Decimals in Maths

Decimals are a way of representing numbers using a decimal point. They can represent whole numbers as well as values between whole numbers, making it easier to express parts of a whole.

Place Value After Decimal Point

In the chart below, you can see that the first digit after the decimal point is in the tenths place, the second digit is in the hundredths place, and the third digit is in the thousandths place. This is because decimal numbers also follow the place value system.

Place value in decimals refers to the position of a digit in relation to the decimal point. Each position represents a power of 10, determining the digit's value in the number.

For example, in the decimal 0.75:

• The 7 is in the "tenths" place because it's one-tenth of the whole.
• The 5 is in the "hundredths" place because it's one-hundredth of the whole.

In 0.75, you have 7 tenths and 5 hundredths.

Expanded Form of Decimals

The expanded form of decimals shows the value of each digit based on its place value.

For example, the expanded form for 56.739,

First, we will write the digits of the given number in the place value chart of decimals, as shown below.

56.739 = 5 × 10 + 6 × 1 + 7 × 0.1 + 3 × 0.01 + 9 × 0.001
OR
56.739 = 5 × 10 + 6 × 1 + 7 × 1/10 + 3 × 1/100 + 9 × 1/1000

Types of Decimals

Decimals can be classified into three main types based on the pattern of digits after the decimal point:

Terminating Decimals

A terminating decimal has a finite number of digits after the decimal point. The decimal expansion ends after a certain number of digits.

For example: 0.5, 0.25, 2.75, 12.125

Non-Terminating Recurring Decimals

A non-terminating recurring decimal has an infinite number of digits after the decimal point, with one or more digits repeating in a fixed pattern.

For example: 1/3​ = 0.333… = 0.\bar{3}; Here, the digit 3 repeats indefinitely.

Another example is: 5/11 ​= 0.454545… = 0.\bar{45}

Non-Terminating Non-Recurring Decimals

A non-terminating non-recurring decimal has an infinite number of digits after the decimal point, but the digits do not repeat in a fixed pattern.

For example: √2 ​= 1.414213562…

The digits continue indefinitely without forming a repeating pattern.

Decimal Fractions

Decimal fractions are numbers that fall between two consecutive integers on the number line and are expressed in decimal form. These numbers have a finite number of digits after the decimal point.

For example, in the decimal fraction 0.75, the digits 7 and 5 represent the fractional part, and there is no recurring or infinite pattern.

Decimal Conversion Examples

Example 1: Convert 1/4 to decimal.

Solution:

To convert the fraction 1/4 into a decimal, you can divide 1 by 4: 1/4 =0.25

Therefore, 1/4 as a decimal is 0.25

Example 2: Express the percentage 25% as a decimal.

Solution:

25% can be written as 25/100 in fraction on solving we get 1/4

1/4 = 0.25

Example 3: Convert the mixed number 2 \frac{1}{4} into a decimal.

Solution:

Convert 2 \frac{1}{4} into whole fraction as [(4 × 2) + 1]/4

= 9/4

Now divide 9 by 4 we get

= 2.25

Example 4. Represent the repeating decimal 1/3 as a fraction.

Solution:

To convert the fraction 1/3 into a decimal, you can divide 1 by 3: 1/3 = 0.3333....

Therefore, 1/3 as a decimal is 0.3333......, this is a non terminating repeating decimal representation.