A Peek Beyond the Point · Lesson 5 of 11
Decimal Place Value
“Extend familiar place value to write and read fractional quantities efficiently.”
• Explain the factor-of-ten relationship between neighbouring places. • Interpret tenths, hundredths, and thousandths in decimal notation. • Use zeros to preserve empty places. • Read decimals and translate between expanded, fractional, and decimal forms. • Regroup counts such as 234 tenths into decimal notation.
Extending the Place-Value System
You already use place value whenever you read 281 as two hundreds, eight tens, and one unit. Moving one place to the left multiplies the value of a single part by ten. Moving one place to the right makes it one-tenth as large. This relationship can continue beyond the units place instead of stopping there.
Immediately to the right of units come tenths. To the right of tenths come hundredths, and then thousandths. Ten thousandths make one hundredth, ten hundredths make one tenth, and ten tenths make one unit. The same relationship explains regrouping on either side of the units place.
One of one thousand equal parts of a unit, written as 1/1000. Splitting one hundredth into ten equal parts produces thousandths.
| Place | Size of one part | How many make one unit? |
|---|---|---|
| Ones | 1 | 1 |
| Tenths | 1/10 | 10 |
| Hundredths | 1/100 | 100 |
| Thousandths | 1/1000 | 1000 |
| Ten-thousandths | 1/10000 | 10000 |
A unit could be divided into four or eight equal parts instead. These are valid fractions too. Dividing by ten repeatedly is especially useful because it extends the same base-ten grouping already used for whole numbers. The decimal system is called decimal because it is based on ten.
How Big?
Ask how many small parts fit into a larger part. The answer depends on the sizes of both. A tenth contains ten hundredths but one hundred thousandths. A whole unit contains a thousand thousandths. Counting these relationships helps us move between descriptions without memorising isolated rules.
Problem
How many thousandths make one tenth, and how many hundredths make one ten?
- 1.One tenth is 1/10 unit. Written in thousandths, it is 100/1000 unit, so 100 thousandths make one tenth.
- 2.One ten is 10 units. Each unit contains 100 hundredths.
- 3.Therefore one ten contains 10 × 100 = 1000 hundredths. The word ten here names a whole-number place, not the fractional place tenth.
Notation, Writing and Reading of Numbers
If we wrote the digits 705 without identifying where whole units end, they might be intended as seven hundreds and five units, seven tens and five tenths, or seven units and five hundredths. These are different quantities. A decimal point fixes the units position and removes that ambiguity.
The point that separates the whole-number part from the fractional part in decimal notation. The ones place is immediately to its left and the tenths place immediately to its right.
| Quantity | Decimal notation | Expanded meaning |
|---|---|---|
| 7 hundreds and 5 units | 705 | 700 + 5 |
| 7 tens and 5 tenths | 70.5 | 70 + 5/10 |
| 7 units and 5 hundredths | 7.05 | 7 + 0/10 + 5/100 |
A digit tells us how many parts of its place are present. In 70.504, the 5 represents five tenths and the 4 represents four thousandths. The zero between them says that there are no hundredths in the grouped form. Removing that zero would move the 4 into the hundredths place and change the value.
Problem
Explain the value of each digit in 0.274 and read it aloud.
- 1.There are 0 whole units, 2 tenths, 7 hundredths, and 4 thousandths.
- 2.Its expanded form is 2/10 + 7/100 + 4/1000.
- 3.Read it as zero point two seven four. Reading the digits after the point individually makes their order clear.
- 4.It can also be described as 274 thousandths, because 200 + 70 + 4 = 274.
Problem
Write 1 hundred, 1 unit, and 1 hundredth in decimal notation.
- 1.The whole-number part is 100 + 1 = 101.
- 2.There are no tenths, so put 0 in the tenths place.
- 3.Put 1 in the hundredths place. The result is 101.01.
- 4.The expanded form 100 + 1 + 1/100 checks the interpretation.
Regrouping Before Writing a Decimal
An instruction such as “234 tenths” gives a count of small parts, not digits already placed in a chart. Convert groups of ten tenths into units before recording the number. If a description includes more than nine parts in any place, exchanges produce the ordinary grouped notation.
Problem
Express 234 tenths in decimal notation.
- 1.234 tenths = 234/10 units.
- 2.Separate the count: 200 tenths + 30 tenths + 4 tenths.
- 3.These are 20 units + 3 units + 4 tenths.
- 4.The grouped decimal is 23.4. By contrast, 234 hundredths is 2.34 because a hundred hundredths make a unit.
Problem
Write 87 units, 5 tenths, and 60 hundredths as a decimal.
- 1.Sixty hundredths equal 6 tenths. Combined with the existing 5 tenths, they give 11 tenths.
- 2.Eleven tenths equal 1 unit and 1 tenth.
- 3.Add the new unit to 87 to obtain 88 units and 1 tenth.
- 4.The decimal is 88.1, also written 88.10. The latter form records zero hundredths without changing the quantity.
Familiar Fractions in Decimal Form
Some familiar fractions can be written using tenths or hundredths by subdividing their parts further. Half a unit covers five of ten equal parts. A quarter covers twenty-five of one hundred equal parts. We can therefore connect those fractions to the decimal places we have already built.
Three quarters contain three groups of twenty-five hundredths, giving 75 hundredths or 0.75. Four fifths contain four groups of two tenths, giving 8 tenths or 0.8. One and a half units can be written as 1.5. These conversions use equivalent fractions rather than introducing a new division procedure.
Build a chart from hundreds through thousandths. Write 2.35, 10.5, 4.06, 101.01, 0.98, 0.05, 0.1, 2.01, 4.1, and 7.007. For each number, name every nonzero digit’s place and read the decimal aloud.
Read 7.05 as seven point zero five. Calling it seven point five would suggest 7.5, which contains five tenths rather than five hundredths. Zeros inside a decimal can be essential placeholders.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
What is the value of the 4 in 70.504?
How is 0.274 read in decimal notation?
What is 234 hundredths?
Which decimal represents 4 units and 6 hundredths?
Which is the decimal form of 3/4?
The 4 is in the third place after the decimal point, the thousandths place.
Practice Problems
- Write 105 tenths in decimal notation and explain your grouping.
- Expand 7.007 using whole units and fractional units.
- Write 12 tens and 12 tenths as a decimal.
- Write 10 tens, 10 units, 10 tenths, and 10 hundredths as a decimal.
- Write 25 tens, 25 units, 25 tenths, and 25 hundredths as a decimal.
- How many thousandths make one hundredth? How many tenths make one ten?
- Write 1/2, 3/2, 1/4, 3/4, 1/5, and 4/5 as decimals.
100 tenths make 10 units and 5 tenths remain. The decimal is 10.5.
Key Takeaways
• Decimal notation extends the same base-ten place-value system used for whole numbers. • Each place to the right is one-tenth as large as its neighbour to the left. • The decimal point separates whole units from fractional units. • Zeros preserve places that contain no parts in the grouped description. • Read digits after the decimal point individually. • Regroup counts larger than nine before writing digits in their places. • Equivalent fractions connect halves, quarters, and fifths with decimals.