How to Convert Any Rational Number to Standard Form with Examples
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)
= ⅖
Hence, the rational number can express 18/45 in its standard form as ⅖.
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)
= ⅕
So, 3/15 deduced to its standard Form is ⅕.
FAQs on Rational Numbers in Standard Form: Explained for Students
1. What does it mean for a rational number to be in its standard form?
A rational number, expressed as p/q, is in its standard form when its denominator 'q' is a positive integer, and the integers 'p' and 'q' have no common factor other than 1. This is also known as the simplest form or lowest terms.
2. What are the two main conditions for a rational number to be in standard form?
For a rational number to be in its standard form, it must satisfy two essential conditions:
The denominator must be a positive integer.
The numerator and the denominator must be co-prime, meaning their highest common factor (HCF) is 1.
3. How do you convert any rational number into its standard form?
To convert a rational number to its standard form, follow these two steps:
Step 1: Check the denominator. If it's negative, multiply both the numerator and the denominator by -1 to make the denominator positive.
Step 2: Find the highest common factor (HCF) of the absolute values of the new numerator and the denominator. Divide both by their HCF to simplify the fraction to its lowest terms.
4. What is the standard form of the rational number 36/-24?
To find the standard form of 36/-24:
First, make the denominator positive: (36 × -1) / (-24 × -1) = -36/24.
Next, find the HCF of 36 and 24, which is 12.
Divide both the numerator and denominator by 12: -36 ÷ 12 = -3 and 24 ÷ 12 = 2.
Therefore, the standard form of 36/-24 is -3/2.
5. Why is it important to use the standard form of rational numbers?
Using the standard form is important because it provides a unique representation for every rational number. This makes it much easier to compare different rational numbers, perform calculations, and avoid confusion. For example, 1/2, 2/4, and 50/100 all represent the same value, but their standard form, 1/2, is the simplest and most consistent way to write it.
6. Is the standard form of a rational number always unique?
Yes, the standard form of any non-zero rational number is always unique. While a rational number can have infinitely many equivalent forms (e.g., 2/3, 4/6, 6/9), it has only one standard form. This uniqueness is a fundamental property that makes the standard form so useful in mathematics.
7. What is the difference between an equivalent rational number and the standard form?
The main difference lies in simplicity and uniqueness. Equivalent rational numbers are different fractions that represent the same value (e.g., 4/8 and 6/12 are equivalent to 1/2). The standard form is the single, simplest version of all these equivalent fractions (in this case, 1/2), where the numerator and denominator have no common factors other than 1 and the denominator is positive.
8. How does a negative denominator affect the process of finding the standard form?
A negative denominator violates one of the primary conditions of the standard form. If a rational number has a negative denominator, the first step is always to make it positive. This is done by multiplying both the numerator and the denominator by -1. This effectively moves the negative sign to the numerator or cancels it out if the numerator was also negative, ensuring the denominator becomes positive before you proceed to simplify the fraction.
9. Can you provide a few examples of rational numbers that are not in standard form and their correct standard form?
Certainly. Here are some examples:
The rational number -8/12 is not in standard form because 8 and 12 have a common factor of 4. Its standard form is -2/3.
The rational number 15/-25 is not in standard form because the denominator is negative and there's a common factor of 5. Its standard form is -3/5.
The rational number -14/-49 is not in standard form because the denominator is negative and they share a common factor of 7. Its standard form is 2/7.
10. Is 0/5 considered a rational number in standard form?
Yes, 0/5 is a rational number. Its value is 0. To check if it's in standard form, we see that the denominator (5) is positive. While 0/5 is valid, its most simplified and universally accepted standard form is often written as just 0 or 0/1, as it meets all conditions in the simplest possible way.
