Introduction

Requirements

To fully appreciate this geometry book, you must have a basic understanding of algebra. If that is what you have really come to learn, then may I suggest you get a copy of Schaum’s Outline of College Algebra. You will learn everything you need and more (things you don’t need to know!)

If you have come to learn geometry, it begins at Chapter one.

As for algebra, you must understand that we can talk about numbers we do not know by assigning them variables like x, y, and A.

You must understand that variables can be combined when they are exactly the same, like x + x = 2x and 3x2 + 11x2 = 14x2, but not when there is any difference, like 3x2y–9xy = 3x2y–9xy.

You should understand the deep importance of the equals sign, which indicates that two things that appear different are actually exactly the same. If 3x = 15, then this means that 3x is just another name for 15. If we do the same thing to both sides of an equation (add the same thing, divide both sides by something, take a square root, etc.), then the result will still be equal.

You must know how to solve an equation like 3x + 8 = 23 by subtracting eight from both sides, 3x + 8 – 8 = 23 – 8 = 15, and then dividing both sides by 3 to get 3x/3 = 15/3 = 5. In this case, the variable was constrained; there was only one possible value and so x would have to be 5.

You must know how to add these sorts of things together, such as (3x + 8) + (9–x) = (3xx) + (8 + 9) = 2x + 17. You don’t need to know that the ability to rearrange the parentheses is called associativity and the ability to change the order is called commutativity.

You must also know how to multiply them: (3x + 8)·(9–x) = 27x–3x2 + 72–8x =–3x2 + 19x + 72

Actually, you might not even need to know that.

You must also be comfortable using more than one variable at a time, such as taking an equation in terms of y like y = x2 + 3 and rearranging the equation to put it in terms of x like y–3 = x2. so Image = Image and thus Image = ± x, so x = ± Image.

You should know about square roots, how Image It is useful to keep in mind that there are many irrational numbers, like Image, which could never be written as a neat ratio or fraction, but only approximated with a number of decimals.

You shouldn’t be scared when there are lots of variables, either, such as Image thus, Fr2 = gM1M2 by cross-multiplication, so Image

Most important of all, you should know how to take a formula like Image and replace values and simplify. If r = 5 cm and h = 8 cm, then

Image (5 cm)2 (8 cm) = Image