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

Чтение книги онлайн.

       You get a square number by multiplying a number by itself, so knowing the square numbers is another handy way to remember part of the multiplication table. Although you probably remember without help that 2 × 2 = 4, you may be sketchy on some of the higher numbers, such as 7 × 7 = 49. Knowing the square numbers gives you another way to etch that multiplication table forever into your brain.

      Square numbers are also a great first step on the way to understanding exponents, which I introduce later in this chapter and explain in more detail in Chapter 5.

      Composing yourself with composite numbers

Some numbers can be placed in rectangular patterns. Mathematicians probably should call numbers like these “rectangular numbers,” but instead they chose the term composite numbers. For example, 12 is a composite number because you can place 12 objects in rectangles of two different shapes, as in Figure 1-2.

      © John Wiley & Sons, Inc.

      FIGURE 1-2: The number 12 laid out in two rectangular patterns.

      As with square numbers, arranging numbers in visual patterns like this tells you something about how multiplication works. In this case, by counting the sides of both rectangles, you find out the following:

Similarly, other numbers such as 8 and 15 can also be arranged in rectangles, as in Figure 1-3.

      As you can see, both these numbers are quite happy being placed in boxes with at least two rows and two columns. And these visual patterns show this:

      © John Wiley & Sons, Inc.

      FIGURE 1-3: Composite numbers, such as 8 and 15, can form rectangles.

      The word composite means that these numbers are composed of smaller numbers. For example, the number 15 is composed of 3 and 5 — that is, when you multiply these two smaller numbers, you get 15. Here are all the composite numbers from 1 to 16:

      Notice that all the square numbers (see the section, “Getting square with square numbers”) also count as composite numbers because you can arrange them in boxes with at least two rows and two columns. Additionally, a lot of other non-square numbers are also composite numbers.

      Stepping out of the box with prime numbers

Some numbers are stubborn. Like certain people you may know, these numbers — called prime numbers — resist being placed in any sort of a box. Look at how Figure 1-4 depicts the number 13, for example.

      © John Wiley & Sons, Inc.

      FIGURE 1-4: Unlucky 13, a prime example 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:

      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:

      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:

      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 СКАЧАТЬ