SAT Hub

Complex Numbers

There is a new “number” that we associate negative numbers with under the square root. These are called COMPLEX NUMBERS, and they are indicated with the letter i, where:* \(i=\sqrt{-1}\)*

We can do regular operations like addition, subtraction, multiplication, and division with complex numbers. Keep the following in mind, to make complex multiplication simpler.

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

\(i^{2}=i\times i=\sqrt{-1}\times \sqrt{-1}=-1\)

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

\(i^{4}=i^{2}\times i^{2}=-1\times -1=1\)

\(i^{5}=i^{4}\times i=i\)

\(i^{6}=i^{4}\times i^{2}=-1\)

\(i^{7}=i^{4}\times i^{3}=-i\)

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

*Solved Example 1*

Find the value of \(i^{203}\)** A.** i

Solution

\(i^{202}\times i^{1}\)

\((i^{2})^{101}\times i\)

\((-1)^{101}\times i\)

-1 x i

= -i

The correct answer is B.

Adding and subtracting complex numbers is similar to adding and subtracting polynomials. Just add or subtract the real and imaginary parts.

**Example:** Solve: (-6 - 5i) - (3 - 4i)

(-6 - 5i) - (3 - 4i) = (-6 - 5i) + (-3 + 4i)

= [-6 + (-3)] + (-5 + 4)i

= -9 - i

Multiplying complex numbers is similar to multiplying polynomials.

**Solve:** (2 + 5i) (6 + 4i)

(2 + 5i) (6 + 4i) = 12 + 8i + 30i + 20i2

= 12 + 38i + 20(-1)

= 12 + 38i - 20

= -8 + 38i

For complex division, we need to know what a conjugate is. For starters, all complex numbers take the form of a + bi, where a is the x-coordinate and b is the y-coordinate. A complex conjugate is changing the + in a + bi to a − bi or vice versa. When we have complex division, we multiply by a clever form of “one,” which will be the conjugate of the denominator.

**Example: **Write each quotient in the form \(a+bi:\frac{6}{7-4i}\)

\(\frac{6}{7-4i}=\frac{6}{7-4i}\times \frac{7+4i}{7+4i}\)

\(\frac{42+24i}{49-16i^{2}}\)

\(\frac{42+24i}{49-16(-1)}\)

\(\frac{42+24i}{65}\)

\(\frac{42}{65}+\frac{25}{65}i\)

The rich text element allows you to create and format headings, paragraphs, blockquotes, images, and video all in one place instead of having to add and format them individually. Just double-click and easily create content.

A rich text element can be used with static or dynamic content. For static content, just drop it into any page and begin editing. For dynamic content, add a rich text field to any collection and then connect a rich text element to that field in the **settings panel. Voila!**

Headings, paragraphs, blockquotes, figures, images, and figure captions can all be styled after a class is added to the rich text element using the "When inside of" nested selector system.

It's important to understand the graphs and charts often used in statistics based questions before we explain the core concepts tested on the SAT Math section.

Checkout some of our ACT Training related resources for you.

View Resources NowPROBABILITY is the likelihood that something will happen. It is a number between 0 and 1 and can be written as a per cent. When you asked about something's probability, you are asking, "How likely is it?" A larger number means there is a greater likelihood that the event will happen.

Checkout some of our ACT Training related resources for you.

View Resources NowAbout 35% of your total SAT Math Test will be made up of word problems. Though the actual math topics can vary, it is important to develop a consistent process for answering them.

Checkout some of our ACT Training related resources for you.

View Resources NowThere are usually 1-2 questions on complex numbers on every SAT Test. If you know what complex numbers are, these questions are pretty straightforward.

Checkout some of our ACT Training related resources for you.

View Resources Now