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2 Tutorials that teach Adding and Subtracting Complex Numbers
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Adding and Subtracting Complex Numbers

Adding and Subtracting Complex Numbers

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In this lesson, students will learn how to add and subtract complex numbers with the imaginary unit i.

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Tutorial
This tutorial covers how to add and subtract imaginary and complex numbers, through the exploration of:
  1. Imaginary and Complex Numbers
  2. Adding and Subtracting Imaginary Numbers
  3. Adding and Subtracting Complex Numbers


1. Imaginary and Complex Numbers

To review, the square root of a negative number is a non-real, or imaginary, number. The imaginary unit i is defined as the square root of -1.

KEY FORMULA
i equals square root of negative 1 end root

A complex number is a value in the form below, in which a and b are real numbers, and i is the imaginary unit. In a complex number, a is the real part, and b times i is the imaginary part.

File:1587-nonreal1.PNG

Complex Number
A value of the form a + bi, where a and b are real numbers and i is the imaginary unit
Complex numbers are used in fields such as engineering and physics.


2. Adding and Subtracting Imaginary Numbers

You may recall that the product property for square roots states that the square root of a times b is equal to the square root of a times the square root of b.

KEY FORMULA
square root of a b end root equals square root of a times square root of b

You can apply the product property of square roots to solve equations involving the square root of a negative number, so that you are able to simplify your solution using imaginary numbers.

For example, you can rewrite the square root of -25 as follows, then apply the product property for square roots. Then you are able to simplify to arrive at your solution, which is an imaginary number.
square root of negative 25 end root equals square root of 25 times negative 1 end root equals square root of 25 times square root of negative 1 end root equals 5 i

The product property of square roots can also be applied when adding and subtracting imaginary numbers. Suppose you are solving the equation:

square root of negative 4 end root plus square root of negative 49 end root minus square root of negative 9 end root

Applying the product property for square roots, you can simplify to:

2 i plus 7 i minus 3 i

Now, 2i, 7i, and 3i are all like terms. Therefore, you can combine them together by adding or subtracting their coefficients, to arrive at your final answer:

2 i plus 7 i minus 3 i equals 6 i


3. Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers is similar to combining like terms. You can add or subtract the real parts together, and add or subtract the coefficients of the imaginary parts together. You can add or subtract complex numbers in this way because of the commutative property of addition.

Suppose you want to add the complex numbers:

left parenthesis 4 plus 8 i right parenthesis plus left parenthesis 2 plus 3 i right parenthesis

You would start by adding your real parts, 4 and 2, together. Then you would add your imaginary parts, 8i and 3i, together.

left parenthesis 4 plus 2 right parenthesis plus left parenthesis 8 i plus 3 i right parenthesis

Combining your real parts together and your imaginary parts together gives you the final answer.

table attributes columnalign left end attributes row cell left parenthesis 4 plus 2 right parenthesis plus left parenthesis 8 i plus 3 i right parenthesis equals end cell row cell 6 plus 11 i end cell end table
Try combining like terms to subtract the complex numbers:
left parenthesis 11 minus 6 i right parenthesis minus left parenthesis 7 plus 9 i right parenthesis
Start by combining and subtracting your real parts, 11 minus 7, then combine and subtract your imaginary parts, -6i minus 9i, to arrive at your final answer.
table attributes columnalign left end attributes row cell left parenthesis 11 minus 7 right parenthesis plus left parenthesis negative 6 i minus 9 i right parenthesis equals end cell row cell 4 minus 15 i end cell end table

Today you reviewed imaginary numbers, recalling that the square root of a negative number is non-real, or an imaginary number; the imaginary unit i is equal to the square root of -1. You also reviewed the definition of a complex number, which is a value in the form a plus bi, where a is the real part, and b times i is the imaginary part of the complex number. You learned how to apply the product property of square roots when adding or subtracting imaginary numbers. You also learned that when adding or subtracting complex numbers, you add or subtract the real parts and add or subtract the coefficients of the imaginary parts.

Source: This work is adapted from Sophia author Colleen Atakpu.

TERMS TO KNOW
  • KEY FORMULA

    i = sqrt(-1)

  • KEY FORMULA

    sqrt(ab) = sqrt(a)sqrt(b)

  • Complex Number

    A value of the form a + bi, where a and b are real numbers and 'i' is the imaginary unit.