## Add and Subtract Decimals

Decimals
The fact that our decimal notation for whole numbers can be extended to represent many rational numbers has many important consequences. Key to such an extension is the use of a decimal point to the right of the digit in the ones place. The place-value rule that each digit has a place value equal to 10 times that of the digit to the right can be extended to digits to the right of the decimal point: In decimals, each digit has a place value equal to one-tenth that of the digit to the left. This can be represented in the following place-value chart on which the decimal 6.0542 is written.

Ones . Tenths Hundredths Thousandths Ten Thousandths
1

100 = 1
× 1
or
10-1 or
0.1
×
or
10-2 or
0.01
10 × 100
or
10-3 or
0.001
10 × 10
or
10-4 or
0.0001
6 . 0 5 4 2

The decimal 6.0542 is read six and five hundred forty-two ten thousandths. Note that the decimal point is read as and. For numbers less than zero, a zero is usually placed to the left of the decimal point, such as in 0.572. However, the zero is not read, thus 0.572 is read as five hundred seventy-two thousandths. Every decimal can be written as a fraction (3.75 = ).

To compare decimals, students begin by comparing digits in each place, starting at the left, until different digits appear. The decimal with the greater digit in that place is the greater number. For example, when comparing 0.238 and 0.2319, students note that the digits in the tenths and hundredths places are the same, but the digits in the thousandths place are different. The fact that 8 > 1 implies that 0.238 > 0.2319. Students should also note that 8 > 1 can be written as 1 < 8, which implies that 0.2319 < 0.238.

Place-value concepts are also applied when rounding decimals. When rounding a decimal to the nearest tenth, students look at the digit in the place to its right, the hundredths place. If that digit is 5 or more, the number is rounded to the next higher tenth; otherwise the digit in the tenths place is not changed.

When rounding to a decimal place, all digits to the right of the rounded place are dropped. For example, 6.0835 rounded to the nearest tenth is 6.1; to the nearest hundredth, 6.08; and to the nearest thousandth, 6.084.

Addition and Subtraction of Decimals
The algorithms that were used for the addition and subtraction of whole numbers can also be used for the addition and subtraction of decimals. Digits are aligned according to place value, which means the decimal points should be aligned. Then computation is completed from right to left.

Example: Add 33.78 + 29.83

11 hundredths =
1 tenth + 1 hundredth

16 tenths =
1 one + 6 tenths

13 ones =
1 ten + 3 ones

When subtracting, appending one or more zeros after the decimal point may make computation easier.

Example: Subtract 17.32 from 31.5

Append a zero.
Regroup.
Subtract the hundredths.

Subtract the tenths.

Regroup.
Subtract the ones.

Subtract the tens.

Teaching Model 11.3: Subtract Decimals