Rational Numbers in Standard Form

Dhristi JEE 2022-24

Define Standard Form of a Rational Number

A rational number is formed when the common factor between the denominator and numerator is only 1. But the combination is that the denominator is always positive, including that a rational number can be standard when the numerator is associated with a positive sign. If these conditions are fulfilled, these numbers can be called rational numbers in standard form.


Based on this definition, some examples, theories, and processes demonstrate rational numbers in their standard form. Hopefully, these examples will better understand the conflict of rational numbers.


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.


What is the Standard Form of a Rational Number?

From the above definition, we can say that generally, x/y can be asserted as a rational number in its standard form. But there must not be any common factor except 1 between the numerator and denominator. Apart from that, the denominator, which is 'y' here, should always 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: First, take a thorough look at the given rational number.


  • Step 2: The condition of the rational number in its standard form is that the denominator should be positive. Find out if your assigned denominator of the fraction is favorable or not. If it is positive, then the condition will be fulfilled, and if it is not clear, you can divide or multiply both the numerator and denominator with -1. By doing that, it will turn the negative denominator into a positive.


  • Step 3: After that, you have to find out the GCD ( Greatest Common Divisor) of the absolute value of both denominator and numerator.


  • Step 4: The final step is to divide the numerator and the denominator with the GCD. You have already got the GCD in the earlier stage. After the division, you will finally get the standard form of the given rational number.


Solved Examples on Rational Numbers Standard Forms

Example1: Identify whether the given Rational Numbers are in Standard Form or Not?

(A)  -8/23

(B)  -13/-39


Solution:

A. -8/23 can be said to be a rational number in standard form. Look closely, and this option fulfills the conditions of rational numbers in standard form. Here both the numerator and the denominator have none but one common factor that is 1. Apart from that, the denominator contains a positive sign. By n analysing this, we can conclude that -8/23 can be called a rational number in its standard form.


B. -13/-39 can not be said to be a rational number in its standard form. Both the denominator and numerator contain a common factor beside one, 13. Apart from that, the denominator here is -39, which means it doesn't have any positive sign. Therefore -13/-39 can not be declared a rational number in its standard form.


Example 2: Show the Rational Number 18/45 in the Standard Form?


Solution:

Here, the rational number is 18/45.

Here we have to give the GCD ( Greatest Common Divisor) of the numerator 18 and denominator 45. As the denominator 45 is not negative. Here we do not need to do anything.

GCD (18, 45) comes out to be 9

GCD (18, 45) = 9

So if you want to convert 18/45 into its standard form, as we get from the rule, we have to divide both numerator and denominator by 9.

= 18/45

= (18÷9)/ (45÷9)

=

Hence, the rational number can express 18/45 in its standard form as.

Example 3: Identify the Standard Form of the Number 12/-18?


Solution:

In the rational number 12/-18, the denominator -18 is negative; hence, we must multiply the numerator and denominator with -1 to be positive.

12/-18 = 12 × (-1)/-18 × (-1)

= -12/18

Now we have to find out the GCD of the absolute value of the numerator and denominator. 

Hence, GCD (12, 18) comes out to be 6

GCD (12, 18) = 6

After that, both the denominator and the numerator should be multiplied and divided by the GCD that is 6.

= -12/18

= (−12)÷6)/(18÷6)(−12)÷6)/(18÷6)(-12) ÷6)/ (18÷6)

= -⅔

Hence, the standard form of Rational Number 12/-18 will be -⅔.


Example 4: How do We Reduce the Number 3/15 to its Standard Form?


Solution:

Here, we are given that the rational number is 3/15.

The denominator 15 is positive, so there is no need to change it.

We have to find out the GCD of numerator and denominator absolute values.

GCD (3,15) comes out to be 3GCD (3,15) = Therefore the numerator and denominator will be divided by the GCD.

3/15 = (3÷3)/ (15÷3)

=

So, 3/15 deduced to its standard Form is .

FAQs on Rational Numbers in Standard Form

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

A rational number will be standard when the denominator has a positive integer. Along with this, both the denominator and numerator must contain only one common fact. If the rational no doesn't follow these conditions, it can be reduced to the standard forms.


For doing this, at first, both the numerator and denominator will be divided with the same non-zero positive integer. After that, by finding GCD, both the numerator and denominator can be divided. Thus we can reduce rational numbers to their standard form.

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

Besides expressing a rational number in its standard form, there must not be any common factor. Along with it, both the numerator and the denominator will carry positive integers. By giving an example, Vedantu is trying to clarify the concept.


Say, if a number can be expressed as p/q form, where p and q are integers and q, the denominator must not be equivalent to zero. This type of number is known as a rational number.


If we take 6/8 as a rational number, its standard form will be ¾, as ¾ can't be broken any longer.


Like that, can transform 30/35 into 6/7, 22/55 can be expressed as ⅖, etc.

3. What is a Numerator?

At the point when numbers are written as a small portion, it very well may be addressed as a/b, where an is the numerator and b is the denominator. For instance, 4/5 is a small portion, and the line isolates the numbers 4 and 5 in the division bar. Here the number over the part bar is the numerator, and the one underneath the division bar is the denominator. A numerator addresses the number of parts out of the entire, which is the denominator.

4. What is a Denominator?

A denominator is the piece of a portion that comes underneath the fragmentary bar. The denominator shows the aggregate sum of parts that make up an entirety.


For any division a/b, an is the numerator and b is the denominator. Divisions address a piece of an entire or, all the more, by and large, quite a few equivalent parts. Numerically, it is shown by the division of two numbers. For instance, 1/4 is one section out of the four equivalent parts made from that something entirely. Here, 1 is the numerator and 4 is the denominator.

5. What is an Integer?

Numbers are generally negative and positive entire numbers, and 0. Numbers or whole number qualities are essential for different numbering frameworks.


Number Definition and Examples

Numbering frameworks are methods of counting and arranging genuine and fanciful items. Numbers are one bunch of numbers or numbering framework you utilize each day.


Normal numbering frameworks you might experience incorporate every one of these:

  • Genuine numbers

  • Regular numbers

  • Whole numbers

  • Fanciful numbers

  • Judicious numbers

  • Silly numbers

  • Complex numbers

We ordinarily compose our numbers utilizing Arabic numerals: 0,1,2,3. In any case, we can likewise address them utilizing Roman numerals (with the exception of 0): I, II, III.

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