## Decimals

In extending our number system from whole numbers to fractions, a new class of numbers called rational numbers are being introduced. The fact that our decimal notation for whole numbers can be extended to represent 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 which states that each digit has a place value equal to 10 times that of the digit to its 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 its left. This can be represented as follows.

Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths |
---|---|---|---|---|---|---|

10 × 100 or 1,000 |
10 × 10 or 100 |
10 × 1 or 10 |
1 1 |
× 1 or or 0.1 |
× or or 0.01 |

The decimal 365.42 is read three hundred sixty-five and forty-two hundredths. Notice that the decimal point is read as **and.**

If there is no whole-number part to a decimal, a zero is usually placed to the left of the decimal point, such as 0.72. However, the zero is not read, so 0.72 is read as seventy-two hundredths.

**Computation With Decimals**

Be sure students understand that the standard algorithms for adding, subtracting, multiplying, and dividing whole numbers also apply to decimals.

To add decimals, students first align digits according to place value, which means the decimal points should be aligned. Then they complete the computation from right to left.

Example: Add. 3.78 + 0.83

Add the hundredths. 11 hundredths = 1 tenth + 1 hundredth |
Add the tenths. 16 tenths = 1 one + 6 tenths |
Add the ones. |

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

Example: Add. 3.5 − 1.32

Append a zero. Rename. Subtract the hundredths. |
Subtract the tenths. | Subtract the ones. |

**Teaching Model 20.5:** Compare and Order Decimals