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How Do You Add Complex Numbers by Graphing in the Complex Plane?

How do you add 2 + 4i and 5 + 3i graphically?

Summary

  1. 'i' is the imaginary unit and is equal to the square root of -1
  2. On the complex plane, the horizontal axis is the real axis
  3. The vertical axis is the imaginary axis
  4. Draw lines from each graphed point to the origin
  5. Starting at 5+3i, draw another line by moving right 2 and up 4
  6. Starting at 2+4i, draw another line by moving right 5 and up 3
  7. The point where the lines meet is the sum of the complex numbers, 7+7i

Notes

    1. The complex plane is the coordinate system we use to graph complex numbers
    2. Just like the coordinate plane, the complex plane has two axes
    3. But the horizontal axis is the real axis and the vertical axis is the imaginary axis
    4. So we graph the real part of the complex number on the real axis and the imaginary part on the imaginary axis
    1. 'i' is the imaginary unit and is equal to the square root of -1
    2. 2 is the real part of the complex number
    3. Since 2 is positive, we move 2 units to the right on the real (horizontal) axis
    4. 4i is the imaginary part of the complex number
    5. Since 4i is also positive, we move 4 units up on the imaginary (vertical) axis
    1. 'i' is the imaginary unit and is equal to the square root of -1
    2. 5 is the real part of the complex number
    3. Since 5 is positive, we move 5 units to the right on the real (horizontal) axis
    4. 3i is the imaginary part of the complex number
    5. Since 3i is also positive, we move 3 units up on the imaginary (vertical) axis
    1. Remember, the origin is the point (0,0)
    2. We'll use these lines later when we actually go to add the complex numbers
    1. Remember, a parallelogram has opposite sides that are parallel and congruent
    2. Congruent sides are the same length
    1. To create a parallelogram, we need to make two more sides that are parallel and congruent to the two we already have
    2. Parallel lines have the same slope, or rise over run
    3. Start at 5+3i, then use the rise over run from 2+4i to draw a new line parallel and congruent to the line we drew to the origin
    4. Then start at 2+4i, and use the rise over run from 5+3i to draw a new line parallel and congruent to the line we drew to the origin
    1. The new point we reached when we made our parallelogram is the sum of the two complex numbers
    2. Counting on the real axis, we've gone right 7 units
    3. That means the real part of the solution will be positive 7
    4. Counting on the imaginary axis, we've gone up 7 units
    5. That means the imaginary part of the solution will be positive 7i
    6. Put the two parts together, and our sum is 7+7i!