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2. Quadratic Functions & Equations
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United States of AmericaFL
Grade 912

2.03 Complex numbers and operations

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Concept summary

We have found that quadratic equations can have non-real solutions, when the discriminant is less than zero. To define these non-real solutions we use complex numbers, which are built on the concept that there is a number, called i, that is equal to the square root of -1.

\begin{aligned} i &=\sqrt{-1} \\\ i^2 &=-1 \end{aligned}

This is not a real number, since the square of any real number is always non-negative. However, using this new number, we can define an entirely new dimension for the number line, which we call the imaginary numbers.

Imaginary number

A number that, when squared, results in a negative real number. It can be expressed as a real multiple of i, defined by i = \sqrt{-1}.

Complex number

A number that can be written in the form a+bi, where a and b are real numbers.

Imaginary part

The value of b of a complex number written as a+bi

A diagram showing the Complex Number System. Under the complex number system is the Real Number System, and numbers like square root of negative 2 and i. The Real Number System is divided into Rational and Irrational. Within Rational is the subset Integers; within Integers is the subset Whole; within Whole is the subset Natural. Examples of each type of number are shown. Speak to your teacher for more information.

We can express the square root of any negative number by taking out a factor of \sqrt{-1}=i to get a imaginary number. For example:\sqrt{-5}=\sqrt{-1\left(5\right)}=\sqrt{-1}\sqrt{5}=i\sqrt{5} \text{ or }\sqrt{5}i

When solving quadratic equations with real coefficients that have non-real roots, we can now find the solutions by expressing them as complex numbers, with the roots being complex conjugates.

Conjugate (of a complex number)

A number resulted from changing the sign of the imaginary part of a complex number. The conjugate of a+bi is a-bi.

The algebraic operations for complex numbers are the same as how we perform operations for rational algebraic expressions, except we sometimes have an extra step to account for the powers of i. For example,

Addition

  • Add in the same way as binomials with like terms:

    • \left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i

Subtraction

  • Subtract in the same way as binomials with like terms:

    • \left(a+bi\right)-\left(c+di\right)=\left(a-c\right)+\left(b-d\right)i

Multiplication

  • Distribute in the same way as binomials, evaluate any powers of i, then combine any like terms.

    • \left(a+bi\right)\left(c+di\right)=ac+adi+bci+bdi^2=ac+\left(ad+bc\right)i+bdi^2=\left(ac-bd\right)+\left(ad+bc\right)i

Division

  • Multiply both the numerator and denominator by the complex conjugate of the denominator, then simplify the products. Doing this makes the denominator a real value.

    • \\ \dfrac{a+bi}{c+di}=\dfrac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=\dfrac{\left(ac+bd\right)+\left(bc-ad\right)i}{c^2+d}=\dfrac{ac+bd}{c^2+d}+\dfrac{bc-ad}{c^2+d}i

Worked examples

Example 1

Simplify each of the following expressions:

a

\left( -3 + 5 i\right) + \left(7-2 i\right)

Approach

When adding or subtracting complex numbers, we can combine like terms by adding the real parts together, and the imaginary parts together.

Solution

\displaystyle \left( -3 + 5 i\right) + \left(7-2 i\right)\displaystyle =\displaystyle -3 + 5 i + 7-2 iDistribute the parentheses
\displaystyle =\displaystyle \left(-3+7\right) + \left(5i -2i\right)Combine the real and imaginary parts
\displaystyle =\displaystyle 4+3iSimplify
b

\left( 2 - 4 i\right) \left(-4+2 i\right)

Approach

When multiplying complex numbers we can use the distributive property to distribute the parentheses: \left(A+B\right)\left(C+D\right)=AB+AD+BC+BD

Solution

\displaystyle \left( 2 - 4 i\right) \left(-4+2 i\right)\displaystyle =\displaystyle 2\left(-4\right)+2\left(2i\right)-4i\left(-4\right)-4i\left(2i\right)Use the distributive property
\displaystyle =\displaystyle -8+4i+16i-8i^2Simplify
\displaystyle =\displaystyle -8+4i+16i-8(-1)Use the identity i^2=-1
\displaystyle =\displaystyle -8+4i+16i+8Simplify
\displaystyle =\displaystyle 20iCombine like terms

Reflection

It is important to take care of the signs when evalutaing expressions involving complex numbers. Notice that -4i\left(2i\right)=8. If we break this into real and imaginary factors, we are evaluating -4\left(2\right)=-8 and i^2=-1; the product of which is 8.

c

\sqrt{-7}\left(\sqrt{-4}\right)

Approach

When multiplying terms with negative radicands we want to first rewrite the radicand as a product of \sqrt{-1} and a positive radical: \sqrt{-a}=\sqrt{-1}\sqrt{a}. This can then be simplified to \sqrt{a}i.

We can then evaluate the product of the real terms and the imaginary terms separately.

Solution

\displaystyle \sqrt{-7}\left(\sqrt{-4}\right)\displaystyle =\displaystyle \sqrt{7}i\left(\sqrt{4}i\right)Express in terms of i
\displaystyle =\displaystyle \sqrt{7}\left(\sqrt{4}\right)i^2Commutative property of multiplication
\displaystyle =\displaystyle \sqrt{28}i^2Simplify the product
\displaystyle =\displaystyle 2\sqrt{7}i^2Simplify the radical
\displaystyle =\displaystyle -2\sqrt{7}Use the identity i^2=-1

Reflection

It is important to not make the mistake of thinking \sqrt{-7}\left(\sqrt{-4}\right)=\sqrt{28} as \sqrt{x}\sqrt{y}=\sqrt{xy} only if x, y are non-negative real numbers.

d

\dfrac{\sqrt{-33}}{\sqrt{-3}}

Approach

When dividing negative radicals, we follow a similar approach to multiplying. We want to rewrite the negative radicals in terms of i and then divide out the common factors as usual.

Solution

\displaystyle \dfrac{\sqrt{-33}}{\sqrt{-5}}\displaystyle =\displaystyle \dfrac{\sqrt{33}i}{\sqrt{3}i}Express in terms of i
\displaystyle =\displaystyle \dfrac{\sqrt{33}}{\sqrt{11}}Divide out the common factor of i
\displaystyle =\displaystyle \sqrt{\dfrac{33}{11}}Combine the quotient of positive radicals
\displaystyle =\displaystyle \sqrt{11}Simplify the radicand

Reflection

Notice that in this case the result is a positive radical, as the imaginary parts are divided out.

e

5 i^{2} - 2 i^{4} + 3 i^{7}

Approach

It is useful to identify what the value of each power of i is equal to.

  • i=\sqrt{-1}
  • i^2=-1
  • i^3=-i
  • i^4=1
  • i^5=\sqrt{-1}
  • i^6=-1
  • \cdots

We can see that there is a cycle of 4, and so we can work out the value of any power of i by working out what the remainder is when the power is divided by 4.

Solution

\displaystyle 5 i^{2} - 2 i^{4} + 3 i^{7}\displaystyle =\displaystyle 5\left(-1\right)-2\left(1\right)+3\left(-i\right)Evaluating powers of i
\displaystyle =\displaystyle -5-2-3iEvaluating products
\displaystyle =\displaystyle -7-3iSimplifying
f

\dfrac{3 + 7 i}{5 + 2 i}

Approach

We first want to multiply both the numerator and the denominator by the conjugate of the denominator.

Since the conjugate of the complex number a+bi is a-bi, the conjugate of 5+2i is 5-2i.

Solution

\displaystyle \frac{3 + 7 i}{5 + 2 i}\displaystyle =\displaystyle \frac{3 + 7 i}{5 + 2 i}\left(\dfrac{5- 2 i}{5 - 2 i}\right)Multiply numerator and denominator by the conjugate
\displaystyle =\displaystyle \frac{\left(3 + 7 i\right)\left(5- 2 i\right)}{\left( 5+ 2 i\right)\left( 5 - 2 i\right)}Combining fractions
\displaystyle =\displaystyle \frac{15-6i+35i-14i^2}{\left( 5+ 2 i\right)\left( 5 - 2 i\right)}Distribute parentheses in numerator
\displaystyle =\displaystyle \frac{15-6i+35i-14i^2}{25-4i^2}Product of a sum and difference in the denominator
\displaystyle =\displaystyle \frac{15-6i+35i+14}{25+4}Simplify using identity i^2=-1
\displaystyle =\displaystyle \frac{29+29i}{29}Combine like terms
\displaystyle =\displaystyle 1+iEvaluate the quotient

Reflection

Multiplying a complex number by its conjugate will always result in a real number.

Using this fact, we can always rationalize a complex quotient by multiplying both the numerator and denominator by the complex conjugate of the denominator.

Outcomes

MA.912.NSO.2.1

Extend previous understanding of the real number system to include the complex number system. Add, subtract, multiply and divide complex numbers.

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