Chapter 1

We’ve Got Your Numbers

IN THIS CHAPTER

Understanding how place value turns digits into numbers

Rounding numbers to the nearest ten, hundred, or thousand

Calculating with the Big Four operations: Adding, subtracting, multiplying, and dividing

Getting comfortable with long division

In this chapter, I give you a review of basic math, and I do mean basic. I bet you know a lot of this stuff already. So, consider this a trip down memory lane, a mini-vacation from whatever math you may be working on right now. With a really strong foundation in these areas, you’ll find the chapters that follow a lot easier.

First, I discuss how the number system you’re familiar with — called the Hindu-Arabic number system (or decimal numbers) — uses digits and place value to express numbers. Next, I show you how to round numbers to the nearest ten, hundred, or thousand.

After that, I discuss the Big Four operations: adding, subtracting, multiplying, and dividing. You see how to use the number line to make sense of all four operations. Then I give you practice doing calculations with larger numbers. To finish up, I make sure you know how to do long division both with and without a remainder.

Some math books use a dot (·) to indicate multiplication. In this book, I use the more familiar times sign .

Getting in Place with Numbers and Digits

The number system used most commonly throughout the world is the Hindu-Arabic number system. This system contains ten digits (also called numerals), which are symbols like the letters A through Z. I’m sure you’re quite familiar with them:

1

2

3

4

5

6

7

8

9

0

Like letters of the alphabet, individual digits aren’t very useful. When used in combination, however, these ten symbols can build numbers as large as you like using place value. Place value assigns each digit a greater or lesser value depending upon where it appears in a number. Each place in a number is ten times greater than the place to its immediate right.

Although the digit 0 adds no value to a number, it can act as a placeholder. When a 0 appears to the right of at least one nonzero digit, it’s a placeholder. Placeholders are important for giving digits their proper place value. In contrast, when a 0 isn’t to the right of any nonzero digit, it’s a leading zero. Leading zeros are unnecessary and can be removed from a number.

Q. In the number 284, identify the ones digit, the tens digit, and the hundreds digit.

A. The ones digit is 4, the tens digit is 8, and the hundreds digit is 2.

Q. Place the number 5,672 in a table that shows the value of each digit. Then use this table and an addition problem to show how this number breaks down digit by digit.

A.

Millions

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Ones

5

6

7

2

The numeral 5 is in the thousands place, 6 is in the hundreds place, 7 is in the tens place, and 2 is in the ones place, so here’s how the number breaks down:

Q. Place the number 040,120 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit. Which 0s are placeholders, and which are leading zeros?

A.

Millions

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Ones

0

4

0

1

2

0

The first 0 is in the hundred-thousands place, 4 is in the ten-thousands place, the next 0 is in the thousands place, 1 is in the hundreds place, 2 is in the tens place, and the last 0 is in the ones place, so

The first 0 is a leading zero, and the remaining 0s are placeholders.

1 In the number 7,359, identify the following digits:

1. The ones digit
2. The tens digit
3. The hundreds digit
4. The thousands digit

2 Place the number 2,136 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit.

Millions

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Ones

3 Place the number 03,809 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit. Which 0 is a placeholder and which is a leading zero?

Millions

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Ones

4 Place the number 0,450,900 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit. Which 0s are placeholders and which are leading zeros?

Millions

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Ones

Rollover: Rounding Numbers Up and Down

Rounding numbers makes long numbers easier to work with. To round a two-digit number to the nearest ten, simply increase it or decrease it to the nearest number that ends in 0:

• When a number ends in 1, 2, 3, or 4, bring it down; in other words, keep the tens digit the same and turn the ones digit into a 0.
• When a number ends in 5, 6, 7, 8, or 9, bring it up; add 1 to the tens digit and turn the ones digit into a 0.

To round a number with more than two digits to the nearest ten, use the same method, focusing only on the ones and tens digits.

After you understand how to round a number to the nearest ten, rounding a number to the nearest hundred, thousand, or beyond is easy. Focus only on two digits: The digit in the place you’re rounding to and the digit to its immediate right, which tells you whether to round up or down. All the digits to the right of the number you’re rounding to change to 0s.

Occasionally when you’re rounding a number up, a small change to the ones and tens digits affects the other digits. This is a lot like when the odometer in your car rolls a bunch of 9s over to 0s, such as when you go from 11,999 miles to 12,000 miles.

Q. Round the numbers 31, 58, and 95 to the nearest ten.

A. 30, 60, and 100.

The number 31 ends in 1, so round it down:

The number 58 ends in 8, so round it up:

The number 95 ends in 5, so round it up:

Q. Round the numbers 742, 3,820, and 61,225 to the nearest ten.

A. 740, 3,820, and 61,230.

The number 742 ends in 2, so round it down:

The number 3,820 already ends in 0, so no rounding is needed:

The number 61,225 ends in 5, so round it up:

5 Round these two-digit numbers to the nearest ten:

1. 29
2. 43
3. 75
4. 97

6 Round these numbers to the nearest ten:

1. 164
2. 765
3. 1,989
4. 9,999,995

7 Round these numbers to the nearest hundred:

1. 439
2. 562
3. 2,950
4. 109,974

8 Round these numbers to the nearest thousand:

1. 5,280
2. 77,777
3. 1,234,567
4. 1,899,999

Using the Number Line with the Big Four

The number line is just a line with numbers marked off at regular intervals. You probably saw your first number line when you were learning how to count to ten. In this section, I show you how to use this trusty tool to perform the Big Four operations (adding, subtracting, multiplying, and dividing) on relatively small numbers.

The number line can be a useful tool for adding and subtracting small numbers:

• When you add, move up the number line, to the right.
• When you subtract, move down the number line, to the left.

To multiply on the number line, start at 0 and count by the first number in the problem...