mathematics and algebra be without numbers? A part of everyday life, numbers are the basic building blocks of algebra. Numbers give you a value to work with. Where would civilization be today if not for numbers? Without numbers to figure the distances, slants, heights, and directions, the pyramids would never have been built. Without numbers to figure out navigational points, the Vikings would never have left Scandinavia. Without numbers to examine distance in space, humankind could not have landed on the moon.
Even the simple tasks and the most common of circumstances require a knowledge of numbers. Suppose that you wanted to figure the amount of gasoline it takes to get from home to work and back each day. You need a number for the total miles between your home and business and another number for the total miles your car can run on a gallon of gasoline.
The different sets of numbers are important because what they look like and how they behave can set the scene for particular situations or help to solve particular problems. It’s sometimes really convenient to declare, “I’m only going to look at whole-number answers,” because whole numbers don’t include fractions or negatives. You could easily end up with a fraction if you’re working through a problem that involves a number of cars or people. Who wants half a car or, heaven forbid, a third of a person?
Algebra uses different sets of numbers, in different circumstances. I describe the different types of numbers here.
Realizing real numbers
Real numbers are just what the name implies. In contrast to imaginary numbers, they represent real values – no pretend or make-believe. Real numbers cover the gamut and can take on any form – fractions or whole numbers, decimal numbers that can go on forever and ever without end, positives and negatives. The variations on the theme are endless.
Counting on natural numbers
A natural number (also called a counting number) is a number that comes naturally. What numbers did you first use? Remember someone asking, “How old are you?” You proudly held up four fingers and said, “Four!” The natural numbers are the numbers starting with 1 and going up by ones: 1, 2, 3, 4, 5, 6, 7, and so on into infinity. You’ll find lots of counting numbers in Chapter 8, where I discuss prime numbers and factorizations.
Dating back to about 2000 B.C. with the Babylonians, algebra seems to have developed in slightly different ways in different cultures. The Babylonians were solving three-term quadratic equations, while the Egyptians were more concerned with linear equations. The Hindus made further advances in about the sixth century A.D. In the seventh century, Brahmagupta of India provided general solutions to quadratic equations and had interesting takes on 0. The Hindus regarded irrational numbers as actual numbers – although not everybody held to that belief.
The sophisticated communication technology that exists in the world now was not available then, but early civilizations still managed to exchange information over the centuries. In A.D. 825, al-Khowarizmi of Baghdad wrote the first algebra textbook. One of the first solutions to an algebra problem, however, is on an Egyptian papyrus that is about 3,500 years old. Known as the Rhind Mathematical Papyrus after the Scotsman who purchased the 1-foot-wide, 18-foot-long papyrus in Egypt in 1858, the artifact is preserved in the British Museum – with a piece of it in the Brooklyn Museum. Scholars determined that in 1650 B.C., the Egyptian scribe Ahmes copied some earlier mathematical works onto the Rhind Mathematical Papyrus.
One of the problems reads, “Aha, its whole, its seventh, it makes 19.” The aha isn’t an exclamation. The word aha designated the unknown. Can you solve this early Egyptian problem? It would be translated, using current algebra symbols, as:
Whittling out whole numbers
Whole numbers aren’t a whole lot different from natural numbers. Whole numbers are just all the natural numbers plus a 0: 0, 1, 2, 3, 4, 5, and so on into infinity.
Whole numbers act like natural numbers and are used when whole amounts (no fractions) are required. Zero can also indicate none. Algebraic problems often require you to round the answer to the nearest whole number. This makes perfect sense when the problem involves people, cars, animals, houses, or anything that shouldn’t be cut into pieces.
Integrating integers
Integers allow you to broaden your horizons a bit. Integers incorporate all the qualities of whole numbers and their opposites (called their additive inverses). Integers can be described as being positive and negative whole numbers: … , –3, –2, –1, 0, 1, 2, 3, …
Integers are popular in algebra. When you solve a long, complicated problem and come up with an integer, you can be joyous because your answer is probably right. After all, it’s not a fraction! This doesn’t mean that answers in algebra can’t be fractions or decimals. It’s just that most textbooks and reference books try to stick with nice answers to increase the comfort level and avoid confusion. This is my plan in this book, too. After all, who wants a messy answer, even though, in real life, that’s more often the case. I use integers in Chapter 14 and those later on, where you find out how to solve equations.
Being reasonable: Rational numbers
Rational numbers act rationally! What does that mean? In this case, acting rationally means that the decimal equivalent of the rational number behaves. The decimal ends somewhere, or it has a repeating pattern to it. That’s what constitutes “behaving.”
Some rational numbers have decimals that end such as: 3.4, 5.77623, –4.5. Other rational numbers have decimals that repeat the same pattern, such as
In all cases, rational numbers can be written as fractions. Each rational number has a fraction that it’s equal to. So one definition of a rational number is any number that can be written as a fraction,
Restraining irrational numbers
Irrational numbers are just what you may expect from their name – the opposite of rational numbers. An irrational number cannot be written as a fraction, and decimal values for irrationals never end and never have a nice pattern to them. Whew! Talk about irrational! For example, π, with its never-ending decimal 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 first prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on. The only prime number that’s
