Numbers and Operations Workbook

Lesson One

Place Value for Whole Numbers and Decimals

In this lesson, we will explore the meaning of the individual digits of a number, whether it is a whole number or a decimal. This skill is very valuable.

The ability to put numbers into words and vice versa is important to our daily lives. Think about what would happen if a bank accidentally dropped decimal points. A person’s bank account of $12,400.05 might be recorded as $1,240,005—a difference of one million dollars! As another example, suppose you pay $2.90 for a gallon of gasoline. Imagine your shock if the price read $290 instead. As a third example, suppose a surgeon was to perform an operation in which he must make a cut of 0.03 inch in a person’s body. If the surgeon mistakenly made a 0.3 inch cut, the result could be serious.

Your Goal: When you have completed this lesson, you should be able to identify the meaning and interpretation of individual digits of any number.

For whole numbers, reading from right to left, the names of the place values are as follows:

This list is endless, but you will rarely see numbers with higher values.

1

Example:How should the number 507 be written in words?

Solution: The 5 represents hundreds, the 0 represents tens, and the 7 represents ones. This number is written as five hundred seven. (It could be written as five hundred and seven. This is correct, but it is not necessary to use the word and.)

2

Example:How should the number 2148 be written in words?

Solution: The 2 represents thousands, the 1 represents hundreds, the 4 represents tens, and the 8 represents ones. This number is written as two thousand one hundred forty-eight.

3

Example:How should the number 325,089 be written in words?

Solution: Instead of identifying each digit separately, the key is to realize that the highest place value, which is the digit 3, is in the hundred-thousands place. Thus, this number is written as three hundred twenty-five thousand eighty-nine.

Any number with a value of at least 10,000 should be written with commas that separate each group of three digits, counting from the right. Therefore, 35078162 would be written as 35,078,162.

4

Example:How should the number 41,590 be written in words?

Solution: Since this number has five digits, the digit 4 of this number represents ten-thousands. Combine the 4 and 1 and write it as forty-one thousand. Thus, we write forty-one thousand five hundred ninety.

A decimal number has at least one non-zero digit to the right of the decimal point. The numbers 0.6, 5.407, and 34.1179 are examples.

For decimal numbers, reading from left to right from the placement of the decimal point, the place values are as follows:

This list is also endless, but you will hardly ever see units of numbers smaller than millionths. If there is no digit to the left of the decimal point, a 0 is placed in that spot. The use of 0 helps the reader to easily recognize the number as a decimal.

5

Example:How should the number 0.49 be written in words?

Solution: The 4 represents tenths, and the 9 represents hundredths. This number is written as forty-nine hundredths. Notice that ths at the end of hundredths signals that positions to the right of the decimal point are occupied by digits (in the case of hundredths, two positions are occupied).

6

Example:How should the number 0.0651 be written in words?

Solution: The first 0 to the right of the decimal point represents tenths, the 6 represents hundredths, the 5 represents thousandths, and the 1 represents ten-thousandths. The number is written as six hundred fifty-one ten-thousandths.

7

Example:How should the number 0.007 be written in words?

Solution: Since there are three placeholders to the right of the decimal point, the answer must contain the word thousandths. The number is written as seven thousandths.

8

Example:How should the number 0.29734 be written in words?

Solution: Instead of identifying each digit separately, note that the rightmost digit, which is 4, is in the hundred-thousandths place. The number is written as twenty-nine thousand seven hundred thirty-four hundred-thousandths.

Rarely will a decimal number have a zero as its rightmost digit. An example would involve rounding off a number, which will be discussed in a future lesson. If such a decimal appears, the position occupied by zero must be stated when writing the words for the number.

Example: 0.180 would be written as one hundred eighty thousandths. Of course, the actual value of 0.180 is equivalent to 0.18, which is written as eighteen hundredths.

Often, numbers are written in words and must be converted back to the corresponding digits. An example in which both words and digits need to be written would be in writing a check.

9

Example:How should three thousand sixteen be written in digits?

Solution: We need four place values to the left of the decimal point, beginning with 3. Since the word sixteen will occupy two place values, we will need a zero in the hundreds place. The answer is 3016.

10

Example:How should eight hundred eighty be written in digits?

Solution: Since the highest place value is hundreds, the number will have three digits. The answer is 880.

11

Example:How should forty-five thousand eight hundred thirty-two be written in digits?

Solution: The phrase forty-five thousand indicates that this number requires five place values, beginning with 4. Forty-five occupies two place values, and eight hundred thirty-two occupies three place values. The answer is 45,832.

12

Example:How should the number nine million three be written in digits?

Solution: Nine million would start in the seventh place value, but three only occupies one place value. Do you see why we will need five fillers? The answer is 9,000,003.

Whole numbers may have many zeros, depending on where the leftmost digit is located. You will not need to put any zeros to the left of the first nonzero digit. For example, if the required answer is 783, there is no need to write 0783. Examples of circumstances in which leading zeros are necessary are identification labels and zip codes.

13

Example:How should the number fifty-one hundredths be written in digits?

Solution: Fifty-one occupies two place values, and hundredths occupies two place values. Therefore, 0.51 is the answer.

14

Example:How should the number one hundred forty-seven ten-thousandths be written in digits?

Solution: One hundred forty-seven will occupy three place values; however, ten-thousandths requires four place values. This means that a zero is needed immediately to the right of the decimal point. Then 0.0147 is the answer.

15

Example:How should the number five thousand three hundred twenty ten-thousandths be written in digits?

Solution: As we noted following Example 6, there are instances in which a decimal may have a zero as its rightmost digit. The phrase five thousand three hundred twenty requires four place values, and ten-thousandths also requires four place values. Thus, 0.5320 is the answer.

16

Example:The width of a very thin pencil point is sixty-five hundred-thousandths of a centimeter. How is this number written in digits?

Solution: A total of five digits to the right of the decimal point are needed. Since sixty-five occupies two place values, we will need three zeros. Then 0.00065 is the answer.

Lesson Two

Rounding Off Whole Numbers

In this lesson, we will explore how to round off whole numbers to a specified place value. You have read and heard about many different settings in which this procedure is done, such as (a) the number of fans at a soccer game, rounded off to the nearest thousand; (b) the weight of a bag of grapes, rounded off to the nearest pound; (c) the number of people living in New Jersey, rounded off to the nearest ten thousand; and (d) the time for a horse to run a mile, rounded off to the nearest second.

Your Goal: When you have completed this lesson, you should be able to approximate any number to a specified place value. In rounding off, you give up a small amount of accuracy to help make the numbers easier to work with.

A whole number is zero or a counting number, such as 4, 17, 587, and so forth. The smallest counting number is 1. There are times when we need information on a number, but not an exact answer. For example, attendance at a baseball game is usually a number with five digits, such as 43,268. But, you may only need to know that the attendance was about 43,000. In this lesson, our objective will be to understand how to round off any counting number to a specified place.

Counting numbers are also called natural numbers.

1

Example:What is 761 rounded off to the nearest