# Decimals

### Place Value and Comparing Decimals

#### Place Value for Decimals

Just as "ones," "tens," and "hundreds" are used to describe place value for whole numbers, there are terms that describe place value for decimals: tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, etc. These terms are used from left to right, starting with the first numeral after the decimal point. For example, the number 0.1234 has a "1" in the tenths place, a "2" in the hundredths place, a "3" in the thousandths place, and a "4" in the ten thousandths place. The number 78.304 has a "7" in the tens place, an "8" in the ones place, a "3" in the tenths place, a "0" in the hundredths place, and a "4" in the thousandths place.

Because our system is base ten, a value of 10 in one place is equal to a value of 1 in the place to the left: 10 thousandths is equivalent to 1 hundredth, 10 hundredths is equivalent to 1 tenth, 10 tenths is equivalent to 1 one, and so on.

Example 1. What are the place values for 67.0534?

67.0534 has a 6 in the tens place, a 7 in the ones place, a in the tenths place, a 5 in the hundredths place, a 3 in the thousandths place, and a 4 in the ten thousandths place.

A decimal does not change when zeros are added at the end. For example, 3.45 is equal to 3.450 is equal to 3.4500, and so on.

#### Comparing Decimals

It is sometimes tricky to determine which decimal is the larger of two decimals. Upon first glance, it appears that 0.304 is larger than 0.62, because 304 is larger than 62. However, this is not the case. In order to determine which decimal is larger, we must compare the two place-by-place, starting from the tenths place, until the decimals differ. 0.304 has a "3" in the tenths place, and 0.62 has a "6" in the tenths place. Since 6 is greater than 3, 0.62 is greater than 0.304--it is not necessary to look to further decimal places.

Example 2. Compare 0.4531 and 0.4507. Which is greater?

0.4531 and 0.4507 both have a "4" in the tenths place.
0.4531 and 0.4507 both have a "5" in the hundredths place.
0.4531 has a "3" in the thousandths place and 0.4507 has a "0" in the thousandths place.
3 is greater than 0, so 0.4531 is greater than 0.4507.

Example 3. Compare 76.7 and 400.05. Which is greater?

Since these numbers have digits to the left of the decimal place (they are greater than 1), we can just compare these digits. 400 is greater than 76, so 400.05 is greater than 76.7. In other words, 400.05 has a "4" in the hundreds place, and 76.7 has a "0" in the hundreds place (since it has no numeral in the hundreds place), making 400.05 greater than 76.7.

Example 4. Compare 45.91 and 45.618. Which is greater?

45.91 and 45.618 have the same numerals to the left of their decimal places (a "4" in the tens place and a "5" in the ones place). Therefore, we must look at their decimals, starting with the tenths place. Since 9 is greater than 6, 45.91 is greater than 45.618.

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