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
B. -i
C. 1
D. -1
Solution: Now, we need to first break up \(i^{203}\: to\: i^{202+1}\)
\(i^{202}\times i^{1}\)
\((i^{2})^{101}\times i\)
\((-1)^{101}\times i\)
-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 \(a+bi:\frac{6}{7-4i}\)
\(\frac{6}{7-4i}=\frac{6}{7-4i}\times \frac{7+4i}{7+4i}\)
\(\frac{42+24i}{49-16i^{2}}\)
\(\frac{42+24i}{49-16(-1)}\)
\(\frac{42+24i}{65}\)
\(\frac{42}{65}+\frac{25}{65}i\)
There are usually 1-2 questions on complex numbers on every ACT Test. If you know what complex numbers are, these questions are pretty straightforward.
About 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.
PROBABILITY is the likelihood that something will happen. It is a number between 0 and 1 and can be written as a percent. 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.
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.
The ACT will probably contain two to three questions on each ACT test from a collection of additional topics. Even though none of the topics will not appear more than once on a test,collectively they could make a significant different to your ACT Math score.
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