Have you ever come across decimal numbers and wondered why not all such numbers have a fixed number of digits after the decimal point? Or how do I convert these numbers into fractions? These are some questions that keep students bothered for a long time.
Vedantu is here to help you. To solve such problems first, you need to know what Repeating Decimal is.
Recurring Decimal numbers are those numbers that keep on repeating the same value after a decimal point. These numbers are also called Repeating Decimals. For example:
1/3 = 0.33333..... (3 repeats forever)
1/7 = 0.142857142857142857....... (14285714 repeat forever)
77/600= 0.128333333...... (3 repeat forever)
To display a repeating digit in a decimal number, often we put a dot or a line over the repeating digit as shown below:
(Image will be uploaded soon)
Non - Recurring Numbers
Non - recurring numbers are those in Mathematics that do not repeat their values after a decimal point. They are also called non-terminating decimals and non-Repeating Decimal numbers. For Example:
√2 = 1.41421356237309504…...
√7 = 2.647568759…...
π = 3.1415926535897932384626….....
e = 2.7182818284590452353602….....
Terminating and Recurring Decimals
When performing the conversion from fractions to decimals, you can formulate whether the decimal will terminate or recur.
Types of Decimal Numbers
Decimal numbers are classified into three types i.e.:
Terminating Decimals: these decimals have a finite number of digits followed by the decimal point.
For example, 0.5, 1.456, 123.456, etc.
Recurring Decimals: These consist of one or more repeating numbers or sequences of numbers followed by the decimal point, which keeps on infinitely.
For example, 5.232323…., 21.123123…, 0.1111….
Irrational: These Decimals go on forever, are never-ending and also never form a repeating pattern. These numbers are known as irrational numbers and cannot be written in the form of a fraction.
For example, 0.45445544…., etc.
Converting Recurring Decimals to Fractions
Conversion of Recurring Decimals into fractions is very useful for students throughout their academic life and also thereafter. It forms part of their basic Mathematical aptitude in various competitive exams in the future.
Let us understand and perform an example to understand how we a Recurring Decimal to a fraction.
Let’s check the steps involved in converting Repeating Decimals to fractions recurring (rational). The steps involved are as given:-
Step I: Let ‘x’ be the Repeating Decimal number that we want to convert into a rational number.
Step II: observe the Repeating Decimal to identify the repeating digits.
Step III: Carefully place the repeating digits to the left of the decimal point.
Step IV: Place the repeating digits to the right of the decimal point.
Step V: Now deduct the left sides of the two equations. Then, do the subtraction on the right side of the two equations. As we subtract, just ensure the differences of both sides will be positive.
Example: Convert 0.7 (one recurring digit) into a fraction.
Solution: Follow the below steps to convert Recurring Decimals to fractions:
Let x= Recurring Decimal
Let n = the number of recurring digits
Multiply the Recurring Decimal by 10
Subtracting (1) from (3) to remove the recurring part
Solve for x, expressing the answer as the fraction in the simplest form
Now, to solve for the given example
x = 0.777777....
10x = 7.77777
10x –x = 7
9x = 7
X = 7/9
Therefore, x = 7/9 is the required rational number.
Example: Convert 1.256 (two recurring digit) into a fraction
x = 0.125656....
10x = 125.6565....
100x – x = 125.6565... -1.256565
99x = 124.4
x = 124.4/99 = 1244/990 = 622/495
Example of Conversion of Repeating Decimal to Fraction
Convert the numerical digit 4.567878….. into a rational fraction.
Converting the given decimal number into a rational fraction can be performed by undertaking the following conversion steps:
Step I: Let x = 4.56787878…
Step II: After analyzing the expression, we identified that the repeating digits are ‘78’.
Step III: Now have to place the repeating digits ‘78’ to the left of the decimal point. To do so, we are required to move the decimal point to the right by 4 places. This can be accomplished by multiplying the given number by’10,000’.
10,000x = 45678.787878
Step IV: Now we would require moving the repeating digits to the left of the decimal point in the original decimal number. For this purpose, we will have to multiply the original number by ‘100’.
100x = 456.787878
Step V: here, the two equations become:
10,000x = 45678.787878, and
100x = 456.787878
Step VI: Now we need to subtract both the left and right-hand sides of the two equations and equate them in such a manner that the equality remains the same.
10,000x - 100x = 45678.787878 - 456.787878
⟹ 9,900x = 45,222
⟹ x = 45222/9900
This rational fraction can further be reduced to
x = 75371650 (dividing both denominator and numerator by 6)
Thus, the rational conversion of the provided decimal number comes out to be 75371650.
All the conversion of this type can be carried out by using the above-mentioned steps carefully.
Fun Facts About Recurring Decimal
Some numbers cannot be expressed as a fraction. For example √2.
Ancient Egyptians only used unit fractions.
Solving Recurring Decimals can touch infinity. To understand this you need to do more complicated Mathematical calculations.
In this topic, you have learned about recurring numbers, recurring numbers, types of recurring numbers and conversion of decimal numbers into fractions. Apart from this, you have also been given a sufficient number of solved examples to illustrate each concept.
After studying these you are in a position to further with other topics in Mathematics that are more complex. But you need to worry about it because at every stage Vedantu is here to help you.