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Chapter 5. Decimals. Introduction to Decimals. 5.1. Decimal Notation. 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 . The decimal 16.734 has three parts. 16.743.

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decimal notation
Decimal Notation

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. The decimal 16.734 has three parts.

16.743

Whole number part

Decimal part

Decimal point

slide4

Place Value

1

1

1

10

1000

100,000

1

1

100

10,000

The position of each digit in a number determines its place value.

hundred-thousandths

hundreds

tens

tenths

hundredths

thousandths

ten-thousandths

ones

1 6 7 3 4

100

10

1

Place Value

decimal point

slide5

Place Value

1

10

Notice that the value of each place is

of the value of the place to its left.

16 734
16.734

Place Value

3

100

The digit 3 is in the hundredths place, so

its value is 3 hundredths or .

writing a decimal in words
Writing a Decimal in Words

Step 1: Write the whole number part in words.

Step 2: Write “and” for the decimal point.

Step 3: Write the decimal part in words as though it were a whole number, followed by the place value of the last digit.

writing a decimal in words1
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

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

slide10

Helpful Hint

thousandths place

When writing a decimal from words to decimal notation, make sure the last digit is in the correct place by inserting 0s after the decimal point if necessary.

For example,

three and fifty-four thousandths is 3.054

slide11

Writing Decimals as Fractions

Once you master writing and reading decimals correctly, then you write a decimal as a fraction using the fractions associated with the words you use when you read it.

0.9

is read “nine tenths” and written as a fraction as

slide12

Writing Decimals as Fractions

21

100

11

1000

0.21 is read as

twenty-one hundredths

and written as a fraction as

0.011 is read as

eleven thousandths

and written as a fraction as

comparing two positive decimals
Comparing Two Positive Decimals

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
Comparing Decimals

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 decimals1
Comparing Two Positive Decimals

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 decimals2
Comparing Two Positive Decimals

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

slide17

Helpful Hint

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
Rounding Decimals

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 decimals1
Rounding Decimals

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
Rounding Decimals to a Place Value

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

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