[Maths Class Notes] on Repeating Decimal to Fraction Conversion Pdf for Exam

A repeating decimal, also known as a recurring decimal, is a decimal representation of a number with periodic digits (values that occur at frequent intervals) and an infinitely repeated part that is not empty. A number can be seen to be rational if and only if its decimal representation repeats or terminates (i.e. all except finitely many digits are zero).

The repetend or reptend is a digit series that can be replicated indefinitely. Since the zeros can be omitted and the decimal terminates before these zeros, this decimal representation is considered a terminating decimal rather than a repeated decimal where the repetend is a zero.

What is a Fraction?

A fraction (from the Latin fractus, which means “broken”) denotes a portion of a whole or, more broadly, any number of equivalent parts. In daily English, a fraction denotes the number of pieces of a certain scale, such as one-half, eight-fifths, or three-quarters. The numerator and denominator of positive general fractions are all natural numbers. The denominator means how many of those components make up a unit or a whole, while the numerator represents a number of equivalent parts. Since zero pieces will never make up a whole, the denominator cannot be zero. The numerator 3 of the fraction 3/8, for example, means that the fraction represents three equal parts, and the denominator 8 indicates that the fraction represents three equal parts.

Types of Decimals

• Recurring Decimal

• Non-recurring Decimal 

What is Terminating and Non Terminating Decimal?

The two main classifications of decimals are terminating and non terminating numbers, let us understand them in detail.

What is a Terminating Decimal?

Terminating Decimal: A decimal number with a finite number of digits after the decimal point is known as a terminating decimal.

What is Non Terminating Decimal?

Non Terminating Decimal: A non-terminating decimal is one that continues indefinitely. To put it another way, a non-terminating decimal is formed when a fraction is represented in decimal form but still has a remainder, regardless of how much the long division procedure is carried out.

What is a Recurring Decimal?

Recurring Decimal: A repeating decimal, also known as a recurring decimal, is a decimal representation of a number with periodic digits (values that occur at frequent intervals) and an infinitely repeated part that is not empty. A number can be seen to be rational if and only if its decimal representation repeats or terminates (i.e. all except finitely many digits are zero).

What is a Non-Recurring Decimal?

Non-recurring Decimal: A non-recurring decimal is one in which the digits do not repeat. The number 0.101001000100001…, for example, is apparently non-recurring. Most numbers that are not specified as fractions of integers, to begin with, have non-recurring decimal expansions; for example, the square root of 2 is 1.4142135623730950488016887242…, and its digits do not repeat. These are referred to as irrational numbers.

Conversion Repeating Decimal to Fraction

Following are the steps to convert repeating decimal to fraction:

• Allow x to be the repeating decimal to convert to a fraction.

• Look for the repeating digit of the repeating decimal(s).

• To the left of the decimal point, place the repeated digit(s).

• To the right of the decimal point, place the repeated digit(s).

• Subtract the left sides of the two equations using the two equations you discovered in steps 3 and 4. Then, take the right sides of the two equations and subtract them.

• Only make sure that the difference is positive on both sides when you deduct.

Non Terminating Repeating Decimals are Rational

The repetitive terms are used in non-repeating decimals. Irrational numbers are described as non-terminating and non-repeating decimals. Non-terminating and repeated decimals are Rational numbers and can be expressed as p/q, where q is greater than zero.

Let us take the example of 0.004 to understand the concept,

Answer: Start by writing the following simple equation to transform 0.004 repeating into a fraction:

n = 0.004 ……..( 1)

Step 2: Since the repeated block (4) has 1 digit, multiply all sides by 1 followed by 1 zero, i.e., by 10.

10 × n = 0.044 …….(2)

Step 3: Deduct equation 1 from equation 2 to eliminate the repeated block (or repetend).

10 × n = 0.044

 1 × n = 0.004

 9 × n = 0.04

The fraction above has a decimal numerator. We would convert it to an integer by multiplying it by 100. We can multiply the denominator by the same sum as we multiply the numerator. As a result,

[frac{0.04}{9}] = [frac{4}{900}]

On simplification, we get,

n = [frac{1}{25}]

Therefore, we can say that 0.004can be represented as [frac{1}{25}] which is in [frac{p}{q}] form, hence repeated non terminating decimals are rational.