SAT 2016

CHAPTER 10

THE SAT MATH: ADDITIONAL TOPICS

Skill 3: Understanding Complex Numbers

Lesson 11: Understanding the imaginary number i and the complex plane

Imaginary Numbers

Images

The imaginary number i is defined as the principal square root of –1.

The square root of a negative number is not on the real number line, because the square of a real number cannot be negative. Therefore, the square roots of negative numbers must reside on their own number line, which we call the “imaginary axis”, which is perpendicular to the real axis, intersecting it at the origin. The plane defined by the real axis and the imaginary axis is called the complex plane.

image

Given that Images, which of the following is equal to Images?

A)   i

B)   −i

C)   1

D)   −1

(Medium-hard) To answer this question, we just need to know the basic exponent rules and the definition of i.

Original expression:

Images

Factor:

Images

Substitute i2 = –1

Images

Multiply by i/i:

Images

Substitute i2 = –1:

–(–i) = i

Therefore, the correct answer is (A).

Lesson 12: Adding, multiplying, dividing, and simplifying complex numbers

Complex Numbers

The sum of a real number and an imaginary number is called a complex number. All complex numbers can be expressed in the form a + bi where a and b are real numbers and Images.

Every complex number a + bi corresponds to the point (a, b) on the complex plane.

image

Adding Complex Numbers

To add complex numbers, just combine “like” terms.

Original expression:

(3 – 2i) + (2 + 6i)

Regroup with Commutative and Associative Laws of Addition:

(3 + 2) + (–2i + 6i)

Simplify:

5 + 4i

Multiplying Complex Numbers

To multiply complex numbers, just FOIL and combine like terms.

Original expression:

(3 – 2i)(2 + 6i)

FOIL:

(3)(2) + (3)(6i) + (–2i)(2) + (–2i)(6i)

Simplify:

6 + 18i – 4i –12i2

Substitute i2 = –1:

6 + 18i – 4i – 12(–1)

Combine like terms:

18 + 14i

Dividing Complex Numbers

To divide complex numbers, express the quotient as a fraction, multiply numerator and denominator by the complex conjugate of the denominator, and simplify. The complex conjugate of a + bi is a – bi.

Original expression:

Images

Multiply numerator and

denominator

by conjugate of denominator:

Images

FOIL:

Images

Simplify:

Images

Distribute division:

Images

Powers of i

The successive powers of i (i1, i2, i3, i4, i5, i6, i7 . . .) cycle counterclockwise around the unit circle in the complex plane.

image

We can verify this by expanding any positive integer power of i.

Expression to be evaluated:

i13

Expand:

(i)(i)(i)(i)(i)(i)(i)(i)(i)(i)(i)(i)(i)

Group in pairs:

(i × i)(i × i)(i × i)(i × i)(i × i)(i × i)(i)

Substitute i2 = –1:

(–1)(–1)(–1)(–1)(–1)(–1)(i)

Simplify:

i

This implies that in = 1 if n is a multiple of 4. (That is, i4 = 1, i8 = 1, i12 = 1, etc.) This gives us a convenient way to simplify large powers of ijust replace the exponent with the remainder when it is divided by 4. For instance, i39 = i3 = –i, because 3 is the remainder when 39 is divided by 4.

If Images, where Images, which of the following is equal to K2?

A)   2i

B)   4i

C)   4 + i

D)   4

image

Therefore, the correct answer is (D).

Which of the following is NOT equal to i6 – i2?

A)   i5 – i

B)   i4

C)   2i3 + 2i

D)   1 + i6

(Medium) Here, we have to use our knowledge about powers of i. Since i6 = (i × i)(i × i)(i × i) = (–1)(–1)(–1) = –1, and i2 = –1, the given expression, i6 – i2, is equal to (–1) – (–1) = 0. Simplifying each choice gives us

A)   i5 – i = i – i = 0

B)   i4 = 1

C)   2i3 + 2i = –2i + 2i = 0

D)   1 + i6 = 1 + (–1) = 0

Therefore, the correct answer is (B).

Exercise Set 4: Complex Numbers (No Calculator)

1

If a + bi = (1 + 2i)(3 – 4i), where a and b are constants and Images, what is the value of a + b?

image

2

If Images, where a and b are constants and Images, what is the value of a?

image

3

For what value of b does (b + i)2 = 80 + 18i?

image

4

The solutions of the equation x2 – 2x + 15 = 0 are Images and Images, where a and b are positive numbers. What is the value of a + b?

image

5

Given that Images, which of the following is equal to Images?

A)   Images

B)   Images

C)   Images

D)   Images

6

Which of the following expressions is equal to (2 + 2i)2?

A)   0

B)   4i

C)   8i

D)   4 – 4i

7

If B(3 + i) = 3 – i, what is the value of B?

A)   Images

B)   Images

C)   Images

D)   Images

8

x2 + kx = –6

If one of the solutions to the equation above is Images, what is the value of k?

A)   –4

B)   –2

C)   2

D)   4

9

If im = –i, which of the following CANNOT be the value of m?

A)   15

B)   18

C)   19

D)   27

EXERCISE SET 4: COMPLEX NUMBERS ANSWER KEY

No Calculator

1.  13

(1 + 2i)(3 – 4i)

FOIL:

(1)(3) + (1)(–4i) + (2i)(3) + (2i)(–4i)

Simplify:

3 – 4i + 6i – 8i2

Substitute i2 = –1:

3 – 4i + 6i –8(–1)

Combine like terms:

11 + 2i

Therefore, a = 11 and b = 2, so a + b = 13.

2.  7/5 or 1.4

Images

Multiply conjugate:

Images

FOIL:

Images

Substitute i2 = –1:

Images

Combine like terms:

Images

Distribute division:

Images

3.  9

(b + i)2

FOIL:

(b + i)(b + i) = b2 + bi + bi + i2

Substitute i2 = –1:

b2 + bi + bi – 1

Combine like terms:

(b2 – 1) + 2bi

Since this must equal 80 + 18i, we can find b by solving either b2 – 1 = 80 or 2b = 18. The solution to both equations is b = 9.

4.  15   The equation we are given is a quadratic equation in which a = 1, b = –2, and c = 15. Therefore, we can use the quadratic formula:

Quadratic Formula:

Images

Substitute:

Images

Simplify:

Images

Simplify:

Images

Distribute division:

Images

Therefore, a = 1 and b = 14, so a + b = 15.

5.  B

Images

FOIL:

Images

Substitute i2 = –1:

Images

Simplify:

Images

Multiply by i/i:

Images

Substitute i2 = -1:

Images

6.  C

(2 + 2i)2

FOIL:

(2 + 2i)(2 + 2i) = 4 + 4i + 4i + 4i2

Substitute i2 = –1:

4 + 8i – 4 = 8i

7.  D

B(3 + i) = 3 – i

Divide by 3 + i:

Images

FOIL:

Images

Substitute i2 = –1:

Images

Simplify:

Images

Distribute division:

Images

8.  B

x2 + kx = –6

Add 6:

x2 + kx + 6 = 0

Substitute Images:

Images

FOIL:

Images

Simplify:

Images

Distribute:

Images

Collect terms:

Images

Therefore, both 2 + k = 0 and Images. Solving either equation gives k = –2.

9.  B As we discussed in Lesson 10, the powers of i are “cyclical,” and im = –i if and only if m is 3 more than a multiple of 4. The only number among the choices that is not 3 more than a multiple of 4 is (B) 18.