Operations with Complex Numbers
We will now explore operations of complex numbers via transformational geometry.
We will now explore operations of complex numbers via transformational geometry.
Addition
Like real numbers, we are able to operate on complex numbers with addition by combining like terms and adding arithmetically. If we have two complex numbers (a + bi) + (c + di), where a, b, c, and d are real numbers, we add the real parts of the two complex numbers and the imaginary numbers of the two complex numbers. This adds the two numbers to be equal to (a + c) + (bi + di). Geometrically adding is translation as visible in Figure 13.
Like real numbers, we are able to operate on complex numbers with addition by combining like terms and adding arithmetically. If we have two complex numbers (a + bi) + (c + di), where a, b, c, and d are real numbers, we add the real parts of the two complex numbers and the imaginary numbers of the two complex numbers. This adds the two numbers to be equal to (a + c) + (bi + di). Geometrically adding is translation as visible in Figure 13.
Here the complex number, z and w are being added together to represent the complex number z +w. We translate z by a vector equal to w in length direction, as demonstrated by the gray line from z to z + w, resulting in a parallelogram formed in Figure 13. Subtraction is the same as addition, but done negating numbers so we are safe to assume that translating is also applied to subtraction. In the complex plane, negated numbers, −z, are represented by moving in the opposite direction of z, but the same length. See Figure 14.
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