Imaginary Numbers

Imaginary Numbers

Unit 1 Lesson 1

Make copies of:

• How do I simplify Powers of i version 2.docx

GPS Standards

• MM2N1a- Write square roots of negative numbers in imaginary form.

• MM2N1b- Write complex numbers in the form a + bi.

Essential Questions:

• How do I write square roots of negative numbers as imaginary numbers?

• How do I simplify powers of i?

Why do we need imaginary numbers?

Think back to when you first learned about numbers…

Number probably meant 0,1,2,3,…. (these are the whole numbers)

Then you came upon a problem like 3 – 5So we had to expand number to include all the negative numbers ….-3,-2,-1,0,1,2,3,...

That was the set of integers, which are also numbers

Let’s look at division…try 3 divided by 5So now our definition of numbers needs to include fractions…this is the set of rational

numbersHow about trying to take a square root of a

number like 2?This means numbers has to include

radicals….these are irrational numbers

So..why do we need imaginary numbers?

Let’s look at an equation:X2 + 1 = 0

Isolate x termX2 = -1

Take the square root of both sides…Can you take the square root of a negative

number??

Let’s investigate…(-4)2 = 16 and 42 = 16

Is there any time that you can square something and get a

positive answer?So…how do we take the square

root of a negative number?We need a new type of

“number”

Imaginary numbers

i is the imaginary number uniti = √-1i2 = -1

Using imaginary numbers

Simplify the following√-9

√-72√50√-17i 3

Test Prep Example

• Express in terms of i: -3√-64

A) -24iB) -24√iC) 24iD) 24√i

Test Prep Example

• Simplify: -10 + √-16 2

A) -5 + 2iB) -5 – 4iC) 20 + 4iD) 30 + 2i

i 2 = -1• i 2 = -1 is the basis of everything you

will ever do with complex numbers.• Simplest form of a complex number

never allows a power of i greater than the 1st power to be present, so ………

Simplifying Powers of i

Simplification Simplest Formi None needed ii2 By definition - 1

i3 i2 x i = -1 x i = - ii4 ( i2)2 = ( -1)2 = 1

i5 ( i2)2 x i = ( -1)2 x i ii6 ( i2)3 = ( -1)3= - 1

i7 ( i2)3 x i = ( -1)3 x i - i i8 ( i2)4 = ( -1)4 = 1

Simplification Examples

• i42 = • Divide exponent by 2 (42 ÷2 = 21 R 0)• Quotient is exponent; Remainder is extra

power of i• Write as power of i (i2)21

• Simplify = (-1)21 = - 1

A Different Twist

Let’s look back at that pattern…i27 =

Divide exponent by 4 (27÷4 = 6 R 3)

Our remainder will determine the answer based on that pattern.

Remainder of 1 = iRemainder of 2 = -1

Remainder of 3 = - i No Remainder = 1

How Do I Simplify Powers of i graphic organizer

• How do I simplify Powers of i version 2.dcx

Examples

Test Prep Example

• Simplifyi4 + i3 + i2 + i

A) 0B) 1C) -1D) i

Test Prep Example

• Simplify(i)237

A) -1B) 1C) iD) -i

Lesson 1 Support Assignment

• Pg. 4: #1-33

Simplifying Square Roots of Negative Numbers

• √ – 9 does not exist in the reals because there is no number that can be squared to give a negative answer. Therefore, you must use i2 to replace the negative.

• √ – 9 = √9i2 = 3i• √ – 20 = √20i2 = √45i2 = 2i√5

Multiplying Square Roots of Negative Numbers

• Any time multiplication of square roots involves the square root of a negative number, you MUST replace the negative with i2 before doing any computation.

• √6 √ – 3 = √6 √3i2 = √18i2 = √92i2 = 3i√2• √– 2 √ – 8 = √2i2 √8i2 = √16i4 = 4i2 = – 4

Examples

Test Prep Example

• (√-4)(√-4)Simplify the expressionA) -4B) 2iC) 2i2

D) 4

Solving Equations in the Complex Numbers

• x2 + 4 = 0• Remember this equation that we used to

show why a sum of two squares never factors in the reals?

• x2 = - 4 √x2 = √-4• x = √-4 = √4i2 = 2i• Complex solutions always come in conjugate

pairs.

Examples