Rational Numbers in Standard Form

Define Standard Form of a Rational Number

A rational number is said to be in its standard form when the common factor between the numerator and the denominator is only 1 while that denominator is always positive. In addition, the standard form of a rational number is satisfied when the numerator contains a positive sign. Such Numbers are what we call Rational Numbers in Standard Form. Below are a few theories and examples which illustrate the process of expressing rational numbers in standard form and will help you get acquainted with the concept even better.

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What is the Standard Form of Rational Number?

Generally, a rational number, say x/y is asserted to be in standard form if it has no common factors except 1 between the numerator and denominator besides that the denominator ‘y’ should be positive.

How To Convert a Rational Number into Standard Form?

Below is a step-by-step listed guideline to express the rational number in standard form. The detailed step-by-step procedure is explained for enhanced understanding of the learners and they are along the lines.

  • Step 1: Take a look at the assigned rational number.

  • Step 2: Next, find out whether the denominator of the given fraction is positive or not. In case, it is not positive, multiplies or divides the numerator and denominator with -1 such that the denominator no longer remains negative.

  • Step 3: Identify the GCD (Greatest Common Divisor) of the absolute values of both the numerator and Denominator.

  • Step 4: Now, divide the numerator and denominator with the GCD you get in the earlier step. Subsequently, the rational number obtained is the standard form of the assigned rational number.

Solved Examples On Rational Numbers Standard Form

Example1:

Identify whether the given Rational Numbers are in Standard Form or Not?

(A)  -8/23

(B)  -13/-39

Solution:

(A). -8/23 is supposed to be in the standard Form seeing that both the numerator and denominator do not contain any common factors besides 1. For a fact, the denominator also has a positive sign. Therefore, the given rational number i.e. -8/23 is concluded to be in its standard form.

(B). -13/-39 cannot be concluded to be in its standard form seeing that it contains common factor 13 together with 1. Additionally, the denominator also does not have a positive sign. Therefore, we can easily declare that the given rational number is not in standard form.


Example2:

How to show the Rational Number 18/45 in the Standard Form?

Solution:

We are given,

Rational Number: 18/45

Now we are needed to check for the denominator in the rational number provided to us. Because it is positive you need not do anything.

Afterwards, identify the GCD of the absolute values of numerator 18 and the denominator 45

GCD (18, 45) comes out to be 9

GCD (18, 45) = 9

Therefore, in order to convert the provided rational number 18/45 into its standard form, we would simply divide both the numerator and denominator by 9

= 18/45

= (18÷9)/ (45÷9)

= 2/5

Hence, 18/45 expressed in standard form will be 2/5.


Example3:

Identify the Standard Form of the number 12/-18?

Solution:

We are given;

Rational Number = 12/-18

Nest we are needed to check for the denominator in the Rational Number given

Seeing that the denominator, -18 is negative, we will multiply both numerator and denominator with -1 in order to make it positive.

12/-18 = 12 × (-1)/-18 × (-1)

= -12/18

Next is to determine the GCD of absolute values of both the numerator and the denominator

GCD (12, 18) comes out to be 6

GCD (12, 18) = 6

In order to convert a given rational number into the standard form, we would require multiplying and dividing both numerator and denominator by 6.

-12/18 = [(-12) ÷6)/ (18÷6)]

= -2/3

Hence, the standard form of Rational Number 12/-18 will be -2/3.


Example4:

How do we reduce the number 3/15 to its Standard Form?

Solution:

We are given;

Rational Numbers = 3/15

Seeing that the denominator has a positive sign, we would require not doing anything in order to change it to positive.

Now, we will identify the GCD of absolute values of the numerator and denominator of the given rational number.

GCD (3, 15) comes out to be 3

GCD (3, 15) = 3

Divide both the numerator and denominator with GCD acquired

3/15 = (3÷3)/ (15÷3)

= 1/5

Hence, 3/15 deduced to its standard Form is 1/5.

FAQs (Frequently Asked Questions)

Q1. How Do We Reduce the Rational Numbers in Their Standard Form?

Answer: A rational number is in the standard form when its denominator has a positive integer and both the numerator and denominator contain no common factor except 1. However, if a rational number is not in the standard form, then we can reduce it to the standard form.


For the purpose of reducing the rational number to its standard form, we would require dividing both the numerator and the denominator of the fraction with the same non-zero positive integer. Also dividing the numerator and denominator of the fraction by their HCF (Highest Common Factor) overlooking the negative sign, if any, we can reduce the rational number to standard form.

Q2. How Do We Use Rational Numbers in Standard Form?

Answer: Expressing a rational number into its standard form typically implies that there is no common factor, besides 1, in its numerator and denominator while the denominator is a positive integer. Numbers that can be expressed in the p/q form, where p and q are integers and q is not equivalent to zero, are known as rational numbers. Thus, if 6/8 is a rational number, then its standard form will be ¾ because we cannot solve 3/4 any longer. 


The standard form of rational number further enables us to identify the value in a more particulate way. Just like, 30/35 can be expressed as 6/7, 22/55 can be expressed as 2/5 and so on.