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

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       This sequence is important for defining place value, the basis of the decimal number system, which I discuss in Chapter 3. It also shows up when I discuss decimals in Chapter 13 and scientific notation in Chapter 17. You find out more about exponents in Chapter 5.

Four Important Sets of Numbers

      In the preceding section, you see how a variety of number sequences extend infinitely. In this section, I provide a quick tour of how numbers fit together as a set of nested systems, one inside the other.

       When I talk about a set of numbers, I’m really just talking about a group of numbers. You can use the number line to deal with four important sets of numbers.

      The sets of counting numbers, integers, rational, and real numbers are nested, one inside another. This nesting of one set inside another is similar to the way that a city (for example, Boston) is inside a state (Massachusetts), which is inside a country (the United States), which is inside a continent (North America). The set of counting numbers is inside the set of integers, which is inside the set of rational numbers, which is inside the set of real numbers.

      Counting on the counting numbers

      The set of counting numbers is the set of numbers you first count with, starting with 1. Because they seem to arise naturally from observing the world, they’re also called the natural numbers:

      The counting numbers are infinite, which means they go on forever.

       When you add two counting numbers, the answer is always another counting number. Similarly, when you multiply two counting numbers, the answer is always a counting number. Another way of saying this is that the set of counting numbers is closed under both addition and multiplication.

      Introducing integers

      The set of integers arises when you try to subtract a larger number from a smaller one. For example, . The set of integers includes the following:

       The counting numbers

       Zero

       The negative counting numbers

Here’s a partial list of the integers:

      Like the counting numbers, the integers are closed under addition and multiplication. Similarly, when you subtract one integer from another, the answer is always an integer. That is, the integers are also closed under subtraction.

      Staying rational

      Here’s the set of rational numbers:

       Integers (which include the counting numbers, zero, and the negative counting numbers)

       Fractions

      Like the integers, the rational numbers are closed under addition, subtraction, and multiplication. Furthermore, when you divide one rational number by another, the answer is always a rational number. Another way to say this is that the rational numbers are closed under division.

      Getting real

      Even if you filled in all the rational numbers, you’d still have points left unlabeled on the number line. These points are the irrational numbers.

      An irrational number is a number that’s neither a whole number nor a fraction. In fact, an irrational number can only be approximated as a non-repeating decimal. In other words, no matter how many decimal places you write down, you can always write down more; furthermore, the digits in this decimal never become repetitive or fall into any pattern. (For more on repeating decimals, see Chapter 13.)

      The most famous irrational number is π (you find out more about π when I discuss the geometry of circles in Chapter 19):

      Together, the rational and irrational numbers make up the real numbers, which comprise every point on the number line. In this book, I don’t spend too much time on irrational numbers, but just remember that they’re there for future reference.

Chapter 2

      The Big Four Operations

      IN THIS CHAPTER

       Identifying the Big Four operations (addition, subtraction, multiplication, and division)

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