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Chapter 4. Decimals. Introduction to Decimals. 4.1. Decimal Notation and Writing Decimals in Words.

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### Decimals

Decimal Notation and Writing Decimals in Words

Like fractional notation, decimal notation is used to denote a part of a whole. Numbers written in decimal notation are called decimal numbers, or simply decimals. Place names and place values for the decimal parts are also shown.

Writing a Decimal in Words

decimal part

whole number part

Write the decimal 143.056 in words.

143.056

one hundred forty-three

and

fifty-six thousandths

- Write the decimals in words.
- 3.6 three and six tenths
- 124. 52 one hundred twenty-four and fifty- two hundredths

A decimal written in words can be written in standard form by reversing the procedure.

Writing Decimals in Standard Form

whole-number part

decimal

decimal part

Write one hundred six and five hundredths in standard form.

one hundred six and five hundredths

5 must be in the

hundredths place

106

.

05

Examples by reversing the procedure.

- Write each decimal in standard form.
- Twelve and six hundredths
- 12.06
- Fifty-four and seventy-two thousandths
- 54.072

Writing Decimals as Fractions by reversing the procedure.

Examples by reversing the procedure.

- Write the decimals as a fraction or mixed number.
- 0.9
- 21.094

1 decimal place

1 zero

3 decimal places

3 zeros

1 decimal place by reversing the procedure.

1 zero

3 decimal places

3 zeros

Examples

- Write each fraction as a decimal.
- = 0.6
- = 36.005

Comparing Two Positive Decimals by reversing the procedure.

37

29

=

0

.

37

=

0

.

029

100

1000

2 zeros

3 decimal places

3 zeros

2 decimal places

Notice that the number of decimal places in a decimal number is the same as the number of zeros in the denominator of the equivalent fraction. We can use this fact to write decimals as fractions.

Comparing Decimals by reversing the procedure.

3

7

10

10

One way to compare decimals is to compare their graphs on a number line. Recall that for any two numbers on a number line, the number to the left is smaller and the number to the right is larger. To compare 0.3 and 0.7 look at their graphs.

0

0.3

0.7

1

0.3 < 0.7 or 0.7 > 0.3

Comparing Two Positive Decimals by reversing the procedure.

Comparing decimals by comparing their graphs on a number line can be time consuming, so we compare the size of decimals by comparing digits in corresponding places.

Comparing Two Positive Decimals by reversing the procedure.

Compare digits in the same places from left to right. When two digits are not equal, the number with the larger digit is the larger decimal. If necessary, insert 0s after the last digit to the right of the decimal point to continue comparing.

Compare hundredths place digits.

35.638

35.657

5

<

3

35.638

<

35.657

Helpful Hint by reversing the procedure.

For any decimal, writing 0s after the last digit to the right of the decimal point does not change the value of the number.

8.5 = 8.50 = 8.500, and so on

When a whole number is written as a decimal, the decimal point is placed to the right of the ones digit.

15 = 15.0 = 15.00, and so on

Rounding Decimals by reversing the procedure.

We round the decimal part of a decimal number in nearly the same way as we round whole numbers. The only difference is that we drop digits to the right of the rounding place, instead of replacing these digits by 0s. For example,

63.782 rounded to the nearest hundredth is

63.78

Rounding Decimals by reversing the procedure.

Step 1: Locate the digit to the right of the given place value.

Step 2: If this digit is 5 or greater, add 1 to the digit in the given place value and drop all digits to the right. If this digit is less than 5, drop all digits to the right of the given place.

Rounding Decimals to a Place Value by reversing the procedure.

tenths place

digit to the right

Round 326.4386 to the nearest tenth.

Locate the digit to the right of the tenths place.

326.4386

Since the digit to the right is less than 5, drop it and all digits to its right.

326.4386 rounded to the nearest tenths is 326.4

Example by reversing the procedure.

Round 0.5942 to the nearest thousandth.

Step 1: Locate the digit to the right of the given place value.

0.5942

2 is to the right of the thousandth place value.

Step 2: Since the digit to the right is less than 5, we delete it and all digits to its right.

Thus 0.5942 rounded to the nearest thousandth is 0.594

Example by reversing the procedure.

Round $0.067 to the nearest cent.

Step 1: Locate the digit to the right of the given place value.

$0.067

7 is to the right of the hundredths place value

Step 2: Since the digit to the right is more than 5, we round up.

Thus $0.067 rounds to $0.07.

Example by reversing the procedure.

An aquarium costs $245.69. Round the price to the nearest dollar.

The nearest dollar means the ones place. Locate the digit to the right of the ones place.

$245.69

6 is to the right of the ones place value

Thus $245.69 rounds to $246.00.

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