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Operations with complex numbers

Operations of addition and multiplication of complex numbers are commutative, associative and distributive. If z_1 = x_1 + y_1 i and z_2 =  x_2 + y_2 i are complex numbers, then their addition, subtraction and product are defined as:

    \[\begin{aligned} z_1 + z_2 &= (x_1 + y_1 i) + (x_2 + y_2 i) \\ &= x_1 + y_1 i + x_2 + y_2 i \\ &= (x_1 + x_2) + (y_1 i + y_2 i) \\ &= (x_1 + x_2) + (y_1 + y_2) i \end{aligned}\]

    \[\begin{aligned} z_1 - z_2 &= (x_1 + y_1 i) - (x_2 + y_2 i) \\ &= x_1 + y_1 i - x_2 - y_2 i \\ &= (x_1 - x_2) + (y_1 i - y_2 i) \\ &= (x_1 - x_2) + (y_1 - y_2) i \end{aligned}\]

    \[\begin{aligned} z_1 \cdot z_2 &= (x_1 + y_1 i) \cdot (x_2 + y_2 i) \\ &= x_1 \cdot  x_2 + x_1 \cdot y_2 \cdot i + y_1 \cdot x_2 \cdot i + y_1 \cdot y_2 \cdot i^2 \\ &= x_1 \cdot x_2 + (x_1 \cdot y_2 + y_1 \cdot x_2 ) i + y_1 \cdot y_2 \cdot (-1) \\ &=(x_1x_2 - y_1 y_2) + (x_1 y_2 + y_1x_2)i \end{aligned}\]

Example 1. Calculate

    \[\frac{1+2i}{4} - \frac{3-5i}{2}.\]

Solution:

    \[\begin{aligned} \frac{1+2i}{4} - \frac{3-5i}{2} &= \frac{(1+2i) - 2 \cdot (3-5i)}{4} \\ &= \frac{1+2i -6 +10i}{4} \\ &= \frac{-5+12i}{4} \\ & = - \frac{5}{4} + 3i.\\ \end{aligned}\]

Division of complex numbers

If  z = x + yi is any complex number, then the number \overline{z} = x - yi is called the complex conjugate of a complex number z.

The pair of complex numbers z and \overline{z} are called the pair of complex conjugate numbers. That is, this is the pair which real parts are equal, and imaginary parts are opposite real numbers.

The product of two complex conjugate numbers is a positive real number:

    \[z \cdot \overline{z} = (x + y) \cdot ( x - y i) = x^2 - (yi)^2 = x^2 + y^2.\]

For the division of complex numbers we will use the rationalization of fractions. Let z_1 = x_1 +y_1i and z_2= x_2 + y_2i be two complex numbers. Then:

    \[\begin{aligned} \frac{z_1}{z_2} &= \frac{x_1 + y_1i}{x_2 + y_2 i} \\ &= \frac{x_1 + y_1i}{x_2 + y_2 i} \cdot \frac{x_2 - y_2i}{x_2 - y_2 i} \\ & = \frac{x_1x_2 + y_1y_2 +(x_2y_1 - x_1 y_2)i}{x_2^2 + y_2^2} \\ & = \frac{x_1x_2 + y_1y_2}{x_2^2 + y_2^2} + \frac{x_2y_1 - x_1y_2}{x_2^2 + y_2^2} i. \end{aligned}\]

Example 2. Calculate

    \[\frac{6 + 9i}{-1 + 4i}.\]

Solution:

    \[\begin{aligned} \frac{6 + 9i}{-1 + 4i} \cdot \frac{-1 - 4i}{-1 - 4i} &= \frac{(6+9i)(-1-4i)}{(-1 + 4i)(-1-4i)} \\ &= \frac{-30 - 33i}{1+16} \\ &= -\frac{30}{17} - \frac{33}{17}i.\\ \end{aligned}\]

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