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Understanding Addition and Subtraction of Complex Numbers Made Easy

This guide explores the processes of adding and subtracting complex numbers. It explains the general formulas for addition and subtraction of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i and (a + bi) - (c + di) = (a - c) + (b - d)i. Examples illustrate how to combine real and imaginary parts, along with graphical interpretations using vectors. Additionally, it discusses the additive inverse and offers practice problems for readers to enhance their understanding of complex number operations.

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Understanding Addition and Subtraction of Complex Numbers Made Easy

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  1. Aim: How do we add and subtract complex numbers? Do Now: Simplify:

  2. Find the sum of Adding Complex Numbers = 7 + 4i = (2 + 5) + (3i + i) (2 + 3i) + (5 + i) In general, addition of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i Combine the real parts and the imaginary parts separately. convert to complex numbers combine reals and imaginary parts separately

  3. Subtract Subtracting Complex Numbers What is the additive inverse of 2 + 3i? -(2 + 3i) or -2 – 3i Subtraction is the addition of an additive inverse = -2 + i = (1 + 3i) + (-3 – 2i) (1 + 3i) – (3 + 2i) In general, subtraction of complex numbers: (a + bi) – (c + di) = (a – c) + (b – d)i change to addition problem combine reals and imaginary parts separately

  4. yi 5i 4i vector: 2 + 3i 3i (2 + 3i) 2i (5 + 3i) (3 + 0i) vector: 3 + 0i i x 1 -5 -4 -3 -2 -1 0 2 3 4 5 6 -i -2i -3i -4i -5i -6i Adding Complex Numbers Graphically (2 + 3i) + (3 + 0i) = (2 + 3) + (3i + 0i) = = 5 + 3i vector: 5 + 3i

  5. S P resultant force OS O R Adding Vectors Vector - a directed line segment that represents directed force notation: The vectors that represent the applied forces form two adjacent sides of a parallelogram, and the vector that represents the resultant force is the diagonal of this parallelogram.

  6. yi 5i 4i 3i (1 + 3i) 2i (3 + 2i) i (-3 – 2i) (-2 + i) x 1 -5 -4 -3 -2 -1 0 2 3 4 5 6 -i -2i -3i -4i -5i -6i Subtracting Complex Numbers Graphically (1 + 3i) – (3 + 2i) = (1 + 3i) + (-3 – 2i) = -2 + i The vector representing the additive inverse is the image of the vector reflected through the origin. Or the image under a rotation about the origin of 1800.

  7. Model Problems Add/Subtract and simplify: (10 + 3i) + (5 + 8i) = 15 + 11i (4 – 2i) + (-3 + 2i) = 1 Express the difference of in form a + bi

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