Basic Math & Pre-Algebra All-in-One For Dummies (+ Chapter Quizzes Online). Mark Zegarelli

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Chapter 1

      Playing the Numbers Game

      IN THIS CHAPTER

       Finding out how numbers were invented

       Looking at a few familiar number sequences

       Examining the number line

       Understanding four important sets of numbers

      One useful characteristic of numbers is that they’re conceptual, which means that, in an important sense, they’re all in your head. (This fact probably won’t get you out of having to know about them, though — nice try!)

      For example, you can picture three of anything: three cats, three baseballs, three tigers, three planets. But just try to picture the concept of three all by itself, and you find it’s impossible. Oh, sure, you can picture the numeral 3, but threeness itself — much like love or beauty or honor — is beyond direct understanding. But when you understand the concept of three (or four, or a million), you have access to an incredibly powerful system for understanding the world: mathematics.

      In this chapter, I give you a brief history of how numbers likely came into being. I discuss a few common number sequences and show you how these connect with simple math operations like addition, subtraction, multiplication, and division.

      After that, I describe how some of these ideas come together with a simple yet powerful tool: the number line. I discuss how numbers are arranged on the number line, and I also show you how to use the number line as a calculator for simple arithmetic. Finally, I describe how the counting numbers (1, 2, 3, …) sparked the invention of more unusual types of numbers, such as negative numbers, fractions, and irrational numbers. I also show you how these sets of numbers are nested — that is, how one set of numbers fits inside another, which fits inside another.

Inventing Numbers

      Historians believe that the first written number systems came into being at the same time as agriculture and commerce. Before that, people in prehistoric, hunter-gatherer societies were pretty much content to identify bunches of things as “a lot” or “a little.” They may have had concepts of small numbers, probably less than five or ten, but lacked a coherent way to think about, for example, the number 42.

      Throughout the ages, the Babylonians, Egyptians, Greeks, Hindus, Romans, Mayans, Arabs, and Chinese (to name just a few) all developed their own systems of writing numbers.

      Although Roman numerals gained wide currency as the Roman Empire expanded throughout Europe and parts of Asia and Africa, the more advanced system that was invented in India and adapted by the Arabs turned out to be more useful. Our own number system, the Hindu-Arabic numbers (also called decimal numbers), is mainly derived from these earlier number systems.

Understanding Number Sequences

      Although humans invented numbers for counting commodities, as I explain in the preceding section, they soon put them to use in a wide range of applications. Numbers were useful for measuring distances, counting money, amassing armies, levying taxes, building pyramids, and lots more.

      But beyond their many uses for understanding the external world, numbers have an internal order all their own. So numbers are not only an invention, but equally a discovery: a landscape reflecting fundamental truths about nature, and how humans think about it, that seems to exist independently, with its own structure, mysteries, and even perils.

      One path into this new and often strange world is the number sequence: an arrangement of numbers according to a rule. In the following sections, I introduce you to a variety of number sequences that are useful for making sense of numbers.

      Evening the odds

      One of the first facts you probably heard about numbers is that all of them are either even or odd. For example, you can split an even number of marbles evenly into two equal piles. But when you try to divide an odd number of marbles the same way, you always have one odd, leftover marble. Here are the first few even numbers:

      You can easily keep the sequence of even numbers going as long as you like. Starting with the number 2, keep adding 2 to get the next number.

Similarly, here are the first few odd numbers:

      The sequence of odd numbers is just as simple to generate. Starting with the number 1, keep adding 2 to get the next number.

      Patterns of even or odd numbers are the simplest number patterns around, which is why kids often figure out the difference between even and odd numbers soon after learning to count.

      Counting by threes, fours, fives, and so on

      When you get used to the concept of counting by numbers greater than 1, you can run with it. For example, here’s what counting by threes, fours, and fives looks like:

       Counting by a given number is a good way to begin learning the multiplication table for that number, especially for the numbers you’re kind of sketchy on. (In general, people seem to have the most trouble multiplying by 7, but 8 and 9 are also unpopular.)

      These types of sequences are also useful for understanding factors СКАЧАТЬ