1.1: Decimal Numbers
Decimal Numbers
Introduction: Fractions and Beyond
In everyday life, we often need to describe parts of a whole—like half a pizza, a quarter of a dollar, or 0.75 miles. Fractions are one way to represent these parts, but another powerful system is the decimal system.
Decimal numbers are built on the base-10 place value system—the same one Alphabitia discovered when moving beyond A, B, C, and D to larger numbers. But instead of just extending to larger whole numbers, decimals extend the system in the “other direction,” allowing us to represent numbers smaller than one.
Place Value: Extending Past the Decimal Point
The decimal point ( • ) is a marker that separates whole-number places from fractional places.
| Place Value | … | Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths | … |
|---|---|---|---|---|---|---|---|---|---|---|
| Power of 10 | … | 10³ | 10² | 10¹ | 10⁰ | 10⁻¹ | 10⁻² | 10⁻³ | … |
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To the left of the decimal point: values get 10 times larger each step.
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To the right of the decimal point: values get 10 times smaller each step.
Examples: Write out what each of the digits in the number 345.678 represent
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345.678
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3 hundreds (3 × 100)
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4 tens (4 × 10)
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5 ones (5 × 1)
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6 tenths (6 × 0.1)
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7 hundredths (7 × 0.01)
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8 thousandths (8 × 0.001)
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Visualizing Decimals with Blocks
We can use the base-10 style blocks to represent decimals. But now- notice that we need the large block (flat) to represent ONE. Since 10 longs make up a flat, this means that each long represents 1/10 or 0.1.
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🟪 (Flat) = 1 whole (a square, or “1”)
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| (Long) = 0.1 (one tenth of a flat)
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(Unit) = 0.01 (one hundredth of a flat)
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Examples. Use Base-10 blocks to make up the following numbers.
Example: 0.47
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4 longs (||||) = 0.4
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7 units (*******) = 0.07
Together: 0.47
Example: 2.36
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2 flats (🟪🟪) = 2
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3 longs (|||) = 0.3
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6 units (******) = 0.06
Together: 2.36
Connections to Fractions
Decimals are closely tied to fractions with denominators of powers of 10.
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0.1 = 1/10
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0.25 = 25/100 = 1/4
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0.75 = 75/100 = 3/4
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0.125 = 125/1000 = 1/8
Classroom Tip: You can give students base-10 blocks or grid paper (100 squares per flat) to color in parts. This helps connect fractions, decimals, and visuals.
The following table format can be helpful for young students when unpacking how to understand what each digit of a number is representing.
Place Value Chart Example
| Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 4 | 5 | 2 | . | 6 | 7 | 8 |
Number: 452.678
Breakdown:
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4 hundreds = 400
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5 tens = 50
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2 ones = 2
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6 tenths = 0.6
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7 hundredths = 0.07
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8 thousandths = 0.008
Total: 452.678
Why Decimals Matter
Decimals are everywhere in real life!
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Money: $3.75 means 3 dollars and 75 cents.
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Measurement: 1.25 meters = 1 meter + 25 centimeters.
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Science: 0.003 seconds measures very small times.
Understanding decimals helps students see the continuity between whole numbers, fractions, and real-world quantities.
