imaginary numbers

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Imaginary Numbers Unit 1 Lesson 1

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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:. - PowerPoint PPT Presentation

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Page 1: Imaginary Numbers

Imaginary Numbers

Unit 1 Lesson 1

Page 2: Imaginary Numbers

Make copies of:

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

Page 3: Imaginary Numbers

GPS Standards

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

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

Page 4: Imaginary Numbers

Essential Questions:

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

• How do I simplify powers of i?

Page 5: Imaginary Numbers

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

Page 6: Imaginary 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

Page 7: Imaginary 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??

Page 8: Imaginary Numbers

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”

Page 9: Imaginary Numbers

Imaginary numbers

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

Page 10: Imaginary Numbers

Using imaginary numbers

Simplify the following√-9

√-72√50√-17i 3

Page 11: Imaginary Numbers

Test Prep Example

• Express in terms of i: -3√-64

A) -24iB) -24√iC) 24iD) 24√i

Page 12: Imaginary Numbers

Test Prep Example

• Simplify: -10 + √-16 2

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

Page 13: Imaginary Numbers

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 ………

Page 14: Imaginary Numbers

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

Page 15: Imaginary Numbers

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

Page 16: Imaginary Numbers

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

Page 17: Imaginary Numbers

How Do I Simplify Powers of i graphic organizer

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

Page 18: Imaginary Numbers

Examples

Page 19: Imaginary Numbers

Test Prep Example

• Simplifyi4 + i3 + i2 + i

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

Page 20: Imaginary Numbers

Test Prep Example

• Simplify(i)237

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

Page 21: Imaginary Numbers

Lesson 1 Support Assignment

• Pg. 4: #1-33

Page 22: Imaginary Numbers

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

Page 23: Imaginary Numbers

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

Page 24: Imaginary Numbers

Examples

Page 25: Imaginary Numbers

Test Prep Example

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

D) 4

Page 26: Imaginary Numbers

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.

Page 27: Imaginary Numbers

Examples