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.

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Presentation transcript:

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 “real number,” often describing physical characteristics that we can detect, observe, and measure. Common applications are circuit analysis in electrical engineering and vibration analysis in mechanical engineering. Want to learn more? numbers-lesson-1/

Fractals: What are they? A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales.self-similarity

Examples of Fractals Examples include clouds, river networks, fault lines, mountain ranges, craters, snow flakes, crystals, lightning, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels, and ocean waves. DNA and heartbeats can be analyzed as fractals.river networks fault linesmountain rangescraters snow flakescrystalslightning cauliflowerbroccoli blood vesselspulmonary vesselsocean wavesDNAheartbeats

5.4 – Complex Numbers Objectives: 1. Be able to find, identify, and name imaginary numbers 2. Be able to add and subtract imaginary numbers 3. Be able to multiply and divide imaginary numbers Vocabulary: imaginary unit - i, complex numbers, standard form, imaginary numbers, pure imaginary numbers, conjugate

Objective #1 – Be able to find, identify, and name imaginary numbers. 5.4 – Complex Numbers

complex number = a + bi a is the real part b is the imaginary part a + 0i are the real numbers a + bi are the imaginary numbers 0 + bi are the pure imaginary numbers Objective #1 – Be able to find, identify, and name imaginary numbers. 5.4 – Complex Numbers

Examples of Complex Numbers 4 + 0i = 4 (Real Number) 2 – 3i (Imaginary Number  Real Number and Imaginary Number Combo) 0 + 6i = 6i (Pure imaginary Number)

The Square Root of a Negative Number Properties

Examples

Examples

Examples

Examples

Objective #2 – Be able to add and subtract imaginary numbers. 5.4 – Complex Numbers Real partImaginary part

Examples

Examples

Examples

Solving Quadratic Equations with i

1. Simplify the radical expression. 1. Solve the equation. 1. Write the expression as a complex number in standard form. Exit Slip Complete the following problems individually. You may use your notes. Remain quiet until all exit slips have been collected.

pg. 277 #18-28 even, Homework Homework…

5.4 – Complex Numbers Our Imaginary Friends

REMEMBER Objective #3 – Be able to multiply and divide imaginary numbers. 5.4 – Complex Numbers

Example

Example

Example

Example This answer is a REAL number!

Conjugates: a + bi and a - bi multiplying conjugates will always give you a real number! Objective #3 – Be able to multiply and divide imaginary numbers. 5.4 – Complex Numbers

Example

Example

Future electrical engineer? Check out pg. 279 #95, 96 Homework pg. 278 #47-62 (every third problem) That lesson was…