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# Complex Algebra

The beauty of Algebra, through complex numbers, fractals, and Euler’s formula.

Now that you've gotten a taste for algebraic intuition, here are three reasons to dive in and complete everything:

**To fully master Algebra.**(Note: this is*not*an introductory Algebra sequence, it is intended for people looking to master Algebra at a challenging and complex level.)**To understand Euler's identity**: \(e^{i \pi} + 1 = 0.\)To explore the applications of complex algebra to art and animation. (

**To learn how to create CGI and fractals.**)

**This next section will introduce each of the three motivations above in more detail and give you a preview of what you can learn here.**

Don't worry if you get a few or even all of these questions wrong; they are questions that demonstrate that there is a lot of **very challenging math** in this topic area, Complex Algebra.

One of the biggest advantages in how this exploration approaches Algebra is that **we’ll be using complex numbers and the complex plane throughout the whole exploration** in order to demonstrate why complex numbers make a lot of Algebra *more* natural, not less.

That said, mastering complex numbers will be a challenge!

What is \(\left(\sqrt{-4}\right)\left(\sqrt{-9}\right)?\)

**Note:** The complex number \(i\) is defined as \(i = \sqrt{-1}.\) For positive \(a,\) \(\sqrt{-a} = i \sqrt{a}.\)

Motivation 1:

To fully **master Algebra**, so as to be able to use it fluently and creatively, not just by applying memorized procedures.

For example, even if you've never seen **polar equations** like the ones below before, you can still solve this puzzle just by thinking about how the images and equations relate to each other:

**Which of the following is the missing rose?**

Motivation 2:

To understand how complex numbers fit into mathematics and what Euler's mind-blowing identity \(e^{i \pi} + 1 = 0\) really means.

True or False?

\[e^{\color{red}{-i \pi}} + 1 = 0\]

**Note:** Careful! Note the negative sign in the exponent on the left hand side of the equation.

Motivation 3:

To extend the mathematics in this exploration to applications creating animated graphics and mathematical art in the form of **fractals.**

Suppose a copper plate on a circuit board is made by removing squares from a square of copper so that it looks like the \(4^\text{th}\) stage of the

Consider the following "proof" that \(1 = -1.\) What is the first step that is incorrect?

**Step 1:** \(\frac{-1}{1} = \frac{1}{-1}.\)

**Step 2:** Take the square root of both sides to obtain \(\sqrt{\frac{-1}{1}} = \sqrt{\frac{1}{-1}}.\)

**Step 3:** Then, \(\frac{\sqrt{-1}}{\sqrt{1}} = \frac{\sqrt{1}}{\sqrt{-1}}.\)

**Step 4**: So, \(\frac{i}{1} = \frac{1}{i}.\)

**Step 5:** Multiply each side by \(i\) to obtain \(\frac{i^2}{1} = \frac{i}{i}.\)

**Step 6:** Simplify each fraction: \( \frac{i^2}{1} = i^2\) and \(\frac{i}{i} = 1.\) Then, since \(i^2 = -1,\) we have the final result:

\[\large -1 = 1.\]

**Note:** The complex number \(i\) is defined as \(i = \sqrt{-1}.\)

Overall, the questions in this exploration will require more thought and creativity than most problems you’d see in Algebra textbooks. Expect some of them to take you a bit longer, and expect to get many wrong--this is a hard field of mathematics to master!

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