Imaginary Numbers ???

In real life, complex numbers are used by engineers and physicists to measure electrical currents, to analyze stresses in structures such as bridges and buildings, and to study the flow of liquids.

Engineers who design speakers use complex numbers. Engineers who design and test the strength of bridges use complex numbers. Scientists who do experiments on ways to make energy using fuel cells, batteries, and solar cells use complex numbers.

Complex numbers are also used for generating fractals, which are geometric objects created by making a repeating pattern. A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self- similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop.

Which is actually very useful because… …by simply accepting that i exists lets us solve things where we need the square root of a negative number.

Example: What is the square root of -25? Answer:

A few questions… Simplify A) B)

Check your answers!

Aim: How can we simplify powers of i? If a whole number exponent is divided by 4, the remainder is 0, 1, 2, or 3. We can simplify powers of i by using the remainders after dividing by 4. Example: Write i82 in simplest terms 82 = 20 remainder 2 4 The remainder equals the power of i Therefore i82 is equivalent to i2. So i82 = -1

Aim: How can we simplify powers of i? Write each given power of I in simplest terms, as 1, i, -1, or -i i12 i91 i49 i54

Aim: How can we simplify powers of i? Write each given power of I in simplest terms, as 1, i, -1, or -i i12 12/4=3 remainder 0 = i0 = 1 2. i91 91/4 =22 remainder 3 = i3 = -i 3. i49 49/4= 12 remainder 1 = i1 = i 4. i 54 54/4 = 13 remainder 2 = i2 = -1

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Quick Practice

Quick Practice

Complex Numbers

Try it!

Graph it! Can you solve it graphically? What do you notice?

Solve Algebraically!

Practice! Simplify.  a)   b) c) d) e) f)

Solve: Try quadratic formula or complete the square!