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5.3.2: The Complex Numbers

  • Page ID
  • \(a + bi\), the sum of a real and an imaginary number. 

    The coldest possible temperature, known as absolute zero is almost –460 degrees Fahrenheit. What is the square root of this number?

    Complex Numbers

    Before this concept, all numbers have been real numbers. 2, -5, \(\sqrt{11}\), and \(\dfrac{1}{3}\) are all examples of real numbers. With what we have previously learned, we cannot find \(\sqrt{−25}\) because you cannot take the square root of a negative number. There is no real number that, when multiplied by itself, equals -25. Let’s simplify \(\sqrt{−25}\).

    \(\sqrt{−25}=\sqrt{25\cdot −1}=5\sqrt{−1}\)

    In order to take the square root of a negative number we are going to assign \(\sqrt{−1}\) a variable, \(i\). \(i\) represents an imaginary number. Now, we can use i to take the square root of a negative number.

    \(\sqrt{−25}=\sqrt{25\cdot −1}=5\sqrt{−1}=5i\)

    All complex numbers have the form a+bi, where a and b are real numbers. a is the real part of the complex number and b is the imaginary part. If b=0, then a is left and the number is a real number. If a=0, then the number is only bi and called a pure imaginary number. If b≠0 and a≠0, the number will be an imaginary number.

    Figure \(\PageIndex{1}\)

    Let's find \(\sqrt{−162} \).

    First pull out the i. Then, simplify \(\sqrt{−162} \).

    \(\sqrt{−162}=\sqrt{−1}\cdot \sqrt{162}=i\sqrt{162}=i \sqrt{81\cdot 2}=9i\sqrt{2}\)

    Powers of i

    In addition to now being able to take the square root of a negative number, i also has some interesting properties. Try to find \(i^2\), \(i^3\), and \(i^4\).

    Step 1: Write out \(i^2\) and simplify. \(i^2=i\cdot i=\sqrt{−1} \cdot \sqrt{−1}=\sqrt{−1}^2=−1\)

    Step 2: Write out \(i^3\) and simplify. \(i^3=i^2\cdot i=−1\cdot i=−i\)

    Step 3: Write out \(i^4\) and simplify. \(i^4=i^2\cdot i^2=−1\cdot −1=1\)

    Step 4: Write out \(i^5\) and simplify. \(i^5=i^4\cdot i=1\cdot i=i\)

    Step 5: Write out \(i^6\) and simplify. \(i^6=i^4\cdot i^2=1\cdot −1=−1\)

    Step 6: Do you see a pattern? Describe it and try to find \(i^{19}\).

    You should see that the powers of i repeat every 4 powers. So, all the powers that are divisible by 4 will be equal to 1. To find i^{19}, divide 19 by 4 and determine the remainder. That will tell you what power it is the same as.

    \(i^{19}=i^{16}\cdot i^3=1\cdot i^3=−i\)

    Now, let's find the following powers of i.

    1. \(i^{32}\)

     32 is divisible by 4, so \(i^{32}=1\).

    1. \(i^{50}\)

    \(50\div 4=12\), with a remainder of 2. Therefore, \(i^{50}=i^2=−1\).

    1. \(i^7\)

    \(7\div 4=1\), with a remainder of 3. Therefore, \(i^7=i^3=−i\)

    Finally, let's simplify the following complex expressions.

    1. \((6−4i)+(5+8i)\)


    1. \(9−(4+i)+(2−7i)\)


    To add or subtract complex numbers, you need to combine like terms. Be careful with negatives and properly distributing them. Your answer should always be in standard form, which is a+bi. 

    Example \(\PageIndex{1}\)

    Earlier, you were asked to find the square root of -460 degrees. 


    We're looking for \(\sqrt{−460}\) .

    First we need to pull out the i. Then, we need to simplify \(\sqrt{−460}\).

    \(\sqrt{−460}=\sqrt{−1}\cdot \sqrt{460}=i\sqrt{460}=i\sqrt{4\cdot 115}= 2i \sqrt{115}\)

    Example \(\PageIndex{2}\)

    Simplify \(\sqrt{−49}\).


    Rewrite \(\sqrt{−49}\) in terms of i and simplify the radical.


    Example \(\PageIndex{3}\)

    Simplify \(\sqrt{−125}\).


    Rewrite \(\sqrt{−125}\) in terms of i and simplify the radical.

    \(\sqrt{−125}=i\sqrt{125}=i\sqrt{25\cdot 5}=5i\sqrt{5}\)

    Example \(\PageIndex{4}\)

    Simplify \(i^{210}\).


    \(210\div 4=52\), with a remainder of 2. Therefore, \(i^{210}=i^2=−1\).

    Example \(\PageIndex{5}\)

    Simplify \((8−3i)−(12−i)\).


    Distribute the negative and combine like terms.



    Simplify each expression and write in standard form.

    1. \(\sqrt{−9}\)
    2. \(\sqrt{−242}\)
    3. \(6\sqrt{−45}\)
    4. \(−12i \sqrt{98}\)
    5. \(\sqrt{−32}\cdot \sqrt{−27}\)
    6. \(7i\sqrt{-126}\)
    7. \(i^8\)
    8. \(16i^{22}\)
    9. \(−9 i^{65}\)
    10. \(i^{365}\)
    11. \(2i^{91}\)
    12. \(-\sqrt{16}{80}\)
    13. \((11−5i)+(6−7i)\)
    14. \((14+2i)−(20+9i)\)
    15. \((8−i)−(3+4i)+15i\)
    16. \(−10i−(1−4i)\)
    17. \((0.2+1.5i)−(−0.6+i)\)
    18. \(6+(18−i)−(2+12i)\)
    19. \(−i+(19+22i)−(8−14i)\)
    20. \(18−(4+6i)+(17−9i)+24i\)

    Answers for Review Problems

    To see the Review answers, open this PDF file and look for section 5.8. 


    Term Definition
    i i is an imaginary number. \(i=\sqrt{−1}\).
    Absolute Value The absolute value of a number is the distance the number is from zero. The absolute value of a complex number is the distance from the complex number on the complex plane to the origin.
    Complex Conjugate Complex conjugates are pairs of complex binomials. The complex conjugate of \(a+bi\) is \(a−bi\). When complex conjugates are multiplied, the result is a single real number.
    i i is an imaginary number. \(i=\sqrt{−1}\).
    Imaginary Number An imaginary number is a number that can be written as the product of a real number and i.
    imaginary part The imaginary part of a complex number \(a+bi\) is \(b\)i.
    Pure Imaginary Numbers The pure imaginary numbers are the subset of complex numbers without real parts, only bi.
    Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
    real part The real part of a complex number \(a+bi\) is \(a\).
    rectangular coordinates A point is written using rectangular coordinates if it is written in terms of \(x\) and \(y\) and can be graphed on the Cartesian plane.
    rectangular form The rectangular form of a point or a curve is given in terms of \(x\) and \(y\) and is graphed on the Cartesian plane.
    standard form The standard form of a complex number is \(a+bi\) where \(a\) and \(b\) are real numbers.

    Additional Resources

    Interactive Element


    Video: Writing Complex Numbers in Standard Form - Example 1

    Practice: The Complex Numbers

    Real World: Getting Lost in a Pineapple

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