SAT Math

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

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SAT Reading

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

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SAT Writing

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

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SAT Essay

Everything you need to know to get a perfect SAT Math 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

SAT General

Everything you need to know to get a perfect SAT Math 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

SAT Math

Everything you need to know to get a perfect SAT 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

SAT Reading

Everything you need to know to get a perfect SAT Math 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 Writing

Everything you need to know to get a perfect SAT Math 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

SAT Essay

Everything you need to know to get a perfect SAT Math 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

SAT General

Everything you need to know to get a perfect SAT Math 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

SAT Math

Everything you need to know to get a perfect SAT 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

SAT Reading

Everything you need to know to get a perfect SAT Math 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 Writing

Everything you need to know to get a perfect SAT Math 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

SAT Essay

Everything you need to know to get a perfect SAT Math 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

SAT General

Everything you need to know to get a perfect SAT Math 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

SAT Math

Everything you need to know to get a perfect SAT 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

SAT Reading

Everything you need to know to get a perfect SAT Math 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 Writing

Everything you need to know to get a perfect SAT Math 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

SAT Essay

Everything you need to know to get a perfect SAT Math 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

SAT General

Everything you need to know to get a perfect SAT Math 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

SAT Math

Everything you need to know to get a perfect SAT 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

SAT Reading

Everything you need to know to get a perfect SAT Math 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 Writing

Everything you need to know to get a perfect SAT Math 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

SAT Essay

Everything you need to know to get a perfect SAT Math 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

SAT General

Everything you need to know to get a perfect SAT Math 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 SAT 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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