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

Скачать книгу

of a number that refuses to fit in a box.

      Try as you may, you just can’t make a rectangle out of 13 objects. (That fact may be one reason why the number 13 got a bad reputation as unlucky.) Here are all the prime numbers less than 20:

math

      As you can see, the list of prime numbers fills the gaps left by the composite numbers (see the preceding section). Therefore, every counting number is either prime or composite. The only exception is the number 1, which is neither prime nor composite. In Chapter 8, I give you a lot more information about prime numbers and show you how to decompose a number — that is, break down a composite number into its prime factors.

      Multiplying quickly with exponents

      Here’s an old question whose answer may surprise you: Suppose you took a job that paid you just 1 penny the first day, 2 pennies the second day, 4 pennies the third day, and so on, doubling the amount every day, like this:

math

      As you can see, in the first ten days of work, you would’ve earned a little more than $10 (actually, $10.23 — but who’s counting?). How much would you earn in 30 days? Your answer may well be, “I wouldn’t take a lousy job like that in the first place.” At first glance, this looks like a good answer, but here’s a glimpse at your second ten days’ earnings:

math

      By the end of the second 10 days, when you add it all up, your total earnings would be over $10,000. And by the end of 30 days, your earnings would top out around $10,000,000! How does this happen? Through the magic of exponents (also called powers). Each new number in the sequence is obtained by multiplying the previous number by 2:

math

      As you can see, the notation math means multiply 2 by itself 4 times.

      You can use exponents on numbers other than 2. Here’s another sequence you may be familiar with:

math

      In this sequence, every number is 10 times greater than the number before it. You can also generate these numbers using exponents:

math

      Remember 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.

      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.

      Tip 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.

       Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4 and going on infinitely

       Integers: The set of counting numbers, zero, and negative counting numbers

       Rational numbers: The set of integers and fractions

       Real numbers: The set of rational and irrational 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:

math

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

      Remember 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, math. The set of integers includes the following:

       The counting numbers

       Zero

       The negative counting numbers

 math

      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

Скачать книгу