ACT Math

Everything you need to know to get a perfect ACT Math score.

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ACT Reading

Everything you need to know to get a perfect ACT Reading score.

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ACT English

Everything you need to know to get a perfect ACT English score.

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ACT Writing

Everything you need to know to get a perfect ACT Writing score.

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ACT Science

Everything you need to know to get a perfect ACT Science score.

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ACT Math

Everything you need to know to get a perfect ACT Math score.

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ACT Reading

Everything you need to know to get a perfect ACT Reading score.

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ACT Engish

Everything you need to know to get a perfect ACT English score.

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ACT Writing

Everything you need to know to get a perfect ACT Writing score.

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ACT Science

Everything you need to know to get a perfect ACT Science score.

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ACT Math

Everything you need to know to get a perfect ACT Math score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Math SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Math Section

ACT Reading

Everything you need to know to get a perfect ACT Reading score.

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ACT English

Everything you need to know to get a perfect ACT English score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Writing SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Writing Section

ACT Writing

Everything you need to know to get a perfect ACT Writing score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Essay SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Essay Section

ACt Science

Everything you need to know to get a perfect ACT Science score.

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ACT Math

Everything you need to know to get a perfect ACT Math score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Math SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Math Section

ACT Reading

Everything you need to know to get a perfect ACT Reading score.

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ACT English

Everything you need to know to get a perfect ACT Writing score.

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ACT Writing

Everything you need to know to get a perfect ACT Writing score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Essay SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Essay Section

ACT Science

Everything you need to know to get a perfect ACT Science score.

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ACT Math

Everything you need to know to get a perfect ACT Math score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Math SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Math Section

ACT Reading

Everything you need to know to get a perfect ACT Reading score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Reading SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Reading Section

SAT English

Everything you need to know to get a perfect ACT English score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Writing SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Writing Section

ACT Essay

Everything you need to know to get a perfect ACT Writing score.

Detailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Essay SectionDetailed SAT and ACT Strategy guide written by experienced SAT tutors to help you with your SAT Essay Section

ACT Science

Everything you need to know to get a perfect ACT Science score.

Detailed SAT and ACT Strategy guide written by experienced ACT tutors to help you with your ACT Science SectionDetailed SAT and ACT Strategy guide written by experienced ACT tutors to help you with your ACT Science Section

Complex Numbers

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.

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.

\(i=\sqrt{-1}\)
\(i^{2}=i\times i=\sqrt{-1}\times \sqrt{-1}=-1\)
\(i^{3}=i\times i\times i=\sqrt{-1}\times \sqrt{-1}\times \sqrt{-1}= -1\times \sqrt{-1}=-i\)
\(i^{4}=i^{2}\times i^{2}=-1\times -1=1\)
\(i^{5}=i^{4}\times i=i\)
\(i^{6}=i^{4}\times i^{2}=-1\)
\(i^{7}=i^{4}\times i^{3}=-i\)
\(i^{8}=i^{4}\times i^{4}=1\)

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.

Add and Subctract Complex Numbers

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

Multiply Complex Numbers

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

Divide Complex Numbers

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 \(a+bi:\frac{6}{7-4i}\)

\(\frac{6}{7-4i}=\frac{6}{7-4i}\times \frac{7+4i}{7+4i}\)
\(\frac{42+24i}{49-16i^{2}}\)
\(\frac{42+24i}{49-16(-1)}\)
\(\frac{42+24i}{65}\)
\(\frac{42}{65}+\frac{25}{65}i\)

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