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Algebra frequently requires the simplification of expressions, and complex numbers—those that contain the imaginary unit i (defined by i² = –1)—can seem intimidating at first glance. However, once you master the fundamental rules, handling complex numbers is straightforward and reliable.
Follow basic algebraic rules—addition, subtraction, multiplication, and division—when working with complex numbers to simplify any expression.
Complex numbers extend the real number system by incorporating the imaginary unit i, the square root of –1. Any complex number can be written in the standard form:
\(z = a + bi\)
Here, a is the real part and b is the imaginary part, each of which may be positive or negative. For example, z = 2 – 4i demonstrates the structure. In fact, ordinary real numbers are simply complex numbers with b = 0, so the complex number system is a natural extension of all numbers.
Addition & Subtraction
When adding or subtracting complex numbers, combine the real parts and the imaginary parts separately. For instance, with z = 2 – 4i and w = 3 + 5i:
\( \begin{aligned} z + w &= (2 – 4i) + (3 + 5i)\\ &=(2 + 3) + (-4 + 5)i\\ &= 5 + i \end{aligned} \)
Subtracting follows the same principle:
\( \begin{aligned} z - w &= (2 – 4i) – (3 + 5i)\\ &= (2 – 3) + (-4 – 5)i\\ &= -1 - 9i \end{aligned} \)
Multiplication
Multiplication is analogous to ordinary algebra, but you must remember that i² = –1. For two simple imaginary numbers, 3i × –4i:
\(3i \times -4i = -12i^2 = -12(-1) = 12\)
With full complex numbers, use the FOIL method:
\( \begin{aligned} z \times w &= (2 - 4i)(3 + 5i)\\ &= (2 \times 3) + (-4i \times 3) + (2 \times 5i) + (-4i \times 5i)\\ &= 6 - 12i + 10i - 20i^2\\ &= 6 - 2i + 20\\ &= 26 + 2i \end{aligned} \)
Division
To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. The conjugate of a complex number z = a + bi is z* = a – bi. For example:
\( \frac{z}{w} = \frac{2 - 4i}{3 + 5i} \)
Multiply by the conjugate of the denominator (3 – 5i):
\( \frac{z}{w} = \frac{(2 - 4i)(3 - 5i)}{(3 + 5i)(3 - 5i)} \)
Compute numerator and denominator separately:
\( \begin{aligned} (2 - 4i)(3 - 5i) &= 6 - 12i - 10i + 20i^2 \newline &= -14 - 22i \newline (3 + 5i)(3 - 5i) &= 9 + 15i - 15i - 25i^2 \newline &= 34 \end{aligned} \)
Thus:
\( \frac{z}{w} = \frac{-14 - 22i}{34} = -\frac{7}{17} - \frac{11}{17}i \)
Apply the rules above to reduce any complex expression. Consider the example:
\(z = \frac{(4 + 2i) + (2 - i)}{(2 + 2i)(2 + i)}\)
First simplify the numerator:
\((4 + 2i) + (2 - i) = 6 + i\)
Then the denominator:
\( \begin{aligned} (2 + 2i)(2 + i) &= 4 + 4i + 2i + 2i^2 \newline &= (4 - 2) + 6i \newline &= 2 + 6i \end{aligned} \)
The fraction becomes:
\(z = \frac{6 + i}{2 + 6i}\)
Multiply numerator and denominator by the conjugate of the denominator (2 – 6i):
\( \begin{aligned} z &= \frac{(6 + i)(2 - 6i)}{(2 + 6i)(2 - 6i)} \newline &= \frac{12 + 2i - 36i - 6i^2}{4 + 12i - 12i - 36i^2} \newline &= \frac{18 - 34i}{40} \newline &= \frac{9}{20} - \frac{17}{20}i \end{aligned} \)
So the simplified form is:
\(z = \frac{9}{20} - \frac{17}{20}i\)
