What is a Decimal? Definition, Properties, Types, Examples

Lesson Overview

• Definition of Decimals
• Types of Decimals
• Properties of Decimals
• Decimals Place Value Chart
• Decimal to Fraction Conversion
• Solved Examples on Decimals

Decimals are essential in mathematics and everyday life, helping us work with numbers that are not whole. They are used in situations like measuring lengths, calculating money, or tracking time. 

Decimals provide precision, making them crucial for comparing, adding, or subtracting values accurately.

Definition of Decimals

In mathematics, decimals are numbers that consist of a whole number and a fractional part, separated by a decimal point. The decimal point acts as a divider between these two parts.

For example, in the number 72.3, the whole number part is 72, and the fractional part is 3. 

The decimal point helps us represent numbers between whole numbers. For instance, 72.3 means a little more than 72 but less than 73.

Another example: 

Here is the number "twelve and five-tenths" written as a decimal number:

The decimal point goes between Ones and Tenths.

12.5 has 1 Ten, 2 Ones, and 5 Tenths.

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Types of Decimals

Decimals can be categorized based on their characteristics. The main types are:

1. Terminating Decimals
   • These decimals have a finite number of digits after the decimal point.
   • Example: 0.75, 3.5, 12.08.
     
2. Non-Terminating Repeating Decimals
   • These decimals go on forever but have a repeating pattern of digits.
   • Example: 0.333... (3 repeats), 1.666... (6 repeats).
     
3. Non-Terminating Non-Repeating Decimals
   • These decimals continue infinitely without any repeating pattern.
   • Example: π (3.14159…), √2 (1.41421…).

Each type of decimal has its unique characteristics, and understanding them helps in solving mathematical problems with precision.

Properties of Decimals

Decimals follow specific rules when multiplied or divided. These properties include:

• Multiplying two decimal numbers in any order gives the same product.
• Multiplying a whole number and a decimal in any order gives the same product.
• Multiplying a decimal by 1 leaves the decimal unchanged.
• Multiplying a decimal by 0 results in 0.
• Dividing a decimal by 1 keeps the decimal unchanged.
• Dividing a decimal by itself (except 0) gives 1.
• Dividing 0 by any decimal results in 0.
• Dividing a decimal by 0 is undefined, as division by 0 is not possible.

Decimals Place Value Chart

The place value system assigns a value to each digit in a number, based on its position. This helps us understand how the number is formed and what it represents.

Example:
Let's consider the number 372.

• The position of 2 is in the Ones place, which means 2 ones (i.e., 2).
• The position of 7 is in the Tens place, which means 7 tens (i.e., 70).
• The position of 3 is in the Hundreds place, which means 3 hundreds (i.e., 300).

As we move left, each place value becomes 10 times bigger:
Ones → Tens → Hundreds

Now, what happens if we go to the right of the Ones place?

We use a decimal point (".") to separate whole numbers from parts smaller than one

Example with Decimal:
Consider 372.48.

• To the left of the decimal point:
  • 3 is in the Hundreds place (300).
  • 7 is in the Tens place (70).
  • 2 is in the Ones place (2).

• To the right of the decimal point:

• 4 is in the Tenths place, meaning 4/10 (i.e., 0.4).
• 8 is in the Hundredths place, meaning 8/100 (i.e., 0.08).

We can expand 372.48 as:
(3 × 100) + (7 × 10) + (2 × 1) + (4 × 0.1) + (8 × 0.01).

Each digit is multiplied by its respective power of 10, showing the importance of the position in determining the value of a number.

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Decimal to Fraction Conversion

Converting between decimals and fractions is easy when you follow simple steps. Let us explore both methods below:

Decimal to Fraction Conversion

Decimals represent tenths, hundredths, thousandths, and so on, depending on the digits after the decimal point. To convert a decimal to a fraction, write the decimal in expanded form and simplify it.

Example
Convert 0.6 to a fraction.

Step 1: Write the decimal as a fraction. 0.6 can be written as 6/10​.

Step 2: Simplify the fraction. Find the greatest common divisor (GCD) of 6 and 10, which is 2.

Now, divide both the numerator and the denominator by 2:

6/10 = 6/10 ÷ 2/2 = 3/5

Step 3: Final Answer. So, 0.6 as a fraction is 3/5.

Fraction to Decimal Conversion

To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number).

Example
Convert 5/4 to a decimal.

Step 1: Perform the division. Divide 5 by 4:

5 ÷ 4 = 1.25

Step 2: Final Answer . So 5/4 as a decimal is 1.25. 

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Decimal Problems

• Example 1: Convert 3/8 into decimal form.

Solution
Divide 3 by 8.
3 divided by 8 equals 0.375.
Thus, 3/8 equals 0.375.

• Example 2: Express 2.5 as a fraction.

Solution
Write 2.5 as 25 times 1/10 = 25/10.
Simplify 25/10 to 5/2.
Thus, 2.5 equals 5/2.

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Solved Examples on Decimals

Example 1: Adding Decimals

• Problem: Lily buys a pencil for $1.25 and an eraser for $0.75. How much does she spend in total?
  
• Solution:
  

1. Line up the decimals:
   
     1.25

+ 0.75

-------

2. Add each place value, starting from the right: 5 + 5 = 10 (write down 0, carry over 1), 2 + 7 + 1 = 10 (write down 0, carry over 1), 1 + 0 + 1 = 2
   
     1.25

+ 0.75

-------

  2.00 

• Answer: Lily spends $2.00 in total.
  

Example 2: Subtracting Decimals

• Problem: A ribbon is 2.5 meters long. After cutting a piece 1.2 meters long, how much ribbon is left?
  
• Solution:
  

1. Line up the decimals:
   
     2.5 

- 1.2

-------

2. Subtract each place value, starting from the right: 5 - 2 = 3, 2 - 1 = 1
   
     2.5 

- 1.2

-------

  1.3

• Answer: There is 1.3 meters of ribbon left.
  

Example 3: Multiplying Decimals

• Problem: A bag of apples weighs 0.5 kilograms. If you buy 3 bags, what is the total weight?
  
• Solution:
  
  1. Ignore the decimal points and multiply: 3 x 5 = 15
  2. Count the total decimal places in the original numbers: 0.5 has one decimal place.
  3. Place the decimal point in the answer: Starting from the right, move the decimal point one place to the left in 15, giving you 1.5
• Answer: The total weight is 1.5 kilograms.
  

Example 4: Dividing Decimals

• Problem: A 4.8-meter long rope is cut into 4 equal pieces. How long is each piece?
  
• Solution:
  
  1. Divide as you would with whole numbers: 4.8 / 4 = 1.2
• Answer: Each piece is 1.2 meters long.
  

Example 5: Converting Decimals to Fractions

• Problem: Convert 0.75 to a fraction.
  
• Solution:
  
  1. Identify the place value: 75 is in the hundredths place.
  2. Write as a fraction: 75/100
  3. Simplify: Divide both numerator and denominator by 25, resulting in 3/4

Answer: 0.75 is equivalent to 3/4.