Warm-up Simplify as far as possible, do not make a decimal. Your answer will still have a square root sign in it. 1. (5 + 4x) + (7 - 2x)    

How do we take the square root of negative numbers? Math II Day 2 (1-5-11) Standard MM2N1 a - Write square roots of negative numbers in imaginary form. b - Write complex numbers in the form a+bi. Today’s Question: How do we take the square root of negative numbers? Standard: MM2N1.a, b, c, d

in a math class far, far away.. A long long time ago, in a math class far, far away.. There was no way to take the square root of a negative number

Every time we squared a negative number We got a positive.

(-1) = 1 (-2) = 4 (-3) = 9

that when multiplied by itself Was there a number, that when multiplied by itself Gave you a negative???

Can we in fact, take the square root of a negative number? WE CAN!!!!

Ladies and Gentlemen of Math II I present to you a NEW number... A number so complex...

It stretches the imagination.. I present to you:

You can't take the square root of a negative number, right? When we were young and still in Math I, no numbers that, when multiplied by themselves, gave us a negative answer.  Squaring a negative number always gives you a positive.   (-1)² = 1. (-2)² = 4 (-3)² = 9

So here’s what the math people did: They used the letter “i” to represent the square root of (-1). “i” stands for “imaginary” So, does really exist?

Examples of how we use

Examples of how we use            

         

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1.3 Powers of i and Multiplying Complex Numbers

*For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example:

$25,000 Pyramid

$25,000 Pyramid i -i 1 i -1 -i -1 -i 1 -1

Multiplying Treat the i’s like variables, then change any that are not to the first power Ex: Ex:

Remember the area method (6x+2)(4x-5) 24x2 + 8x -30x -10 =24x2 – 22x -10 4x -5 6x 24x2 -30x 2 8x -10

Your Turn!

Your Turn!

Assignment Page 4 – 18 - 21 Page 8 – 20 - 24 Page 14 – 4 - 11

Complex Numbers A complex number has a real part & an imaginary part. Standard form is: Real part Imaginary part Example: 5+4i

The Complex Plane Real Axis Imaginary Axis

Graphing in the complex plane

Absolute Value of a Complex Number The distance the complex number is from the origin on the complex plane. If you have a complex number the absolute value can be found using:

Graphing in the complex plane 5 2

Examples 1. 2. Which of these 2 complex numbers is closest to the origin? -2+5i

Try These!!! 1. 2. Which of these 2 complex numbers is closest to the origin? 3i