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Decimals

Parts of a whole are represented by decimals, which are numbers that contain a dot known as the decimal point. They help us express values that fall between whole numbers, such as weights, measurements, and money. For instance, three and a half is represented by the number 3.5. Decimals are crucial in both everyday life and complex mathematics because they increase calculation accuracy.

Table of Content

What Are Decimals?
Place Value System in Decimals
Ordering and Comparing Decimals
Converting Fractions to Decimals
Types of Decimals
Operations with Decimals
Rounding Decimals
Decimals on a Number Line
Word Problems Involving Decimals
Decimal Patterns and Sequences
Decimals in Science and Technology
Errors to Avoid
Fun Facts About Decimals
Practice Exercises
Conclusion
FAQs on Decimals

What Are Decimals?

Definition: A decimal is a form of writing non-whole numbers, employing a decimal point to distinguish between the whole portion and the fractional portion

Decimal point: Dot "." used to separate the whole number part from the part less than one

Examples:

0.5
2.75
19.003

Reading decimals:

0.7 → "zero point seven"
3.25 → "three point two five"

Decimals are another representation of fractions:

0.5 = ½
0.25 = ¼
0.75 = ¾

Importance of decimals:

Link fractions to real life
Applied in money, measurements, science, and technology

Place Value System in Decimals

How place value functions with decimals:

Digits to the left of the decimal point:

Units, tens, hundreds, etc.

Digits to the right of the decimal point:

Tenths (1/10)
Hundredths (1/100)
Thousandths (1/1000)
And so on

Example:

In 23.456:

2 → tens place

3 → ones place

4 → tenths place

5 → place of hundredths

6 → place of thousandths

Where to locate decimals on a chart of places

Ordering and Comparing Decimals

How to order two decimal numbers:

Order whole number part first

If whole numbers are the same, order digit by digit after the decimal

Example:

3.17 > 3.071

Listing decimals in order from smallest to largest

Common errors:

Assuming that 0.5 is bigger than 0.75 because "5 is greater than 75"

Practice problems:

Put 0.2, 0.15, and 0.05 in order from smallest to largest

Converting Fractions to Decimals

Converting fractions to decimals:

Divide the numerator by the denominator

Example:

¾ = 3 ÷ 4 = 0.75

Decimal to fraction:

Determine the place value

Express the decimal as a fraction and reduce

Example:

0.2 = 2/10 = 1/5

Repeating decimals:

Infinitely repeating numbers

Example:

0.333… = ⅓

Terminating decimals:

Decimals that terminate

Example:

0.75 = ¾

Types of Decimals

Terminating Decimals

Definition:

Decimals terminating after a finite number of digits

Examples:

0.25
4.75

How to identify them:

Have a limited number of digits after the decimal point

Non-Terminating Repeating Decimals

Definition:

Decimals that keep repeating a pattern forever

Examples:

0.666…
2.131313…

Representation:

Use bar notation

0.666… = 0.6̅
2.131313… = 2.13̅

Non-Terminating Non-Repeating Decimals

Definition:

Decimals that never terminate and never repeat

Examples:

π = 3.1415926535…
√2 = 1.414213562…

These are irrational numbers

Operations with Decimals

Adding and Subtracting Decimals

Steps:

Line up decimal points

Add or subtract digits as usual

Leave the decimal point in the same position in the answer

Example:

3.25 + 1.7 = 4.95

Common mistakes:

Not aligning the decimal points

Multiplying Decimals

Multiply as whole numbers

Count total number of decimal places in each number

Put decimal in product in appropriate place

Example:

2.5 × 0.4 = 1.0

Dividing Decimals

Moving the decimal:

Shift the decimal point to make the divisor a whole number

Example:

6.4 ÷ 0.8 → shift → 64 ÷ 8 = 8

Dividing smaller numbers:

0.72 ÷ 0.6 = 1.2

Rounding Decimals

Why rounding is important:

Simplifies numbers for easier use

Steps:

Look at the digit right after the place you’re rounding to

If it's 5 or more, round up

Example:

Round 5.678 to tenths → 5.7

Decimals on a Number Line

Understanding decimals between whole numbers:

0.5 is halfway between 0 and 1

Locate decimals such as 0.3, 0.7, 1.2

Practice:

Put 0.65 on a number line between 0 and 1

Word Problems Involving Decimals

Money problems

"If something costs $4.75 and you purchase 3, how much does it cost altogether?"

Measurement problems:

Lengths, weights, distances

Real-life situations:

Budgeting
Cooking
Sports statistics

Practice:

Do problems with more than one step with decimals

Decimal Patterns and Sequences

Identifying patterns:

Adding 0.1

Multiplying decimals in a pattern

Example:

0.2, 0.4, 0.6, 0.8, …

Decimals in Science and Technology

Accurate measurements:

Laboratory results
Scientific devices

Representation of data:

Computing programming
Financial information

Significant figures comprehension:

Decimals represent how accurate a measurement is

Errors to Avoid

Omitting zeros:

Assuming 0.4 and 0.40 are not the same

Calculation misalignment

Assuming decimals with more than one digit are larger:

Example:

0.65 vs. 0.7 → 0.7 is larger, although it contains fewer digits

Misreading repeating decimals:

Failure to understand that 0.999… = 1

Fun Facts About Decimals

The word decimal is from the Latin "decimus," which means tenth
Our system of decimals is also known as the base-10 system
The Chinese and Egyptians employed decimal systems for measurement in ancient times
The digit π has been computed to more than 100 trillion digits
Decimals aid in dividing things into very small, exact pieces - such as dividing a dollar into cents

Practice Exercises

1. Express these fractions as decimals:

¼, ⅝, ⅓

2. Express these decimals in fraction form:

0.125, 0.6, 0.02

3. Add:

4.35 + 0.125

4. Subtract:

7.8 − 2.57

5. Multiply:

3.2 × 0.4

6. Divide:

5.6 ÷ 0.7

7. Order these decimals from least to greatest:

0.82, 0.09, 0.7, 0.35

8. Round:

8.764 to the nearest hundredth

9. Find:

What is half of 2.8?

Conclusion

Decimals are ubiquitous - in finances, measurement, science, and everyday life. Decimals are a confidence builder in dealing with numbers other than whole numbers. From reading decimals and putting them on the number line, through doing math computations and working on real-life problems using decimals, being a master of decimals is a key stepping stone to success in math and real life.

The more that you work, the more familiar you'll be with decimals. With a good grasp of decimals, you'll be more ready for higher concepts such as percentages, algebra, and data analysis.

Related Links:

percentage questions : Solve percentage questions easily with practical examples and clear methods from Orchids The International School.

Types of Fraction: Understand the types of fractions and master them with examples and fun activities.