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11.2 Arithmetic with Complex Numbers

  • Page ID
    985
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    The idea of a complex number can be hard to comprehend, especially when you start thinking about absolute value. In the past you may have thought of the absolute value of a number as just the number itself or its positive version. How should you think about the absolute value of a complex number?

    Arithmetic Operations with Complex Numbers

    Complex numbers follow all the same rules as real numbers for the operations of adding, subtracting, multiplying and dividing. There are a few important ideas to remember when working with complex numbers:

    1. When simplifying, you must remember to combine imaginary parts with imaginary parts and real parts with real parts. For example, \(4+5 i+2-3 i=6+2 i\)

    2. If you end up with a complex number in the denominator of a fraction, eliminate it by multiplying both the numerator and denominator by the complex conjugate of the denominator.

    3. The powers of \(i\) are:

    • \(i=\sqrt{-1}\)
    • \(i^{2}=-1\)
    • \(i^{3}=-\sqrt{-1}=-i\)
    • \(i^{4}=1\)
    • \(i^{5}=i\)
    • \(\ldots\) and the pattern repeats

    Consider this complex expression:

    \((2+3 i)(1-5 i)-3 i+8\)

    First, multiply the two binomials and then combine the imaginary parts with imaginary parts and real parts with real parts.

    \(=2-10 i+3 i-15 i^{2}-3 i+8\)

    \(=10-10 i+15\)

    \(=25-10 i\)

    Note that a power higher than 1 of \(i\) can be simplified using the pattern above.

    The complex plane is set up in the same way as the regular \(x, y\) plane, except that real numbers are counted horizontally and complex numbers are counted vertically. The following is the number \(4+3 i\) plotted in the complex number plane. Notice how the point is four units over and three units up.

    The absolute value of a complex number like \(|4+3 i|\) is defined as the distance from the complex number to the origin. You can use the Pythagorean Theorem to get the absolute value. In this case, \(|4+3 i|=\sqrt{4^{2}+3^{2}}=\sqrt{25}=5\).

    Examples

    Example 1

    Earlier, you were asked how to think about the absolute value of a complex number. A good way to think about the absolute value for all numbers is to define it as the distance from a number to zero. In the case of complex numbers where an individual number is actually a coordinate on a plane, zero is the origin.

    Example 2

    Compute the following power by hand and use your calculator to support your work.

    \((\sqrt{3}+2 i)^{3}\)

    \((\sqrt{3}+2 i) \cdot(\sqrt{3}+2 i) \cdot(\sqrt{3}+2 i)\)

    \(=(3+4 i \sqrt{3}-4)(\sqrt{3}+2 i)\)

    \(=(-1+4 i \sqrt{3})(\sqrt{3}+2 i)\)

    \(=-\sqrt{3}-2 i+12 i-8 \sqrt{3}\)

    \(=-9 \sqrt{3}+10 i\)

    A TI-84 can be switched to imaginary mode and then compute exactly what you just did. Note that the calculator will give a decimal approximation for \(-9 \sqrt{3}\).

    Example 3

    Simplify the following complex expression.

    \(\frac{7-9 i}{4-3 i}+\frac{3-5 i}{2 i}\)

    To add fractions you need to find a common denominator.

    \(\frac{(7-9 i) \cdot 2 i}{(4-3 i) \cdot 2 i}+\frac{(3-5 i) \cdot(4-3 i)}{2 i \cdot(4-3 i)}\)

    \(=\frac{14 i+18}{8 i+6}+\frac{12-20 i-9 i-15}{8 i+6}\)

    \(=\frac{15-15 i}{8 i+6}\)

    Lastly, eliminate the imaginary component from the denominator by using the conjugate.

    \(=\frac{(15-15 i) \cdot(8 i-6)}{(8 i+6) \cdot(8 i-6)}\)

    \(=\frac{120 i-90+120+90 i}{100}\)

    \(=\frac{30 i+30}{100}\)

    \(=\frac{3 i+3}{10}\)

    Example 4

    Simplify the following complex number.

    \(i^{2013}\)

    When simplifying complex numbers, \(i\) should not have a power greater than 1 . The powers of \(i\) repeat in a four part cycle:

    \(i^{5}=i=\sqrt{-1}\)

    \(i^{6}=i^{2}=-1\)

    \(i^{7}=i^{3}=-\sqrt{-1}=-i\)

    \(i^{8}=i^{4}=1\)

    Therefore, you just need to determine where 2013 is in the cycle. To do this, determine the remainder when you divide 2013 by 4 . The remainder is 1 so \(i^{2013}=i\).

    Example 5

    Plot the following complex number on the complex coordinate plane and determine its absolute value.

    \(-12+5 i\)

    The sides of the right triangle are 5 and \(12,\) which you should recognize as a Pythagorean

    triple with a hypotenuse of 13. \(|-12+5 i|=13\).

    Review

    Simplify the following complex numbers.

    1. \(i^{252}\)

    2. \(i^{312}\)

    3. \(i^{411}\)

    4. \(i^{2345}\)

    For each of the following, plot the complex number on the complex coordinate plane and

    determine its absolute value.

    5. \(6-8 i\)

    6. \(2+i\)

    7. \(4-2 i\)

    8. \(-5 i+1\)

    Let \(c=2+7 i\) and \(d=3-5 i\)

    9. What is \(c+d ?\)

    10. What is \(c-d ?\)

    11. What is \(c \cdot d ?\)

    12. What is \(2 c-4 d ?\)

    13. What is \(2 c \cdot 4 d ?\)

    14. What is \(\frac{c}{d}\) ?

    15. What is \(c^{2}-d^{2} ?\)


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