What Is a Decimal?

A decimal is a way of representing numbers that include parts of a whole.

Consider:

3.7

The number contains:

• 3 whole units
• 7 tenths of another unit

The decimal point separates the whole-number part from the fractional part.

In:

3.7

the 3 is to the left of the decimal point and represents:

3 ones

The 7 is to the right and represents:

7 tenths

So:

3.7 = 3 + 7/10

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The Decimal Point

The decimal point is extremely important because it determines the place value of every digit.

Compare:

25

2.5

0.25

The digits 2 and 5 appear in each number, but their values change because their positions change.

For example, the digit 2 represents:

• 20 in 25
• 2 in 2.5
• 0.2 in 0.25

Position determines value.

---

Place Value to the Left of the Decimal

Whole-number place values increase as we move left.

For example:

4,382

contains:

• 4 thousands
• 3 hundreds
• 8 tens
• 2 ones

Each position is worth:

10 times

the position immediately to its right.

---

Place Value to the Right of the Decimal

The pattern continues to the right of the decimal point.

The first position is:

tenths

The second is:

hundredths

The third is:

thousandths

The fourth is:

ten-thousandths

The fifth is:

hundred-thousandths

and so on.

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The Place Value Pattern

The place values around the decimal point follow this pattern:

Thousands | Hundreds | Tens | Ones | decimal point | Tenths | Hundredths | Thousandths

Notice that there is no "oneths" place.

We move directly from:

ones

to:

tenths

---

Each Place Is Ten Times Smaller

Moving one position to the right divides the place value by:

10

For example:

1

One tenth of 1 is:

0.1

One tenth of 0.1 is:

0.01

One tenth of 0.01 is:

0.001

So:

1 → 0.1 → 0.01 → 0.001

Each step to the right makes the place value 10 times smaller.

---

Each Place Is Ten Times Larger

Moving one position to the left multiplies the place value by:

10

For example:

0.001 → 0.01 → 0.1 → 1 → 10 → 100

Each step to the left makes the place value:

10 times larger

This base-ten pattern is the foundation of our decimal number system.

---

Tenths

The first position to the right of the decimal point is the:

tenths place

One tenth can be written as:

1/10

or:

0.1

For example:

0.7

means:

7 tenths

Therefore:

0.7 = 7/10

---

Hundredths

The second position to the right of the decimal point is the:

hundredths place

One hundredth is:

1/100

or:

0.01

For example:

0.36

contains:

• 3 tenths
• 6 hundredths

So:

0.36 = 3/10 + 6/100

It can also be written:

36/100

---

Thousandths

The third position to the right is the:

thousandths place

One thousandth is:

1/1000

or:

0.001

For example:

0.428

contains:

• 4 tenths
• 2 hundredths
• 8 thousandths

Therefore:

0.428 = 4/10 + 2/100 + 8/1000

It can also be written:

428/1000

---

Using a Decimal Place Value Chart

Consider:

3,482.675

A place value chart would show:

This means:

• 3 thousands = 3,000
• 4 hundreds = 400
• 8 tens = 80
• 2 ones = 2
• 6 tenths = 0.6
• 7 hundredths = 0.07
• 5 thousandths = 0.005

---

Expanded Form

Decimals can be written in expanded form.

For:

3,482.675

expanded form is:

3,000 + 400 + 80 + 2 + 0.6 + 0.07 + 0.005

We could also use fractions:

3,000 + 400 + 80 + 2 + 6/10 + 7/100 + 5/1000

Expanded form makes the value of each digit clear.

---

Worked Example 1

Consider:

27.483

Identify each digit.

2 is in the tens place:

20

7 is in the ones place:

7

4 is in the tenths place:

0.4

8 is in the hundredths place:

0.08

3 is in the thousandths place:

0.003

Therefore:

27.483 = 20 + 7 + 0.4 + 0.08 + 0.003

---

The Difference Between a Digit and Its Value

A digit is one of the symbols:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

But the value of a digit depends on where it appears.

Consider the digit:

5

In:

500

the 5 has a value of:

500

In:

50

its value is:

50

In:

5

its value is:

5

In:

0.5

its value is:

0.5

In:

0.05

its value is:

0.05

In:

0.005

its value is:

0.005

The digit stays the same, but its value changes.

---

Zeros as Placeholders

Zeros are important in decimal numbers.

Compare:

0.5

and:

0.05

These are not equal.

0.5 = 5/10

while:

0.05 = 5/100

Therefore:

0.5 > 0.05

The zero in 0.05 shows that there are:

0 tenths

and:

5 hundredths

---

Another Zero Example

Consider:

4.307

This means:

• 4 ones
• 3 tenths
• 0 hundredths
• 7 thousandths

Expanded:

4 + 0.3 + 0.007

The zero keeps the 7 in the correct:

thousandths place

---

Zeros at the End of a Decimal

Zeros added to the right end of a decimal do not change its value.

For example:

0.5 = 0.50 = 0.500

Why?

Because:

5/10 = 50/100 = 500/1000

These are equivalent fractions.

Similarly:

3.7 = 3.70 = 3.700

---

Reading Decimal Numbers

One way to read decimals is to read the whole-number part, say and for the decimal point, and then name the fractional part using its final place value.

For example:

4.7

can be read:

four and seven tenths

---

Reading Hundredths

6.25

is:

six and twenty-five hundredths

because:

0.25 = 25/100

---

Reading Thousandths

12.438

is:

twelve and four hundred thirty-eight thousandths

because:

0.438 = 438/1000

---

Reading Numbers with Zeros

Consider:

5.007

This is:

five and seven thousandths

It is not:

five and seven hundredths

because the 7 is in the:

thousandths place

The zeros are important.

---

Writing Decimals from Words

Suppose we have:

eight and three tenths

Write:

8.3

---

Worked Example 2

Write:

twelve and forty-five hundredths

The whole-number part is:

12

The fractional part is:

45/100 = 0.45

Therefore:

12.45

---

Worked Example 3

Write:

six and nine thousandths

Nine thousandths is:

0.009

Therefore:

6.009

Notice the two zeros before the 9.

---

Standard Form, Word Form, and Expanded Form

The same number can be represented in several ways.

Consider:

24.306

Standard form:

24.306

Word form:

twenty-four and three hundred six thousandths

Expanded form:

20 + 4 + 0.3 + 0.006

Different forms highlight different information about the same number.

---

Decimals as Fractions

Every terminating decimal can be written as a fraction.

For example:

0.4 = 4/10

0.27 = 27/100

0.583 = 583/1000

The decimal place tells us the denominator.

---

Tenths and Fractions

If a decimal ends in the tenths place, we can initially write it over:

10

For example:

0.6 = 6/10

Simplify:

6/10 = 3/5

Therefore:

0.6 = 3/5

---

Hundredths and Fractions

If a decimal ends in the hundredths place, write it over:

100

For example:

0.25 = 25/100

Simplify:

25/100 = 1/4

Therefore:

0.25 = 1/4

---

Thousandths and Fractions

If a decimal ends in the thousandths place, write it over:

1000

For example:

0.125 = 125/1000

Simplify:

125/1000 = 1/8

Therefore:

0.125 = 1/8

---

Mixed Numbers and Decimals

Decimals greater than 1 can also be related to mixed numbers.

For example:

2.5

means:

2 + 5/10

Since:

5/10 = 1/2

we have:

2.5 = 2 1/2

---

Another Mixed Number Example

Consider:

3.75

This means:

3 + 75/100

Simplify:

75/100 = 3/4

Therefore:

3.75 = 3 3/4

---

Visualizing Tenths

Imagine a rectangle divided into:

10 equal sections

If 3 sections are shaded, the shaded portion represents:

3/10

or:

0.3

This is why the first decimal place represents tenths.

---

Visualizing Hundredths

Now imagine a square divided into:

100 equal smaller squares

If 37 squares are shaded, the shaded amount is:

37/100

or:

0.37

Hundred grids are useful for visualizing:

• decimals
• fractions
• percentages

---

Visualizing Thousandths

A cube can be divided into:

1,000 equal small cubes

Each small cube represents:

1/1000

or:

0.001

Ten small cubes represent:

0.010 = 0.01

One hundred small cubes represent:

0.100 = 0.1

This shows how tenths, hundredths, and thousandths are connected.

---

Decimal Place Value and Money

Money provides a familiar example of decimal place value.

Consider:

$4.75

The:

4

represents 4 dollars.

The:

7

represents 7 tenths of a dollar, or 70 cents.

The:

5

represents 5 hundredths of a dollar, or 5 cents.

Therefore:

$4.75 = 4 dollars and 75 cents

Money commonly uses two decimal places because:

$1 = 100 cents

---

Decimal Place Value in Measurement

Decimals are also common in measurements.

For example:

2.75 m

means:

2 metres + 0.75 metre

Since:

0.75 = 3/4

this is:

2 3/4 metres

Decimals allow measurements to be expressed more precisely than whole numbers alone.

---

Decimal Place Value in Science

Measurements in science frequently contain decimals.

Examples include:

12.4 g

3.75 m

0.025 L

18.6°C

The position of every digit matters because moving a digit by one place changes its value by a factor of:

10

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Decimals on a Number Line

Decimals can be located between whole numbers.

For example:

0.5

is halfway between:

0 and 1

The decimal:

1.5

is halfway between:

1 and 2

Decimals allow us to describe positions between whole numbers.

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Zooming In on a Number Line

Suppose the distance from:

0 to 1

is divided into 10 equal intervals.

Each interval represents:

0.1

If one of those intervals is divided into another 10 equal pieces, each smaller interval represents:

0.01

This pattern can continue:

0.001

0.0001

and beyond.

There is always another decimal place available.

---

Worked Example 4

Consider:

8.542

What is the value of the digit 5?

The 5 is in the:

tenths place

Therefore, its value is:

0.5

or:

5/10

---

Worked Example 5

Consider:

8.542

What is the value of the digit 4?

The 4 is in the:

hundredths place

Therefore:

0.04

or:

4/100

---

Worked Example 6

Consider:

8.542

What is the value of the digit 2?

The 2 is in the:

thousandths place

Therefore:

0.002

or:

2/1000

---

Worked Example 7

Write:

5 + 0.7 + 0.03 + 0.009

in standard form.

Combine the place values:

5.739

Therefore:

5 + 0.7 + 0.03 + 0.009 = 5.739

---

Worked Example 8

Write:

16.405

in expanded form.

The number contains:

• 1 ten
• 6 ones
• 4 tenths
• 0 hundredths
• 5 thousandths

Therefore:

16.405 = 10 + 6 + 0.4 + 0.005

---

Worked Example 9

Convert:

0.48

to a fraction.

Since 8 is in the hundredths place:

0.48 = 48/100

Simplify by dividing by 4:

48/100 = 12/25

Therefore:

0.48 = 12/25

---

Worked Example 10

Convert:

2.125

to a mixed number.

Separate the whole-number part:

2 + 0.125

Convert:

0.125 = 125/1000

Simplify:

125/1000 = 1/8

Therefore:

2.125 = 2 1/8

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Comparing the Size of Decimal Places

Consider the digit 7 in each number:

7

0.7

0.07

0.007

Its values are:

7

7/10

7/100

7/1000

Each time the digit moves one place to the right, its value becomes:

1/10 as large

---

Multiplying and Dividing by 10

Place value also explains what happens when we multiply or divide by powers of 10.

Consider:

3.47 × 10 = 34.7

The digits become worth ten times as much.

Consider:

3.47 ÷ 10 = 0.347

The digits become worth one tenth as much.

It is more accurate to think about the digits changing place value than to say that the decimal point itself moves.

---

Equivalent Decimals

Decimals can look different but have the same value.

For example:

0.6 = 0.60 = 0.600

Why?

Because:

6/10 = 60/100 = 600/1000

Trailing zeros do not change the value of a decimal.

---

But Leading Zeros Matter

Compare:

0.6

and:

0.06

These are not equivalent.

0.6 = 6/10

0.06 = 6/100

Therefore:

0.6 = 10 × 0.06

Zeros between the decimal point and a nonzero digit determine the digit's place value.

---

Common Mistakes

Mistake 1: Thinking 0.5 and 0.05 are equal

They are not.

0.5 = 5/10

0.05 = 5/100

Therefore:

0.5 > 0.05

---

Mistake 2: Thinking more digits means a larger number

For example:

0.125

has more digits than:

0.9

but:

0.125 < 0.9

Place value determines size.

---

Mistake 3: Ignoring zeros

In:

4.007

the 7 represents:

7 thousandths

not:

7 hundredths

---

Mistake 4: Calling 0.34 "thirty-four tenths"

The final digit is in the hundredths place.

Therefore:

0.34 = thirty-four hundredths

---

Mistake 5: Forgetting that the decimal system is base ten

Each place is:

10 times

the place to its right.

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A Reliable Place Value Strategy

When analyzing a decimal:

Step 1: Locate the decimal point.

Step 2: Identify the whole-number places to the left.

Step 3: Identify tenths, hundredths, thousandths, and further places to the right.

Step 4: Determine the value of each digit.

Step 5: Use expanded form if necessary.

Step 6: Relate the decimal portion to a fraction with denominator 10, 100, 1000, and so on.

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Did You Know?

Our decimal system is called a base-ten system because each place is based on powers of 10.

Whole-number places include:

1, 10, 100, 1,000, ...

Decimal places include:

1/10, 1/100, 1/1000, ...

This means the decimal point is not the beginning of a completely different number system.

It is simply the point where our base-ten place-value pattern moves from whole units into fractions of a unit.

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Key Terms

• Decimal: Number written using a decimal point to represent whole units and parts of a whole.
• Decimal point: Symbol separating the whole-number and fractional parts of a decimal.
• Place value: Value of a digit based on its position.
• Digit: One of the symbols 0 through 9.
• Tenths: First place to the right of the decimal point.
• Hundredths: Second place to the right of the decimal point.
• Thousandths: Third place to the right of the decimal point.
• Standard form: Usual numerical way of writing a number.
• Word form: Writing a number using words.
• Expanded form: Writing a number as the sum of its place-value parts.
• Equivalent decimals: Different decimal representations with the same value.
• Fraction: Number representing part of a whole or a division of quantities.
• Base ten: Number system in which each place is ten times the value of the place immediately to its right.
• Placeholder: A zero used to maintain the correct position and value of other digits.

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Key Relationships

1 tenth = 1/10 = 0.1

1 hundredth = 1/100 = 0.01

1 thousandth = 1/1000 = 0.001

10 tenths = 1 whole

10 hundredths = 1 tenth

10 thousandths = 1 hundredth

Examples:

0.7 = 7/10

0.35 = 35/100

0.428 = 428/1000

Equivalent decimals:

0.5 = 0.50 = 0.500

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Key Takeaways

• Decimals represent whole numbers and parts of a whole.
• The decimal point separates the whole-number part from the fractional part.
• The first place to the right of the decimal point is tenths.
• The second place is hundredths.
• The third place is thousandths.
• There is no "oneths" place.
• Each place to the left is ten times the value of the place immediately to its right.
• Each place to the right is one tenth the value of the place immediately to its left.
• The value of a digit depends on its position.
• Place value charts help organize and interpret decimal numbers.
• Zeros can act as important placeholders.
• Trailing zeros do not change the value of a decimal, so 0.5 = 0.50 = 0.500.
• Zeros between the decimal point and a nonzero digit do affect place value, so 0.5 ≠ 0.05.
• Decimals can be written in standard, word, and expanded forms.
• Tenths can be written as fractions with denominator 10.
• Hundredths can be written as fractions with denominator 100.
• Thousandths can be written as fractions with denominator 1000.
• Decimal fractions can often be simplified to equivalent fractions.
• Decimals greater than one can be related to mixed numbers.
• Decimal place value is used extensively in money, measurement, science, engineering, and everyday life.
• Understanding decimal place value provides the foundation for comparing, rounding, adding, subtracting, multiplying, and dividing decimals.