A.6 Complex Numbers Solving Equations. Imaginary Numbers.

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A.6 Complex Numbers & Solving Equations

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Complex numbers are numbers of the form where a is called the real part and b is the imaginary part Complex Numbers

Transcript of A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Page 1: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

A.6 Complex Numbers & Solving

Equations

Page 2: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Imaginary Numbers

1i

i2 1

i3 i

i4 1 ____5 i

____19 i

Page 3: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Complex numbers are numbers of the form

where a is called the real part and b is the imaginary part

Complex Numbers

a bi

Page 4: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Complex NumbersTo add complex numbers, add the real parts and add the imaginary parts together (same for subtraction).

(a bi) (c di)(a c) (b d)i

(a bi) (c di)(a c) (b d)i

To multiply complex numbers, follow the usual rules for multiplying two binomials.

(a bi)(c di)a(c di) bi(c di)

DOUBLE DISTRIBUTION!!!

i2 1

Page 5: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Given the following complex numbers, perform the indicated operations.

Complex Numbers

w z

w 5 3i

z 1 4i

w z

w z

1)

2)

3)

Page 6: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

If is a complex number, then its conjugate, denoted by , is defined as

Complex Numbers

z a bi

z

z a bi a bi

zz a2 b2

Page 7: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Let’s find the conjugate of the following complex numbers and then multiply by the conjugate.

Complex Numbers

z 3 4i

z 1 8i

z 2i

1)

2)

3)

Page 8: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

We can apply multiplying conjugates to quotients of complex numbers. Let’s write the following in standard form!

Complex Numbers

2 3i4 3i

Page 9: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Complex NumbersGiven the following complex numbers, perform the indicated operations.

w 2 3i

z 5 2i

w z

w z

w z

1)

2)

3)

wz

4)

Page 10: A.6 Complex Numbers Solving Equations. Imaginary Numbers.

Homework#8: p.1008 #9 – 45 Odd

A.6 Complex Numbers & Solving

Equations


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4.8 Complex Numbers Ch 4...Algebra 2 Notes, Chapter 4 1 4.8 Complex Numbers A. Imaginary Numbers Consider: 25 2 Has no REAL solution because (?) = -25 Imaginary Numbers were created

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Section 2.1 Complex Numbers. The Imaginary Unit i.

Section 2.1 Complex Numbers. The Imaginary Unit i.

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5.9: Imaginary + Complex Numbers -Defining i -Simplifying negative radicands -Powers of i -Solving equations -Complex numbers -Operations with complex.

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Imaginary & Complex Numbers. Once upon a time… -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented.

Imaginary & Complex Numbers. Once upon a time… -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented.

Imaginary Numbers Aren't Real

Imaginary Numbers Aren't Real

Complex Numbers 2-4. Imaginary Numbers 1.Designed so negative numbers can have square roots., i 2 = -1 2.Imaginary numbers consist of all numbers bi,

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5.4 – Complex Numbers Our Imaginary Friends Uses of Imaginary Numbers Contrary to its name, the “imaginary number” exists in a similar fashion to the.

5.4 – Complex Numbers Our Imaginary Friends Uses of Imaginary Numbers Contrary to its name, the “imaginary number” exists in a similar fashion to the.

Review on imaginary and complex numbers

Review on imaginary and complex numbers

Integrated Design for Solving Imaginary Complexity in Design · Integrated Design for Solving Imaginary Complexity in Design K. Pimapunsri 1, S. Tichkiewitch 2, 1 Department of Industrial

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Complex numbers i or j. Complex numbers An Imaginary Number, when squared, gives a negative result. imaginary 2 = negative.

Complex numbers i or j. Complex numbers An Imaginary Number, when squared, gives a negative result. imaginary 2 = negative.

PROBLEM SOLVING - Institute for Student Achievement · Web viewHow do the properties of real numbers apply to or not apply to imaginary numbers. Why? Example Project/Research: Sum

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Imaginary numbers a book of poems

Imaginary numbers a book of poems

Imaginary Numbers are not Real

Imaginary Numbers are not Real

Complex Numbers Operations with Complex Numbers What is the Imaginary Unit?

Complex Numbers Operations with Complex Numbers What is the Imaginary Unit?

§ 7.7 Complex Numbers. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.7 Complex Numbers The Imaginary Unit i The imaginary unit i is defined.

§ 7.7 Complex Numbers. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.7 Complex Numbers The Imaginary Unit i The imaginary unit i is defined.

7.7 Complex Numbers. Imaginary Numbers Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution”

7.7 Complex Numbers. Imaginary Numbers Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution”

Imaginary Numbers

Imaginary Numbers

Complex Numbers. Definition of pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

Complex Numbers. Definition of pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.