# Precalculus - A Khan academy revision guide

(under construction ... abandoned until I have students who need this)

The purpose of this note is provide a quick access to learning or reviewing precalculus without having to go through all videos and exercises of Khan academy.

The exercises are those that you really need to be able to do in order to be able to study calculus. American Precalculus contains much more such as trigonometry, conic sections, division of polynomials, rational functions. All beautiful and important topics, but not strictly needed for calculus.

There are also links to exercises in paragraphs entitled "Need to know", but if you feel confident that you understand what the paragraph says, you may skip them.

Of course, if you want to learn more, the excellent videos of Khan are highly recommended. But I believe the course can be done without. Strictly speaking, all you need to know to solve the exercises is on this page (assuming that you have learned the subjects on which precalculus builds ... appropriate references for revision will be added in the future.)

## Prerequisites

[Pre-algebra and Algebra](https://hackmd.io/@m5rnD-8SSPuuSHTKgXvMjg/S1mc96a8r), as Khan calls it, are the most important prerequisites. I collected a few exercises to brush up any rusty skills.

## Complex Numbers

**Exercise:** To [simplify roots of complex numbers](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:imaginary-intro/e/simplify-square-roots-of-negative-numbers?modal=1) you only need to know that $i\cdot i = -1$ (equivalently $i=\sqrt{-1}$) and otherwise follow the usual rules.

*Remark:* If you wonder what precisely is meant by the "usual rules" then you can refer to the laws of the "classic definition" of a [field](https://en.wikipedia.org/wiki/Field_(mathematics)#Classic_definition). That is all there is.

*Need to know:* Numbers $x+iy$ are called complex, $x$ is the real part and $y$ is the imaginary part. You need to know the notation for [real part and imaginary part](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-intro/e/real-and-imaginary-parts-of-complex-numbers?modal=1) $${\rm Re}(x+iy)=x \ , \ {\rm Im}(x+iy)=y.$$

*Need to know:* [Plotting complex numbers](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-plane/e/the_complex_plane?modal=1) works as in geometry with $x+iy$ corresponding to the pair $(x,y)$. Possibly revise [link missing]().

*Need to know:* When we write $iy$ or $yi$ this is an abbreviation of $i*y$ or $y*i$, where the multiplication $*$ follows the usual rules. For example,
\begin{align}(3+2i)-(2-3i)& =
3+2i-2+3i\\
& = 3+2i-2+3i\\
& = 3+(-2)+2i+3i\\
& = 1+(2+3)i\\
& = 1+5i
\end{align}

applying one of the usual rules for calculating with numbers. Make sure you understand all these steps. Can you say which rule was applied in each step?

**Exercise:** [Adding and subtracting](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-add-sub/e/adding_and_subtracting_complex_numbers) works the same as we know from algebra.

**Exercise:** The [distance](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-distance-midpoint/e/distance-and-midpoint-on-the-complex-plane?modal=1) from $x+yi$ to $a+bi$ is $\sqrt{(a-x)^2+(b-y)^2}$. (If you remember Pythagoras you already know this formula.)

**Exercise:** The [**midpoint**](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-distance-midpoint/e/find-the-midpoint-of-two-complex-numbers?modal=1) or **average** of $x+yi$ and $a+bi$ is $\frac{(x+a)+(y+b)i}{2}$.

**Exercise:** To [multiply complex numbers](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-mul/e/multiplying_complex_numbers?modal=1) you only need to know that $i\cdot i = -1$ and otherwise follow the usual rules.

*Need to know:* The [conjugate](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-div/e/find-conjugates-of-complex-numbers?modal=1) of $x+yi$ is $x-yi$.

**Exercise:** To [divide complex numbers](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-div/e/dividing_complex_numbers?modal=1) you only need to know that $i\cdot i = -1$ and otherwise follow the usual rules. Possibly revise [link missing]().

**Exercise:** To [factor polynomials](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-id/e/equivalent-forms-of-expressions-with-complex-numbers?modal=1) you only need to know that $i\cdot i = -1$ and otherwise follow the usual rules. Possibly revise [link missing]().

**Exercise:** The [absolute value]() of $x+yi$ is $\sqrt{x^2 + y^2}$. (This is Pythagoras again).

## To be continued ...

...