Chapter 1

Pre-Pre-Calculus

IN THIS CHAPTER

Refreshing your memory on numbers and variables

Accepting the importance of graphing

Preparing for pre-calculus by grabbing a graphing calculator

Pre-calculus is the bridge (drawbridge, suspension bridge, covered bridge) between Algebra II and calculus. In its scope, you review concepts you've seen before in math, but then you quickly build on them. You see some brand-new ideas, but even those build on the material you've seen before; the main difference is that the problems get much more challenging (for example, going from linear systems to nonlinear systems). You keep on building until the end of the bridge span, which doubles as the beginning of calculus. Have no fear! What you find here will help you cross the bridge (toll free).

Because you've probably already taken Algebra I, Algebra II, and geometry, it's assumed throughout this book that you already know how to do certain things. Just to make sure, though, I address some particular items in this chapter in a little more detail before moving on to the material that is pre-calculus.

If there is any topic in this chapter that you're not familiar with, don't remember how to do, or don't feel comfortable doing, I suggest that you pick up another For Dummies math book and start there. If you need to do this, don't feel like a failure in math. Even pros have to look up things from time to time. Use these books like you use encyclopedias or the Internet - if you don't know the material, just look it up and get going from there.

Pre-Calculus: An Overview

Don't you just love movie previews and trailers? Some people show up early to movies just to see what's coming out in the future. Well, consider this section a trailer that you see a couple months before the Pre-Calculus For Dummies movie comes out! The following list, presents some items you've learned before in math, and some examples of where pre-calculus will take you next:

• Algebra I and II: Dealing with real numbers and solving equations and inequalities.

  Pre-calculus: Expressing inequalities in a new way called interval notation.

  You may have seen solutions to inequalities in set notation, such as . This is read, in inequality notation as . In pre-calculus, you often express this solution as an interval: . (For more, see Chapter 2.)

• Geometry: Solving right triangles, whose sides are all positive.

  Pre-calculus: Solving non-right triangles, whose sides aren't always represented by positive numbers.

  You've learned that a length can never be negative. Well, in pre-calculus you sometimes use negative numbers for the lengths of the sides of triangles. This is to show where these triangles lie in the coordinate plane (they can be in any of the four quadrants).

• Geometry/trigonometry: Using the Pythagorean Theorem to find the lengths of a triangle's sides.

  Pre-calculus: Organizing some frequently used angles and their trig function values into one nice, neat package known as the unit circle (see Part 2).

  In this book, you discover a handy shortcut to finding the sides of triangles - a shortcut that is even handier for finding the trig values for the angles in those triangles.

• Algebra I and II: Graphing equations on a coordinate plane.

  Pre-calculus: Graphing in a brand-new way with the polar coordinate system (see Chapter 12).

  Say goodbye to the good old days of graphing on the Cartesian coordinate plane. You have a new way to graph, and it involves goin' round in circles. I'm not trying to make you dizzy; actually, polar coordinates can make you some pretty pictures.

• Algebra II: Dealing with imaginary numbers.

  Pre-calculus: Adding, subtracting, multiplying, and dividing complex numbers gets boring when the complex numbers are in rectangular form . In pre-calculus, you become familiar with something new called polar form and use that to find solutions to equations you didn't even know existed.

All the Number Basics (No, Not How to Count Them!)

When entering pre-calculus, you should be comfy with sets of numbers (natural, integer, rational, and so on). By this point in your math career, you should also know how to perform operations with numbers. You can find a quick review of these concepts in this section. Also, certain properties hold true for all sets of numbers, and it's helpful to know them by name. I review them in this section, too.

The multitude of number types: Terms to know

Mathematicians love to name things simply because they can; it makes them feel special. In this spirit, mathematicians attach names to many sets of numbers to set them apart and cement their places in math students' heads for all time:

• The set of natural or counting numbers: {1, 2, 3 .}. Notice that the set of natural numbers doesn't include 0.
• The set of whole numbers: {0, 1, 2, 3 .}. The set of whole numbers does include the number 0, however.

• The set of integers: {. -3, -2, -1, 0, 1, 2, 3 .}. The set of integers includes positives, negatives, and 0.

  Dealing with integers is like dealing with money: Think of positives as having it and negatives as owing it. This becomes important when operating on numbers (see the next section).

• The set of rational numbers: the numbers that can be expressed as a fraction where the numerator and the denominator are both integers. The word rational comes from the idea of a ratio (fraction or division) of two integers.

  Examples of rational numbers include (but in no way are limited to) , , and 0.23. A rational number is any number in the form where p and q are integers, but q is never 0. If you look at any rational number in decimal form, you notice that the decimal either stops or repeats.

  Adding and subtracting fractions is all about finding a common denominator. And roots must be like terms in order to add and subtract them. For example, you can add and , but you can't add and .

• The set of irrational numbers: all numbers that can't be expressed as fractions. Examples of irrational numbers include , , and .

• The set of all real numbers: all the sets of numbers previously discussed. For an example of a real number, think of a number . any number. Whatever it is, it's real. Any number from the previous bullets works as an example. The numbers that aren't real numbers are imaginary.

  Like telemarketers and pop-up ads on the Net, real numbers are everywhere; you can't get away from them - not even in pre-calculus. Why? Because they include all numbers except the following:

  • A fraction with a zero as the denominator: Such numbers don't exist and are called undefined.
  • The square root of a negative number: These numbers are part of complex numbers; the negative root is the imaginary part (see Chapter 12). And this extends to any even root of a negative number.
  • Infinity: Infinity is a concept, not an actual number.
• The set of imaginary numbers: square roots of negative numbers. Imaginary numbers have an imaginary unit, like i, 4i, and -2i. Imaginary numbers used to be considered to be made-up numbers, but mathematicians soon realized that these numbers pop up in the real world. They are still called imaginary because they're square roots of negative numbers, but they are a part of the language of mathematics. The imaginary unit is defined as . (For more on these numbers, head to Chapter 12.)

• The set of complex numbers: the sum or difference of a real number and an imaginary number. Complex numbers appear like these examples: , , and . However, they also cover all the previous lists, including the real numbers (3 is the same thing as ) and imaginary numbers (2i is the same thing as ).

  The set of complex numbers is the most complete set of numbers in the math vocabulary, because it includes real numbers (any number you can possibly think of), imaginary numbers (i), and any combination of the two.

The fundamental operations you can perform on numbers

From positives and negatives to fractions, decimals, and square roots, you should know how to perform all the basic operations on all real numbers. These operations include adding, subtracting, multiplying, dividing, taking powers of, and taking roots of numbers. The order of operations is the way in which you perform these operations.

The mnemonic device used most frequently to remember the order is PEMDAS, which stands for

1. Parentheses (and other grouping devices)
2. Exponents (and roots, which can be written as exponents)
3. Multiplication and Division (whichever is first, from left to right)
4. Addition and Subtraction (whichever is first, from left to right)

One type of operation that is often overlooked or forgotten about: absolute value. Absolute value gives you...