Terminating and Non-Terminating Decimals
The decimal numbers are used to express both whole numbers and fractions. Depending on the type of digits after the decimal point, decimals can be categorized into several types such as repeating, non-repeating, ending, or un-ending (infinite digits after the decimal point). There are two types of decimals they are terminating and non-terminating decimals.
- A terminating decimal is a decimal number with a finite number of digits after the decimal point.
- A non-terminating decimal will never end but may predictably repeat one or more values after the decimal point.
Terminating Decimal:
Numbers with a fixed or finite number of digits following the decimal point are known as terminating decimals. In the same way that fractions represent the partial amount of a whole, decimal numbers represent a whole.
Example: 0.2, 0.125, 0.35
These terminating decimals can be expressed in the form p/q.
Example: Express 0.2 in the form of p/q.
Solution:
Non-terminating Decimal:
A non-terminating decimal has infinite decimal places. The digits after the decimal point will not terminate. A non-terminating decimal can be repeating or non-repeating. A non-terminating, non-repeating decimal is a decimal number with no repeating digits and continues indefinitely.
The non-terminating and non-recurring or non-repeating decimals are irrational numbers. Because it’s an irrational number, this decimal can’t be stated as a fraction.
When we split a fraction expressed in decimal form, we receive any remainder. The decimal is non-terminating if the dividing technique does not result in a remainder equal to zero.
In some circumstances, a single digit or a group of digits in the decimal component repeats. A sort of non-terminating repeating decimal is pure repeated decimals, which are also known as non-terminating repeating decimals. To symbolize these decimal numbers, a bar is put on the replicated portion.
Example:
There are two forms of non-terminating decimal expansions, they are:
(i) Non-terminating recurring decimal expansion
(ii) Non-terminating non-recurring decimal expansion
Non-Terminating Repeating Decimal Expansion
A non-terminating decimal is a decimal with an infinite number of digits after the decimal point.
A non-terminating, recurring decimal is a decimal in which some digits after the decimal point repeat without terminating. A non-terminating, recurring decimal can be expressed as p/q form.
Non-Terminating Non-Recurring Decimal Expansion
A non-terminating, non-recurring decimal in which the digits after the decimal point do not repeat and do not terminate.
Conversion of Non-Terminating Decimal to Rational Number
As seen in the previous section, a non-terminating recurring decimal can be converted into a rational number. A rational number is defined as the ratio of two integers p and q and is represented as p/q where q ≠ 0. Let us take an example to understand the conversion of a non-terminating recurring decimal to a rational number.
Steps to Convert Non Terminating Recurring Decimal to Rational Number
Let us understand the steps to convert a non-terminating recurring decimal to a rational number by taking an example.
- Step 1: Assume the repeating decimal to be equal to some variable x.
- Step 2: Write the number without using a bar and equal to x. (Bar is for digits that repeat in the same pattern)
- Step 3: Determine the number of digits having a bar on their heads or the number of digits before the bar for mixed recurring decimal.
- Step 4: If the repeating number is the same digit after decimal such as 0.2222… then multiply by 10, if repetition of the digits is in pairs of two numbers such as 0.7878… then multiply by 100 and so on.
- Step 5: Subtract the equation formed by step 2 and step 4.
- Step 6: Then find the value of x in the simplest form.
Let’s take an example of a non-terminating recurring decimal number 0.777…
Let, x = 0.777… ————– (1)
Multiplying 10 on both the sides, we get,
10x = 7.777.. —————– (2)
(This has to be chosen in such a way that on subtracting we get rid of the decimal)
Subtracting the two equations,
10x – x = 7.777 – 0.777
9x = 7
x = 7/9
Let’s take another example to understand this. Convert a non-terminating decimal 0.6565… to a rational number.
Let x = 0.6565… ——————— (1)
Multiplying 100 on both the sides,
100x = 65.6565… ——————– (2)
Subtracting the above equations, we get,
100x – x = 65.6565 – 0.6565
99x = 65
x = 65/99
Thus, we have understood the steps to convert a non-terminating recurring decimal to a rational number.