Algebra I All-in-One For Dummies. Mary Jane Sterling

places, is irrational. Irrational numbers are often created when using the quadratic formula, as you see in Chapter 16, because you find the square roots of numbers that are not perfect squares, such as: .

      Picking out primes and composites

      A number is considered to be prime if it can be divided evenly only by 1 and by itself. The prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on. The only prime number that’s even is 2, the first prime number. Mathematicians have been studying prime numbers for centuries, and prime numbers have them stumped. No one has ever found a formula for producing all the primes. Mathematicians just assume that prime numbers go on forever.

      A number is composite if it isn’t prime — if it can be divided by at least one number other than 1 and itself. So the number 12 is composite because it’s divisible by 1, 2, 3, 4, 6, and 12. Chapter 8 deals with primes, but you also see them throughout the chapters, where I show you how to factor primes out of expressions.

      Numbers can be classified in more than one way, the same way that a person can be classified as male or female, tall or short, blonde or brunette, and so on. The number –3 is negative, it’s an integer, it’s an odd number, it’s rational, and it’s real. The number –3 is also a negative prime number. You should be familiar with all these classifications so that you can read mathematics correctly.

      Zero: It’s Complicated

      Zero is a very special number. It wasn’t really used in any of the earliest counting systems. In fact, there is no symbol for zero in the Roman numerals!

      Zero is a very useful number, but it also comes with its challenges. You can’t divide by zero, but you can add zero to a number and multiply a number by 0. You’ll find zero popping up in the most interesting places!

      Imagining imaginary numbers

      Yes, there are imaginary numbers in mathematics. These numbers were actually created by mathematicians who didn’t like not finishing a problem! They would be trying to solve a quadratic equation and be stumped by the situation where they needed the square root of a negative number. There was no way to deal with this.

      So some clever mathematicians came up with a solution. They declared that must be equal to i. Yes, the i stands for “imaginary.” You’ll see how this works in Chapter 16.

      Coping with complex numbers

      A complex number isn’t really all that mysterious. This is just a designation that allows for you to deal with both real and imaginary parts of a number. A complex number has some of each! Complex numbers have the general format of , where a and b are real numbers, and the i is that imaginary number, .

      

Q. Using the choices: natural, whole, integer, rational, irrational, prime, and imaginary, which of these can be used to describe the number 8?

      A. Natural, whole, integer, rational. The number 8 fits all of these descriptions. It is rational, because you can write it as a fraction such as or .

      Q. Using the choices: natural, whole, integer, rational, irrational, prime, and imaginary, which of these can be used to describe the number ?

      A. Rational. This is written as a fraction but cannot be reduced to create an integer.

      Q. Using the choices: natural, whole, integer, rational, irrational, prime, and imaginary, which of these can be used to describe the number ?

      A. Irrational. The number 17 isn’t a perfect square, so the decimal equivalence of is a decimal that goes on forever without repeating or terminating.

      Q. Using the choices: natural, whole, integer, rational, irrational, prime, and imaginary, which of these can be used to describe the number ?

      A. Imaginary. Even though 9 is a perfect square, so you can write the number as and then simplify it to read or , this number stays imaginary.

      1 Identify which of the following numbers are natural numbers:

      2 Identify which of the following numbers are integers:

      3 Identify which of the following numbers are rational numbers:

      4 Identify which of the following numbers are irrational numbers:

      5 Identify which of the following numbers are prime numbers:

      6 Identify which of the following numbers are imaginary numbers:

Placing Numbers on the Number Line

      A number line is labeled with numbers that increase as you move from left to right. And numbers are listed with an equal amount or value between any two consecutive numbers.

Numbers are placed on a number line to give you a visual picture of how they compare, how far apart they are, and what is missing between them. The two number lines shown here are examples of some versions that are possible. In Figure 1-1, you see the half-way mark indicated between units. And in Figure 1-2, the negative and positive integers are shown, with 0 in the middle.