Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)

Part I. Getting Started with Basic Math and Pre-Algebra

Chapter 2. It’s All in the Fingers: Numbers and Digits

IN THIS CHAPTER

Understanding how place value turns digits into numbers

Distinguishing whether zeros are important placeholders or meaningless leading zeros

Reading and writing long numbers

Understanding how to round numbers and estimate values

When you’re counting, ten seems to be a natural stopping point — a nice, round number. The fact that our ten fingers match up so nicely with numbers may seem like a happy accident. But of course, it’s no accident at all. Fingers were the first calculator that humans possessed. Our number system — Hindu-Arabic numbers — is based on the number ten because humans have 10 fingers instead of 8 or 12. In fact, the very word digit has two meanings: numerical symbol and finger.

In this chapter, I show you how place value turns digits into numbers. I also show you when 0 is an important placeholder in a number and why leading zeros don’t change the value of a number. And I show you how to read and write long numbers. After that, I discuss two important skills: rounding numbers and estimating values.

Knowing Your Place Value

The number system you’re most familiar with — Hindu-Arabic numbers — has ten familiar digits:

0 1 2 3 4 5 6 7 8 9

Yet with only ten digits, you can express numbers as high as you care to go. In this section, I show you how it happens.

TELLING THE DIFFERENCE BETWEEN NUMBERS AND DIGITS

Sometimes people confuse numbers and digits. For the record, here’s the difference:

· A digit is a single numerical symbol, from 0 to 9.

· A number is a string of one or more digits.

For example, 7 is both a digit and a number. In fact, it’s a one-digit number. However, 15 is a string of two digits, so it’s a number — a two-digit number. And 426 is a three-digit number. You get the idea.

In a sense, a digit is like a letter of the alphabet. By themselves, the uses of 26 letters, A through Z, are limited. (How much can you do with a single letter such as K or W?) Only when you begin using strings of letters as building blocks to spell words does the power of letters become apparent. Similarly, the ten digits, 0 through 9, have limited usefulness until you begin building strings of digits — that is, numbers.

Counting to ten and beyond

The ten digits in our number system allow you to count from 0 to 9. All higher numbers are produced using place value. Place value assigns a digit a greater or lesser value, depending on where it appears in a number. Each place in a number is ten times greater than the place to its immediate right.

To understand how a whole number gets its value, suppose you write the number 45,019 all the way to the right in Table 2-1, one digit per cell, and add up the numbers you get.

TABLE 2-1 45,019 Displayed in a Place-Value Chart

You have 4 ten thousands, 5 thousands, 0 hundreds, 1 ten, and 9 ones. The chart shows you that the number breaks down as follows:

45,019 = 40,000 + 5,000 + 0 + 10 + 9

In this example, notice that the presence of the digit 0 in the hundreds place means that zero hundreds are added to the number.

Telling placeholders from leading zeros

Although the digit 0 adds no value to a number, it acts as a placeholder to keep the other digits in their proper places. For example, the number 5,001,000 breaks down into 5,000,000 + 1,000. Suppose, however, you decide to leave all the 0s out of the chart. Table 2-2 shows what you’d get.

TABLE 2-2 5,001,000 Displayed Incorrectly without Placeholding Zeros

The chart tells you that 5,001,000 = 50 + 1. Clearly, this answer is wrong!

As a rule, when a 0 appears to the right of at least one digit other than 0, it’s a placeholder. Placeholding zeros are important — always include them when you write a number. However, when a 0 appears to the left of every digit other than 0, it’s a leading zero. Leading zeros serve no purpose in a number, so dropping them is customary. For example, place the number 003,040,070 on the chart (see Table 2-3).

TABLE 2-3 3,040,070 Displayed with Two Leading Zeros

The first two 0s in the number are leading zeros because they appear to the left of the 3. You can drop these 0s from the number, leaving you with 3,040,070. The remaining 0s are all to the right of the 3, so they’re placeholders — be sure to write them in.

Reading long numbers

When you write a long number, you use commas to separate groups of three numbers. For example, here’s about as long a number as you’ll ever see:

234,845,021,349,230,467,304

Table 2-4 shows a larger version of the place-value chart.

TABLE 2-4 A Place-Value Chart Separated by Commas

This version of the chart helps you read the number. Begin all the way to the left and read, “Two hundred thirty-four quintillion, eight hundred forty-five quadrillion, twenty-one trillion, three hundred forty-nine billion, two hundred thirty million, four hundred sixty-seven thousand, three hundred four.”

When you read and write whole numbers, don’t say the word and. In math, the word and means you have a decimal point. That’s why, when you write a check, you save the word and for the number of cents, which is usually expressed as a decimal or sometimes as a fraction. (I discuss decimals in Chapter 11.)

Close Enough for Rock ’n’ Roll: Rounding and Estimating

As numbers get longer, calculations become tedious, and you’re more likely to make a mistake or just give up. When you’re working with long numbers, simplifying your work by rounding numbers and estimating values is sometimes helpful.

When you round a number, you change some of its digits to placeholding zeros. And when you estimate a value, you work with rounded numbers to find an approximate answer to a problem. In this section, you build both skills.

Rounding numbers

Rounding numbers makes long numbers easier to work with. In this section, I show you how to round numbers to the nearest ten, hundred, thousand, and beyond.

Rounding numbers to the nearest ten

The simplest kind of rounding you can do is with two-digit numbers. When you round a two-digit number to the nearest ten, you simply bring it up or down to the nearest number that ends in 0. For example,

Even though numbers ending in 5 are in the middle, always round them up to the next-highest number that ends in 0:

Numbers in the upper 90s get rounded up to 100:

When you know how to round a two-digit number, you can round just about any number. For example, to round most longer numbers to the nearest ten, just focus on the ones and tens digits:

Occasionally, a small change to the ones and tens digits affects the other digits. (This situation is a lot like when the odometer in your car rolls a bunch of 9s over to 0s.) For example:

Rounding numbers to the nearest hundred and beyond

To round numbers to the nearest hundred, thousand, or beyond, focus only on two digits: the digit in the place you’re rounding to and the digit to its immediate right. Change all other digits to the right of these two digits to 0s. For example, suppose you want to round 642 to the nearest hundred. Focus on the hundreds digit (6) and the digit to its immediate right (4):

642

I’ve underlined these two digits. Now just round these two digits as if you were rounding to the nearest ten, and change the digit to the right of them to a 0:

Here are a few more examples of rounding numbers to the nearest hundred:

When rounding numbers to the nearest thousand, underline the thousands digit and the digit to its immediate right. Round the number by focusing only on the two underlined digits and, when you’re done, change all digits to the right of these to 0s:

Even when rounding to the nearest million, the same rules apply:

Estimating value to make problems easier

When you know how to round numbers, you can use this skill in estimating values. Estimating saves you time by allowing you to avoid complicated computations and still get an approximate answer to a problem.

When you get an approximate answer, you don’t use an equals sign; instead, you use this wavy symbol, which means is approximately equal to: ≈.

Suppose you want to add these numbers: 722 + 506 + 383 + 1,279 + 91 + 811. This computation is tedious, and you may make a mistake. But you can make the addition easier by first rounding all the numbers to the nearest hundred and then adding:

The approximate answer is 3,800. This answer isn’t far off from the exact answer, which is 3,792.