# 2.4 Complex numbers  (Page 4/8)

 Page 4 / 8

## Simplifying powers of i

The powers of $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ are cyclic. Let’s look at what happens when we raise $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ to increasing powers.

$\begin{array}{l}{i}^{1}=i\hfill \\ {i}^{2}=-1\hfill \\ {i}^{3}={i}^{2}\cdot i=-1\cdot i=-i\hfill \\ {i}^{4}={i}^{3}\cdot i=-i\cdot i=-{i}^{2}=-\left(-1\right)=1\hfill \\ {i}^{5}={i}^{4}\cdot i=1\cdot i=i\hfill \end{array}$

We can see that when we get to the fifth power of $\text{\hspace{0.17em}}i,$ it is equal to the first power. As we continue to multiply $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ by increasing powers, we will see a cycle of four. Let’s examine the next four powers of $\text{\hspace{0.17em}}i.$

$\begin{array}{l}{i}^{6}={i}^{5}\cdot i=i\cdot i={i}^{2}=-1\hfill \\ {i}^{7}={i}^{6}\cdot i={i}^{2}\cdot i={i}^{3}=-i\hfill \\ {i}^{8}={i}^{7}\cdot i={i}^{3}\cdot i={i}^{4}=1\hfill \\ {i}^{9}={i}^{8}\cdot i={i}^{4}\cdot i={i}^{5}=i\hfill \end{array}$

The cycle is repeated continuously: $\text{\hspace{0.17em}}i,-1,-i,1,$ every four powers.

## Simplifying powers of $\text{\hspace{0.17em}}i$

Evaluate: $\text{\hspace{0.17em}}{i}^{35}.$

Since $\text{\hspace{0.17em}}{i}^{4}=1,$ we can simplify the problem by factoring out as many factors of $\text{\hspace{0.17em}}{i}^{4}\text{\hspace{0.17em}}$ as possible. To do so, first determine how many times 4 goes into 35: $\text{\hspace{0.17em}}35=4\cdot 8+3.$

${i}^{35}={i}^{4\cdot 8+3}={i}^{4\cdot 8}\cdot {i}^{3}={\left({i}^{4}\right)}^{8}\cdot {i}^{3}={1}^{8}\cdot {i}^{3}={i}^{3}=-i$

Evaluate: $\text{\hspace{0.17em}}{i}^{18}$

$-1$

Can we write $\text{\hspace{0.17em}}{i}^{35}\text{\hspace{0.17em}}$ in other helpful ways?

As we saw in [link] , we reduced $\text{\hspace{0.17em}}{i}^{35}\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}{i}^{3}\text{\hspace{0.17em}}$ by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of $\text{\hspace{0.17em}}{i}^{35}\text{\hspace{0.17em}}$ may be more useful. [link] shows some other possible factorizations.

 Factorization of $\text{\hspace{0.17em}}{i}^{35}$ ${i}^{34}\cdot i$ ${i}^{33}\cdot {i}^{2}$ ${i}^{31}\cdot {i}^{4}$ ${i}^{19}\cdot {i}^{16}$ Reduced form ${\left({i}^{2}\right)}^{17}\cdot i$ ${i}^{33}\cdot \left(-1\right)$ ${i}^{31}\cdot 1$ ${i}^{19}\cdot {\left({i}^{4}\right)}^{4}$ Simplified form ${\left(-1\right)}^{17}\cdot i$ $-{i}^{33}$ ${i}^{31}$ ${i}^{19}$

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.

Access these online resources for additional instruction and practice with complex numbers.

## Key concepts

• The square root of any negative number can be written as a multiple of $\text{\hspace{0.17em}}i.\text{\hspace{0.17em}}$ See [link] .
• To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See [link] .
• Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See [link] .
• Complex numbers can be multiplied and divided.
• To multiply complex numbers, distribute just as with polynomials. See [link] and [link] .
• To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See [link] and [link] .
• The powers of $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ are cyclic, repeating every fourth one. See [link] .

## Verbal

Explain how to add complex numbers.

Add the real parts together and the imaginary parts together.

What is the basic principle in multiplication of complex numbers?

Give an example to show that the product of two imaginary numbers is not always imaginary.

Possible answer: $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ times $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ equals 1, which is not imaginary.

What is a characteristic of the plot of a real number in the complex plane?

## Algebraic

For the following exercises, evaluate the algebraic expressions.

If $\text{\hspace{0.17em}}y={x}^{2}+x-4,$ evaluate $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}x=2i.$

$-8+2i$

If $\text{\hspace{0.17em}}y={x}^{3}-2,$ evaluate $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}x=i.$

If $\text{\hspace{0.17em}}y={x}^{2}+3x+5,$ evaluate $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}x=2+i.$

$14+7i$

If $\text{\hspace{0.17em}}y=2{x}^{2}+x-3,$ evaluate $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}x=2-3i.$

If $\text{\hspace{0.17em}}y=\frac{x+1}{2-x},$ evaluate $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}x=5i.$

$-\frac{23}{29}+\frac{15}{29}i$

If $\text{\hspace{0.17em}}y=\frac{1+2x}{x+3},$ evaluate $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}x=4i.$

## Graphical

For the following exercises, plot the complex numbers on the complex plane.

$1-2i$

$-2+3i$

$i$

$-3-4i$

## Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

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

$8-i$

$\left(-2-4i\right)+\left(1+6i\right)$

$\left(-5+3i\right)-\left(6-i\right)$

$-11+4i$

$\left(2-3i\right)-\left(3+2i\right)$

$\left(-4+4i\right)-\left(-6+9i\right)$

$2-5i$

$\left(2+3i\right)\left(4i\right)$

$\left(5-2i\right)\left(3i\right)$

$6+15i$

$\left(6-2i\right)\left(5\right)$

$\left(-2+4i\right)\left(8\right)$

$-16+32i$

$\left(2+3i\right)\left(4-i\right)$

$\left(-1+2i\right)\left(-2+3i\right)$

$-4-7i$

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

$\left(3+4i\right)\left(3-4i\right)$

25

$\frac{3+4i}{2}$

$\frac{6-2i}{3}$

$2-\frac{2}{3}i$

$\frac{-5+3i}{2i}$

$\frac{6+4i}{i}$

$4-6i$

$\frac{2-3i}{4+3i}$

$\frac{3+4i}{2-i}$

$\frac{2}{5}+\frac{11}{5}i$

$\frac{2+3i}{2-3i}$

$\sqrt{-9}+3\sqrt{-16}$

$15i$

$-\sqrt{-4}-4\sqrt{-25}$

$\frac{2+\sqrt{-12}}{2}$

$1+i\sqrt{3}$

$\frac{4+\sqrt{-20}}{2}$

${i}^{8}$

$1$

${i}^{15}$

${i}^{22}$

$-1$

## Technology

For the following exercises, use a calculator to help answer the questions.

Evaluate $\text{\hspace{0.17em}}{\left(1+i\right)}^{k}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}k=4,8,\text{and}\text{\hspace{0.17em}}12.\text{\hspace{0.17em}}$ Predict the value if $\text{\hspace{0.17em}}k=16.$

Evaluate $\text{\hspace{0.17em}}{\left(1-i\right)}^{k}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}k=2,6,\text{and}\text{\hspace{0.17em}}10.\text{\hspace{0.17em}}$ Predict the value if $\text{\hspace{0.17em}}k=14.$

128i

Evaluate ${\left(\text{l}+i\right)}^{k}-{\left(\text{l}-i\right)}^{k}$ for $\text{\hspace{0.17em}}k=4,8,\text{and}\text{\hspace{0.17em}}12.\text{\hspace{0.17em}}$ Predict the value for $\text{\hspace{0.17em}}k=16.$

Show that a solution of $\text{\hspace{0.17em}}{x}^{6}+1=0\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\frac{\sqrt{3}}{2}+\frac{1}{2}i.$

${\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)}^{6}=-1$

Show that a solution of $\text{\hspace{0.17em}}{x}^{8}-1=0\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i.$

## Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{1}{i}+\frac{4}{{i}^{3}}$

$3i$

$\frac{1}{{i}^{11}}-\frac{1}{{i}^{21}}$

${i}^{7}\left(1+{i}^{2}\right)$

0

${i}^{-3}+5{i}^{7}$

$\frac{\left(2+i\right)\left(4-2i\right)}{\left(1+i\right)}$

$5-5i$

$\frac{\left(1+3i\right)\left(2-4i\right)}{\left(1+2i\right)}$

$\frac{{\left(3+i\right)}^{2}}{{\left(1+2i\right)}^{2}}$

$-2i$

$\frac{3+2i}{2+i}+\left(4+3i\right)$

$\frac{4+i}{i}+\frac{3-4i}{1-i}$

$\frac{9}{2}-\frac{9}{2}i$

$\frac{3+2i}{1+2i}-\frac{2-3i}{3+i}$

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
Ajith
exponential series
Naveen
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
e power cos hyperbolic (x+iy)
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
why {2kπ} union {kπ}={kπ}?
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
what is complex numbers
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Solve 2cos x + 3sin x = 0.5