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 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.
Examples
Q. Given the numbers: 0, 8, –11,
A. Only the 8 is a natural number. The number 0 is a whole number, but not considered to be a counting number. Natural numbers are positive, so –11 isn’t natural. And the fraction and radical have decimal equivalents that don’t round to natural numbers.
Q. Given the numbers: 0, 8, –11,