Decimals

Navigate the world of decimal numbers with confidence

If fractions made you a little nervous, you might be bracing yourself for decimals. But here’s the good news: you already use decimals every single day. When you check a price tag that says $4.99, glance at your GPA of 3.5, or notice that your phone battery is at 87%, you’re reading decimals. Decimals are simply another way to write fractions, and in many ways, they’re easier to work with because they follow the same patterns as our regular counting system.

So take a breath. You’ve got this.

Core Concepts

What Are Decimals?

A decimal number is any number that uses a decimal point to separate the whole number part from the fractional part. The word “decimal” comes from the Latin word for “ten,” and that’s the key insight: decimals are based on powers of ten, just like our regular number system.

When you write 3.75, you’re really saying “3 whole things plus 75 hundredths of another thing.” It’s the same as the fraction $3\frac{75}{100}$, but written in a more compact way.

Place Value: The Secret to Understanding Decimals

You already understand place value for whole numbers. In the number 456, the 4 represents four hundreds, the 5 represents five tens, and the 6 represents six ones. Each place is worth ten times more than the place to its right.

Decimals extend this pattern to the right of the decimal point, but now each place is worth one-tenth of the place to its left:

Place	Value	As a Fraction
Ones	1	1
Decimal Point	.
Tenths	0.1	$\frac{1}{10}$
Hundredths	0.01	$\frac{1}{100}$
Thousandths	0.001	$\frac{1}{1000}$
Ten-thousandths	0.0001	$\frac{1}{10000}$

So in the number 3.752:

The 3 is in the ones place (3 whole things)
The 7 is in the tenths place ($\frac{7}{10}$)
The 5 is in the hundredths place ($\frac{5}{100}$)
The 2 is in the thousandths place ($\frac{2}{1000}$)

Notation and Terminology

Decimal point: The dot that separates the whole number part from the fractional part. In some countries, a comma is used instead.

Decimal places: The number of digits after the decimal point. The number 3.14159 has five decimal places.

Terminating decimal: A decimal that ends, like 0.5 or 0.125.

Repeating decimal: A decimal where one or more digits repeat forever, written with a bar over the repeating part. For example, $\frac{1}{3} = 0.\overline{3}$ (the 3 repeats forever: 0.333333…).

Leading zeros: Zeros that appear after the decimal point but before the first non-zero digit, like the zeros in 0.007. These matter because they show us how small the number is.

Trailing zeros: Zeros at the end of a decimal, like in 3.50. These don’t change the value (3.50 = 3.5) but sometimes indicate precision in measurements.

Examples

Example 1: Reading and Writing Decimals

Problem: Write “seven and three hundred forty-two thousandths” as a decimal.

Solution: The word “and” tells us where the decimal point goes. Everything before “and” is the whole number part (7), and everything after describes the decimal part.

“Three hundred forty-two thousandths” means $\frac{342}{1000}$.

Since thousandths is the third decimal place, we need three digits after the decimal point:

$$7.342$$

To read this aloud: “seven and three hundred forty-two thousandths.”

Example 2: Comparing Decimals

Problem: Which is larger: 0.35 or 0.287?

Solution: The trick to comparing decimals is to line up the decimal points and compare digit by digit from left to right.

$$0.350$$ $$0.287$$

(We added a zero to 0.35 to make comparison easier. Remember, 0.35 = 0.350.)

Starting from the left:

Both have 0 in the ones place
Tenths place: 3 vs 2. Since 3 > 2, we’re done!

$$0.35 > 0.287$$

This might seem counterintuitive because 287 > 35 as whole numbers. But remember: 0.35 means 35 hundredths, while 0.287 means 287 thousandths. Hundredths are bigger pieces than thousandths!

Example 3: Adding and Subtracting Decimals

Problem: Calculate $23.7 + 8.456$

Solution: The golden rule for adding (or subtracting) decimals: line up the decimal points.

Once the decimal points are aligned, you can fill in empty spaces with zeros and add normally:

$$\begin{array}{r} 23.700 \ + \phantom{0}8.456 \ \hline 32.156 \end{array}$$

The answer is $32.156$.

Think of it like adding money: you wouldn’t add the cents from one amount to the dollars of another. The decimal point keeps everything in its proper column.

Example 4: Multiplying Decimals

Problem: Calculate $2.4 \times 0.35$

Solution: For multiplication, you don’t need to line up decimal points. Instead:

Ignore the decimal points and multiply as if they were whole numbers
Count the total decimal places in both original numbers
Put that many decimal places in your answer

Step 1: Multiply 24 times 35 $$24 \times 35 = 840$$

Step 2: Count decimal places

2.4 has 1 decimal place
0.35 has 2 decimal places
Total: 1 + 2 = 3 decimal places

Step 3: Place the decimal point 3 places from the right $$840 \rightarrow 0.840 = 0.84$$

So $2.4 \times 0.35 = 0.84$.

Why does this work? When you multiply $2.4 \times 0.35$, you’re really multiplying $\frac{24}{10} \times \frac{35}{100} = \frac{840}{1000} = 0.840$.

Example 5: Dividing Decimals

Problem: Calculate $7.2 \div 0.04$

Solution: Division with decimals has a special trick: make the divisor (the number you’re dividing by) a whole number by moving the decimal point.

Step 1: Move the decimal point in the divisor (0.04) to make it a whole number. We need to move it 2 places to the right: $0.04 \rightarrow 4$

Step 2: Move the decimal point in the dividend (7.2) the same number of places. Move it 2 places to the right: $7.2 \rightarrow 720$

Step 3: Divide the new numbers. $$720 \div 4 = 180$$

So $7.2 \div 0.04 = 180$.

Why does this work? Moving the decimal point in both numbers is the same as multiplying both by the same power of 10. Since $\frac{a}{b} = \frac{a \times 100}{b \times 100}$, we haven’t changed the value of the division, just made it easier to compute.

Key Properties and Rules

Comparing Decimals

Line up the decimal points
Add trailing zeros if needed to make the numbers the same length
Compare digit by digit from left to right

Adding and Subtracting Decimals

Line up the decimal points
Fill empty spaces with zeros
Add or subtract as usual
Bring the decimal point straight down

Multiplying Decimals

Multiply as if there were no decimal points
Count total decimal places in both factors
Place the decimal point that many places from the right in the answer

Dividing Decimals

Move the decimal point in the divisor to make it a whole number
Move the decimal point in the dividend the same number of places
Divide normally
Place the decimal point directly above its position in the dividend

Converting Fractions to Decimals

Divide the numerator by the denominator: $$\frac{3}{4} = 3 \div 4 = 0.75$$

Converting Decimals to Fractions

Write the decimal as a fraction with a power of 10 as the denominator
Simplify if possible

$$0.625 = \frac{625}{1000} = \frac{5}{8}$$

Rounding Decimals

Find the place you’re rounding to
Look at the digit to its right
If it’s 5 or greater, round up; if it’s less than 5, round down

To round 3.847 to the nearest hundredth:

Hundredths place has 4
The digit to its right is 7 (which is greater than or equal to 5)
Round up: $3.847 \rightarrow 3.85$

Real-World Applications

Money: The most common use of decimals in daily life. Dollars and cents are written as decimals, with cents representing hundredths of a dollar. When you see $19.99, you’re looking at 19 dollars and 99 cents, or $19\frac{99}{100}$.

Measurements: Scientists, engineers, and even cooks use decimals constantly. A recipe might call for 1.5 cups of flour. A carpenter might measure a board at 4.75 inches. Your body temperature is about 98.6 degrees Fahrenheit.

Statistics and Averages: Your GPA, batting averages in baseball (0.300 is excellent!), and percentages all rely on decimals. When you hear that a basketball player shoots 0.85 from the free-throw line, that’s 85% converted to a decimal.

Technology: Computers process decimals constantly. Your phone’s GPS coordinates are decimals (like 40.7128, -74.0060 for New York City). File sizes are often shown as decimals (a 2.5 GB movie).

Gas Prices: Ever wonder why gas is priced at $3.459 per gallon with that tiny 9 at the end? That’s using thousandths to make prices appear lower while collecting a bit more revenue.

Self-Test Problems

Problem 1 (Easy): Write 0.0073 in words and as a fraction.

Show Answer

In words: “seventy-three ten-thousandths”

As a fraction: $\frac{73}{10000}$

The last digit (3) is in the ten-thousandths place, which tells us the denominator.

Problem 2 (Easy): Put these decimals in order from least to greatest: 0.5, 0.509, 0.49, 0.051

Show Answer

From least to greatest: $0.051 < 0.49 < 0.5 < 0.509$

Adding trailing zeros helps: 0.500, 0.509, 0.490, 0.051. Now compare digit by digit from the left.

Problem 3 (Medium): Calculate $15.8 - 7.235$

Show Answer

Line up the decimal points and fill with zeros:

$$\begin{array}{r} 15.800 \ - \phantom{0}7.235 \ \hline \phantom{0}8.565 \end{array}$$

The answer is $8.565$.

Problem 4 (Medium): Calculate $0.6 \times 0.15$

Show Answer

Step 1: Multiply without decimals: $6 \times 15 = 90$

Step 2: Count decimal places: 1 + 2 = 3 decimal places

Step 3: Place the decimal: $90 \rightarrow 0.090 = 0.09$

The answer is $0.09$.

Problem 5 (Hard): Convert $\frac{5}{8}$ to a decimal, then round to the nearest tenth.

Show Answer

Divide 5 by 8: $$5 \div 8 = 0.625$$

To round to the nearest tenth, look at the hundredths place (2). Since 2 < 5, round down.

$$0.625 \rightarrow 0.6$$

The answer is $0.6$.

Summary

Decimals extend place value to the right of the decimal point, representing fractions with denominators that are powers of 10.
Each place to the right of the decimal represents tenths, hundredths, thousandths, and so on.
To compare decimals, line up the decimal points and compare digit by digit from left to right.
To add or subtract decimals, line up the decimal points, then proceed as with whole numbers.
To multiply decimals, ignore the decimal points, multiply, then count total decimal places and place the point in the answer.
To divide decimals, move the decimal point in the divisor to make it a whole number, then move it the same number of places in the dividend.
Fractions become decimals when you divide the numerator by the denominator.
Decimals become fractions by using the appropriate power of 10 as the denominator.
Rounding looks at the digit to the right of your target place: 5 or more rounds up, less than 5 rounds down.

Remember: decimals and fractions are just two different ways to write the same thing. Once you’re comfortable moving between them, you’ll have a powerful toolkit for handling numbers in any form they appear.