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Complex Numbers

a + bi, the sum of a real and an imaginary number.

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Imaginary and Complex Numbers

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Complete the chart.
Word Definition
_______________ Any number with an \begin{align*}i\end{align*} associated with it
Complex Number _____________________________________________
_______________ An imaginary number without a real part, only \begin{align*}bi\end{align*}
Simplifying the Radical _____________________________________________
Discriminant _____________________________________________
_______________ a complex number that, when used as an input ( ) value, results in an output () value of zero
_______________ binomial terms which are equal aside from inverse operations between them, 
Complex Conjugates _____________________________________________


What does \begin{align*}i = \end{align*} ? __________

What form do complex numbers have? _______________

What is the discriminant used for? ___________________________________________

Imaginary Numbers

Powers of \begin{align*}i\end{align*} repeat every 4 powers, so all the powers that are divisible by 4 will be equal to 1. Use the remainder to determine the answer.



  1. \begin{align*}\sqrt{-25}\end{align*}
  2. \begin{align*}\sqrt{-19}\end{align*}
  3. \begin{align*}\sqrt{4-27}\end{align*}
  4. \begin{align*}i^3\end{align*}
  5. \begin{align*}17^{23}\end{align*}
  6. \begin{align*}i^{148}\end{align*}
  7. \begin{align*}(0.2+1.5i)-(-0.6+i)\end{align*}


  1. \begin{align*}3i^3 \cdot 4i^7\end{align*}
  2. \begin{align*}4\sqrt{-75} \cdot 8\sqrt{-6}\end{align*}
  3. \begin{align*}3\sqrt{-13} \cdot 11\sqrt{-21}\end{align*}

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Quadratic Formula and Complex Sums

The discriminant is used to determine how many roots a quadratic function has.

If - 4 ac > 0 then there are two unequal real solutions.

If - 4 ac = 0 then there are two equal real solutions.

If - 4 ac < 0 then there are two unequal complex solutions.


Solve each equation and express the result as a complex number.

  1. \begin{align*}12x\frac{4}{5}x = 3\frac{1}{5}x^2 + 16\end{align*}
  2. \begin{align*}3x^2 - 6x + 15 = 0\end{align*}
  3. \begin{align*}8x^2 - 5x + 11 = 0\end{align*}
  4. When the sum of -4 + 8i and 2 - 9i is graphed, in which quadrant does it lie?
  5. If \begin{align*}z_1 = -3 + 2i\end{align*} and \begin{align*}z_2 = 4 - 3i\end{align*} , in which quadrant does the graph of \begin{align*}(z_2 - z_1)\end{align*} lie?
  6. On a graph, if point A represents \begin{align*}2 - 3i\end{align*} and point B represents \begin{align*}-2 - 5i\end{align*} , which quadrant contains \begin{align*}A - B\end{align*}?
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Products and Quotients of Complex Numbers

Multiplying Complex Numbers

Multiply as you normally would without imaginary numbers, then deal with the \begin{align*}i\end{align*}. Recall that \begin{align*}i\end{align*} has a cycle of 4 forms, so simplify accordingly.


Dividing Complex Numbers

Divide also as you would rational numbers. Remember that the procedure was to find the irrational conjugate of the denominator and then multiply both the numerator and the denominator by that conjugate. Because in complex numbers you want to elliminate the complex numbers from the denominator, you find the complex conjugate of the denominator and multiply BOTH the numerator AND the denominator by it.


Multiply the complex numbers.

  1. \begin{align*}(4i - 7)(12i^3 + 3)\end{align*}
  2. \begin{align*}(14\sqrt{-9}) \cdot (8\sqrt{-4})\end{align*}
  3. \begin{align*}(16\sqrt{-1}) \cdot (21 \sqrt{-81})\end{align*}


Divide the complex numbers.

  1. \begin{align*}\frac{4i^2 + 4}{-6i^2 - 1}\end{align*}
  2. \begin{align*}\frac{5i^2 - 5}{-8i^2 + 3}\end{align*}
  3. \begin{align*}\frac{-4\sqrt{-60}}{-2\sqrt{-20}}\end{align*}


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