Decimal numbers, also simply known as decimals, represent a part or a portion of a whole, similar to fractions as outlined in the previous sections.
Decimal numbers are used in situations that require more precision than whole numbers can provide. We use decimal numbers frequently in our daily lives; a good example of this is money. For example, a nickel is worth 5¢, which is equal to $0.05, and bus fare for a city may be $3.25; 0.05 and 3.25 are both examples of decimal numbers.
A decimal number contains a whole number portion and a decimal portion. The decimal point (.) is used to separate these two portions: the whole number portion is comprised of the digits to the left of the decimal point, and the decimal portion is comprised of the digits to the right of the decimal point. The decimal portion represents a value less than 1.
For example,
The decimal portion of a decimal number can be represented as a fraction with a denominator that is a power of 10 (i.e., 10, 100, 1,000, etc.). These fractions are referred to as decimal fractions.
For example,
The decimal number 0.3 is [latex] \displaystyle{\frac{3}{10}} [/latex] as a decimal fraction.
The decimal number 0.07 is [latex] \displaystyle{\frac{7}{100}} [/latex] as a decimal fraction.
The decimal portion of the decimal number 345.678 is [latex] \displaystyle{\frac{678}{1,000}} [/latex] as a decimal fraction.
When decimal numbers are expressed as a decimal fraction with a denominator that is a power of 10, we do not reduce to their lowest terms.
For example, [latex] \displaystyle{\frac{678}{1,000}} [/latex] if reduced to [latex] \displaystyle{\frac{339}{500}} [/latex]is no longer expressed with a denominator that is a power of 10, and therefore is not a decimal fraction.
Similarly,
[latex] 1.2 = 1\frac{2}{10} [/latex]
[latex] 23.45 = 23\frac{45}{100} [/latex]
[latex] 75.378 = 75\frac{378}{1,000} [/latex]
Every whole number can be written as a decimal number by placing a decimal point to the right of the units digits.
For example, the whole number 5 written as a decimal number is 5. or 5.0 or 5.00, etc.
The number of decimal places in a decimal number is the number of digits written to the right of the decimal point.
For example,
5. No decimal places
5.0 One decimal place
5.00 Two decimal places
1.250 Three decimal places
2.0050 Four decimal places
There are three different types of decimal numbers.
Non-repeating, terminating decimals numbers:
For example, 0.2, 0.3767, 0.86452
Repeating, non-terminating decimal numbers:
For example, 0.222222.... (0.2), 0.255555.... (0.25), 0.867867.... (0.867)
Non-repeating, non-terminating decimal numbers:
For example, 0.453740...., π (3.141592...), e (2.718281...)
The position of each digit in a decimal number determines the place value of the digit. Exhibit 2.3 illustrates the place value of the five-digit decimal number: 0.35796.
The place value of each digit as you move right from the decimal point is found by decreasing powers of 10. The first place value to the right of the decimal point is the tenths place, the second place value is the hundredths place, and so on, as shown in Table 2.3.
[latex] 10^{-1} = \displaystyle{\frac{1}{10}} [/latex] | [latex] 10^{-2} = \displaystyle{\frac{1}{100}} [/latex] | [latex] 10^{-3} = \displaystyle{\frac{1}{1,000}} [/latex] | [latex] 10^{-4} = \displaystyle{\frac{1}{10,000}} [/latex] | [latex] 10^{-5} = \displaystyle{\frac{1}{100,000}} [/latex] |
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 |
Tenths | Hundredths | Thousandths | Ten-thousandths | Hundred-thousandths |
The five-digit number in Exhibit 2.3 is written as 0.35796 in its standard form.
0. | 3 | 5 | 7 | 9 | 6 |
Tenths | Hundredths | Thousandths | Ten-thousandths | Hundred-thousandths |
The decimal number 0.35796 can also be written in expanded form as follows:
0.3 + 0.05 + 0.007 + 0.0009 + 0.00006
Or,
3 tenths + 5 hundredths + 7 thousandths + 9 ten-thousandths + 6 hundred-thousandths
Or,
[latex] \displaystyle{\frac{3}{10} + \frac{5}{100} + \frac{7}{1,000} + \frac{9}{10,000} + \frac{6}{100,000}} [/latex] (0.35796 as a decimal fraction is [latex] \displaystyle{\frac{35,796}{100,000}} [/latex])
Follow these steps to read and write decimal numbers in word form:
Read or write the number to the left of the decimal point as a whole number.
Read or write the decimal point as “and”.
Read or write the number to the right of the decimal point also as a whole number, but followed by the name of the place value occupied by the right-most digit.
For example, 745.023 is written in word form as:
There are other ways of reading and writing decimal numbers as noted below.
Use the word “point” to indicate the decimal point and thereafter, read or write each digit individually.
For example, 745.023 can also be read or written as: Seven hundred forty-five point zero, two, three.
Ignore the decimal point of the decimal number and read or write the number as a whole number followed by the name of the place value occupied by the right-most digit of the decimal portion.
For example, 745.023 can also be read or written as: Seven hundred forty-five thousand, twenty-three thousandths (i.e., [latex] \displaystyle{\frac{745,023}{1,000}} [/latex]).
Note: The above two representations are not used in the examples and exercise questions within this chapter.
A hyphen ( - ) is used to express the two-digit numbers, 21 to 29, 31 to 39, 41 to 49, … 91 to 99, in each group in their word form.
A hyphen ( - ) is also used while expressing the place value portion of a decimal number, such as ten-thousandths, hundred-thousandths, ten-millionths, hundred-millionths, and so on.
The following examples illustrate the use of hyphens to express numbers in their word form:
0.893 Eight hundred ninety-three thousandths
0.0506 Five hundred six ten-thousandths
0.00145 One hundred forty-five hundred-thousandths
Write the following decimal numbers in standard form:
Two hundred and thirty-five hundredths
Three and seven tenths
Eighty-four thousandths
200
and
[latex] \displaystyle{\frac{35}{100} = 0.35} [/latex]
3
and
[latex] \displaystyle{\frac{7}{10} = 0.7} [/latex]
0
and
[latex] \displaystyle{\frac{84}{1,000} = 0.084} [/latex]
Write the following decimal numbers in word form:
23.125
7.43
20.3
0.2345
23.125
The last digit, 5, is in the thousandths place.
[latex] \displaystyle{= 23\frac{125}{1,000}} [/latex]
Twenty-three and one hundred twenty-five thousandths
7.43
The last digit, 3, is in the hundredths place.
[latex] \displaystyle{= 7\frac{43}{100}} [/latex]
Seven and forty-three hundredths
20.3
The last digit, 3, is in the tenths place.
[latex] \displaystyle{= 20\frac{3}{10}} [/latex]
Twenty and three tenths
0.2345
The last digit, 5, is in the ten-thousandths place.
[latex] \displaystyle{= \frac{2,345}{10,000}} [/latex]
Two thousand, three hundred forty-five ten-thousandths
Rounding decimal numbers refers to changing the value of the decimal number to the nearest whole number, tenth, hundredth, thousandth, etc. It is also referred to as rounding to a specific number of decimal places, indicating the number of decimal places that will be left when the rounding is complete.
Rounding to the nearest whole number is the same as rounding without any decimals.
Rounding to the nearest tenth is the same as rounding to one decimal place.
Rounding to the nearest hundredth is the same as rounding to two decimal places.
Rounding to the nearest cent refers to rounding the amount to the nearest hundredth, which is the same as rounding to two decimal places.
Follow these steps to round decimal numbers:
Identify the digit to be rounded (this is the place value for which the rounding is required).
If the digit to the immediate right of the identified rounding digit is less than 5 (0, 1, 2, 3, 4), do not change the value of the rounding digit.
If the digit to the immediate right of the identified rounding digit is 5 or greater than 5 (5, 6, 7, 8, 9), increase the value of the rounding digit by one (i.e., round up by one number).
Drop all digits to the right of the rounding digit.
Round the following decimal numbers to the indicated place value:
268.143 to the nearest tenth
489.679 to the nearest hundredth
$39.9985 to the nearest cent
Rounding 268.143 to the nearest tenth:
1 is the rounding digit in the tenths place: 268.143.
The digit to the immediate right of the rounding digit is less than 5; therefore, do not change the value of the rounding digit. Drop all of the digits to the right of the rounding digit. This will result in 268.1.
Therefore, 268.143 rounded to the nearest tenth is 268.1.
Rounding 489.679 to the nearest hundredth:
7 is the rounding digit in the hundredths place: 489.679.
The digit to the immediate right of the rounding digit is greater than 5; therefore, increase the value of the rounding digit by one, from 7 to 8, and drop all of the digits to the right of the rounding digit. This will result in 489.68.
Therefore, 489.679 rounded to the nearest hundredth is 489.68.
Rounding $39.9985 to the nearest cent:
9 is the rounding digit in the hundredths place: $39.9985.
The digit to the immediate right of the rounding digit is greater than 5; therefore, increase the value of the rounding digit by one, from 9 to 10, by replacing the rounding digit 9 with 0 and carrying the one to the tenths place, then to the ones, and then to the tens, to increase the digit 3 to 4. Finally, drop all digits that are to the right of the rounding digit. This will result in $40.00.
Therefore, $39.9985 rounded to the nearest cent is $40.00.
For Problems 1 to 8, express the decimal fractions as decimal numbers.
a. [latex] \displaystyle{\frac{9}{10,000}} [/latex] b. [latex] \displaystyle{\frac{41}{1,000}} [/latex]
a. [latex] \displaystyle{\frac{6}{10}} [/latex] b. [latex] \displaystyle{\frac{7}{1,000}} [/latex]
a. [latex] \displaystyle{\frac{75}{100}} [/latex] b. [latex] \displaystyle{\frac{3}{10}} [/latex]
a. [latex] \displaystyle{\frac{12}{100}} [/latex] b. [latex] \displaystyle{\frac{29}{1,000}} [/latex]
a. [latex] 9\frac{3}{10} [/latex] b. [latex] 6\frac{207}{1,000} [/latex]
a. [latex] 7\frac{5}{10} [/latex] b. [latex] 9\frac{503}{1,000} [/latex]
a. [latex] \displaystyle{\frac{475}{10}} [/latex] b. [latex] \displaystyle{\frac{2,972}{1,000}} [/latex]
a. [latex] \displaystyle{\frac{367}{100}} [/latex] b. [latex] \displaystyle{\frac{2,567}{1,000}} [/latex]
For Problems 9 to 24, write the numbers in (i) standard form and (ii) expanded form.
Thirty-five and seven tenths
Eighty-seven and two tenths
Nine and seven hundredths
Three and four hundredths
Two hundred eight thousandths
Four hundred one ten-thousandths
Fifty-two and three hundred five thousandths
Eighty-nine and six hundred twenty-five ten-thousandths
Seven thousand, two hundred sixty and fifteen thousandths
One thousand, seven hundred eighty-seven and twentyfive thousandths
Nine hundred eighty-seven and twenty hundredths
Four hundred twelve and sixty-five hundredths
Six million, two hundred seventeen thousand and five hundredths
One million, six hundred thousand and two hundredths
Twenty-nine hundredths
Twenty-three and five tenths
For Problems 25 to 32, express the decimal numbers in their word form.
a. 7.998 b. 12.77
a. 42.55 b. 734.125
a. 0.987 b. 311.2
a. 0.25 b. 9.5
a. 11.09 b. 9.006
a. 7.07 b. 15.002
a. 0.031 b. 0.073
a. 0.062 b. 0.054
Arrange the following decimal numbers from greatest to least.
1.014, 1.011, 1.104, 1.041
Arrange the following decimal numbers from least to greatest.
0.034, 0.403, 0.043, 0.304
For Problems 35 to 42, round the numbers to one decimal place (nearest tenth).
7.8725
415.1654
25.5742
264.1545
112.1255
24.1575
0.9753
10.3756
For Problems 43 to 50, round the numbers to two decimal places (nearest hundredth, or nearest cent).
0.0645
14.3585
19.6916
181.1267
$10.954
$16.775
$24.995
$9.987