This will result in a complex number in the numerator and a real number only in the denominator.

Watch the following videos on complex conjugates and dividing complex numbers to see a more in-depth explanation.

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