Decimal Place Value & Expanded Form - Sixth Grade Mathematics

Decimals extend our base-ten positional system to fractional quantities, providing a seamless framework for expressing measurements, money, and scientific data.

The Symmetry of the Base-Ten System

The decimal point is the central mathematical anchor of our number system. It separates the whole number values on the left from the fractional parts on the right. As you move left from the ones place, each place value is multiplied by 10. As you move right from the ones place, each place value is divided by 10 (multiplied by 1/10).

+---------------------------------------------------------------------------------------+
|                              DECIMAL PLACE VALUE CHART                                |
+-----------------------+-----+---------------------------------------------------------+
|     WHOLE NUMBERS     |  .  |                    DECIMAL PARTS                        |
+-----+-----+-----+-----+  .  +-----------+-------------+------------+------------------+
| Th  |  H  |  T  |  O  |  .  | Tenths    | Hundredths  | Thousandths| Ten-Thousandths  |
+-----+-----+-----+-----+  .  +-----------+-------------+------------+------------------+
| 1000| 100 | 10  |  1  |  .  | 1/10 (0.1)| 1/100 (0.01)|1/1000(.001)| 1/10000 (0.0001) |
+-----+-----+-----+-----+  .  +-----------+-------------+------------+------------------+

Notice the Lack of an "Oneths" Place

A frequent point of confusion is expecting an "oneths" column to mirror the tens column. The symmetry of the base-ten system is centered around the ones column, not the decimal point. One step to the left of ones is tens (10); one step to the right of ones is tenths (1/10). Two steps to the left is hundreds (100); two steps to the right is hundredths (1/100). Every column to the right of the decimal point ends with the suffix "-ths".

Annexing Zeros: Equivalent Decimals

Annexing (adding) trailing zeros to the extreme right of a decimal does not change its value. In physical terms, 0.4 means 4 tenths. If you partition each tenth into 10 smaller slices, you have 40 hundredths (0.40). Partitioning each hundredth into 10 smaller slices yields 400 thousandths (0.400). All three numerals represent the identical amount: 0.4 = 0.40 = 0.400.

Forms of Decimal Numbers

A decimal number can be represented in standard form, word form, and expanded form.

Standard Form:    43.085
Word Form:        Forty-three and eighty-five thousandths
Expanded Form (Decimals):  40 + 3 + 0.08 + 0.005
Expanded Form (Fractions): (4 * 10) + (3 * 1) + (8 * 1/100) + (5 * 1/1000)

Rules for Writing Word Form

To express a decimal in word form: First, read the whole number part to the left of the decimal point as a normal whole number. Second, read the decimal point as the word "and". (Never use "and" anywhere else in the number). Third, read the entire sequence of digits to the right of the decimal point as if it were a single whole number. Fourth, state the place value name of the last digit on the far right. For example, 0.0028 is read as "twenty-eight ten-thousandths."

Expanded Form with Powers of Ten and Fractions

Expanded form demonstrates the exact contribution of each non-zero digit. For the numeral 7.306: Fractional Expanded Form: (7 * 1) + (3 * 1/10) + (6 * 1/1000). Decimal Expanded Form: 7 + 0.3 + 0.006. Notice that the zero in the hundredths place contributes zero value and does not need to be written in expanded form.

Value Relationships Between Adjacent Decimal Columns

Just as with whole numbers, the multiplicative relationship between adjacent columns remains constant across the decimal point.

In the numeral 0.55:
- The 5 in the tenths place represents 0.5.
- The 5 in the hundredths place represents 0.05.
- 0.5 is 10 times greater than 0.05.
- 0.05 is 1/10 the value of 0.5.

Exercises

Exercise 1. Identify the place, face value, and place value of the digit 7 in the numeral 48.3752.

Exercise 2. Write the numeral 205.048 in word form and in both fractional and decimal expanded forms.

Exercise 3. Express "six hundred four and nine ten-thousandths" in standard decimal form.

Solutions

Solution to Exercise 1. In the numeral 48.3752, the digit 7 occupies the hundredths place (the second digit to the right of the decimal point). Its face value is 7. Its place is hundredths. Its total place value is 7 * 0.01 = 0.07 (or 7/100).

Solution to Exercise 2. In word form, 205.048 is written as "two hundred five and forty-eight thousandths." In decimal expanded form, it is written as 200 + 5 + 0.04 + 0.008. In fractional expanded form, it is written as (2 * 100) + (5 * 1) + (4 * 1/100) + (8 * 1/1000).

Solution to Exercise 3. The whole number part before "and" is "six hundred four", which is 604. The word "and" indicates the decimal point. The fractional part is "nine ten-thousandths." The ten-thousandths column is the fourth place to the right of the decimal point (tenths, hundredths, thousandths, ten-thousandths). To place 9 in the ten-thousandths column, insert three placeholder zeros: 0.0009. Combining gives the standard form: 604.0009.