DecimalsDecimals

Chapter FiveChapter Five

Introduction to DecimalsIntroduction to Decimals

Section 5.1

Whole- number part

Decimal point

Decimal part

16.743

Like fractional notation, decimal notation Like fractional notation, decimal notation is used to denote a part of a whole. is used to denote a part of a whole. Numbers written in decimal notation are Numbers written in decimal notation are called called decimal numbersdecimal numbers, or simply , or simply decimalsdecimals. The decimal 16.734 has three . The decimal 16.734 has three parts.parts.

3

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

1 6 7 3 4 o

nes

tho

usa

nd

ths

hu

nd

red

s

ten

s

ten

ths

hu

nd

red

ths

ten

-th

ou

san

dth

s

hu

nd

red

-th

ou

san

dth

s

Place Value

decimal point

100 10 11

1001

10,000

110

1100,000

11000

4

Notice that the value of each place is Notice that the value of each place is

of the value of the place to its left.of the value of the place to its left.

110

5Martin-Gay, Prealgebra, 5ed

16.73416.734

The digit 3 is in the hundredths place, soThe digit 3 is in the hundredths place, so

its value is 3 hundredths or .its value is 3 hundredths or .3

100

6Martin-Gay, Prealgebra, 5ed

Writing (or Reading) a Decimal Writing (or Reading) a Decimal in Wordsin Words

Step 1. Write the whole-number part in words.Step 1. Write the whole-number part in words.Step 2. Write “and” for the decimal point.Step 2. Write “and” for the decimal point.Step 3. Write the decimal part in words as Step 3. Write the decimal part in words as

though it were a whole number, though it were a whole number, followed by the place value of the last followed by the place value of the last digit.digit.

7Martin-Gay, Prealgebra, 5ed

Writing a Decimal in WordsWriting a Decimal in Words

Write the decimal 143.056 in words.Write the decimal 143.056 in words.

143.056

one hundred forty-three and fifty-six thousandths

whole-number part decimal part

8Martin-Gay, Prealgebra, 5ed

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

Writing Decimals in Writing Decimals in Standard FormStandard Form

Write one hundred six and five hundredths in Write one hundred six and five hundredths in standard form.standard form.

one hundred six and five hundredths

106 . 05

decimal partwhole-number part decimal

5 must be in thehundredths place

5 must be in thehundredths place

9

Helpful Hint

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

For example, For example,

three and fifty-four thousandths is 3.054 three and fifty-four thousandths is 3.054

thousandths place

10Martin-Gay, Prealgebra, 5ed

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

0.90.9

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

9

10

Writing Decimals as FractionsWriting Decimals as Fractions

11Martin-Gay, Prealgebra, 5ed

twenty-one hundredthstwenty-one hundredths

and written as a fraction asand written as a fraction as

2121

100100

0.011 is read as 0.011 is read as

eleven thousandthseleven thousandths

and written as a fraction asand written as a fraction as

1111

10001000

0.21 is read as0.21 is read as

12Martin-Gay, Prealgebra, 5ed

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

0 3737

100.

2 decimal places

2 zeros

0 02929

1000.

3 decimal places

3 zeros

13

Comparing DecimalsComparing Decimals

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

0 10.3

3

10

7

10

0.70.3 < 0.7 or

0.7 > 0.314

Martin-Gay, Prealgebra, 5ed

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

15Martin-Gay, Prealgebra, 5ed

Comparing Two Positive Comparing Two Positive DecimalsDecimals

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

Compare hundredths place digits Compare hundredths place digits

3 5<35.638 35.657<

35.638 35.657

16Martin-Gay, Prealgebra, 5ed

Helpful Hint

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

8.5 8.5 == 8.50 8.50 == 8.500, and so on 8.500, and so on

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

15 15 == 15.0 15.0 == 15.00, and so on 15.00, and so on 17Martin-Gay, Prealgebra, 5ed

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

Rounding DecimalsRounding Decimals

63.782 rounded to the nearest hundredth is63.782 rounded to the nearest hundredth is

63.7863.7818

Martin-Gay, Prealgebra, 5ed

Rounding Decimals To a Place Value to Rounding Decimals To a Place Value to the Right of the Decimal Pointthe Right of the Decimal Point

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

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

19Martin-Gay, Prealgebra, 5ed

Rounding Decimals to a Place ValueRounding Decimals to a Place Value

Round 326.4386 to the nearest tenth.Round 326.4386 to the nearest tenth.Locate the digit to the right of the tenths place.

326.4386

tenths place

digit to the right

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 326.420Martin-Gay, Prealgebra, 5ed