Rational Numbers, Definition, Properties, Types & Solved Questions
Rational Numbers
Rational Numbers: The rational number can be related to the word ratio. Do you know about ratios? The ratio is always represented in the form of fractions, for example, 7/5, 8/3, 9/3 etc. These are also rational numbers. There is often a little bit of confusion between fractions and rational numbers because the basic form of representation of both types of numbers is in the form p/q. The difference between fractions and rational numbers is that fractions are the combinations of whole numbers that cover a broad category including integers, terminating decimals, and repeating decimals. whereas a rational number is a combination of integers. Look at these examples to understand the basic difference between a ratio, rational numbers and fractions,
- If we talk about 9, writing it in the form of 9/1, then 9 is a rational number.
- If we consider 8/7, which is a rational number and can also be written in the form of a fraction.
- We can write 0.333….. in the form of 1/3. Which is in the form p/q, so, 0.333 is a rational number
- 2.5 can also be written as the ratio of 5/2. Which is the correct form of a rational number, Hence, 2.5 is also a rational number.
Now, you may get a basic understanding of all the minor differences between rational numbers, fractions and ratio. In this article, we focus mainly on rational numbers, the definition of rational numbers, examples, properties and their uses.
What are Rational Numbers?
The rational number is defined as a number represented in the form of a ratio as p/q, where p and q are integers and q is not equal to 0.
I.e the basic form of a rational number is pq, Where p and q are integers, q≠0.
There are infinite numbers of rational numbers present in between two rational numbers. if we have any two rational numbers like 10 and 20. Then by simply changing the denominators like 10/1, 10/2, 10/3, 10/4, 10/5 and so on, by continuing this process we can easily obtain an infinite number of rational numbers between any two given numbers.
A rational number can be written in many different ways keeping in mind the basic form of writing a rational number. For example, 4, 41, 332, 0.25, -158 , -0.11, 7/4, -8/3, -1.1, etc these all are rational numbers.
But, when we consider 4/0, then it is not in the form of rational numbers because here q=0.
How do identify Rational Numbers?
We can conclude that a number that can be a rational number is:
- A rational number can be identified by its form i.e a rational number is written in the form p/q where p is the numerator containing integers, q is the denominator containing integers that is not equal to zero. ½, ⅖, ¼, -⅛, 8/7.
- An integer like 1, 2, 44, 88, -1, -2, -93 etc are also a rational number as it can be written as 1/1, 2/1, 44/1, 88/1, -1/1, -2/1, -93/1. Here 1 is a denominator.
- Decimals can be terminating like 0.25, 0.15, 0.125 or non-terminating with repeating patterns only like 0.3333.., 0.1111…, 0.54444…. Are also rational numbers.
Arithmetic Operation of Rational Numbers
As we can perform the basic arithmetic operation on fractions, likewise a rational number can also be added, subtracted, multiplied and divided.
Let’s understand how these operations are performed.
1. For Addition and Subtraction
The denominator of a rational number must be the same while adding or subtracting any rational number. So, if we want to add or subtract
pq and xy then, the denominator of both the number will be the same and can be written in the form (py ± qx)qy.
For example, ⅞ + ½ can be written in the form (7×2) +(8×1)/(8×2) so, 14+8/16
=22/16, or 11/8
2. For Multiplication and Division
For multiplication and division of two rational numbers the denominator need not be the same, the rational number is directly multiplied or divided.
So, if want to multiply pq and xy then, it can be written in the form (px)qy
Now putting the equation in an example, ⅘ x ⅔ is written as 4×2/5×3
That is now, 8/15.
If we divide pq and xy then, it can be written by cross multiplying both rational numbers (py)qx, Let’s understand this with an example,
8/2 ÷ 9/3 so, 8×3/2×9 = 24/18 or, 4/3
Properties of Rational Numbers
- The addition, multiplication and subtraction of rational numbers are always rational numbers.
- When both the numerator and denominator of a rational number are multiplied or divided by the same factor then the resulting rational number remains the same.
- If we add or subtract (0)zero to a rational number then there are no changes in the resulting rational numbers, that is we will get the same rational number itself.
Positive and Negative Rational Numbers
If you have a basic understanding of a rational number, then it will be much easier for you to differentiate the positive and negative rational numbers, these are written in the same form as pq, Where q≠0. But with different signs, A positive rational number has a positive value of both the numerator and denominator with a value greater than zero, while the negative rational number has anyone with a negative value, either a numerator or a denominator with a value less than zero. A positive rational number has the same sign for both numerator and denominator while a negative has the opposite sign.
Examples of positive rational numbers are 8/3, 11/4, -5/-2, 36/7, 62/11 etc.
Examples of negative rational numbers are:8/-15, -⅘, -4/7, 13/-14.
Important points on Rational Numbers
- 0 (Zero) is also a rational number.
- The denominator of a rational number will never be zero.
- Between two rational numbers, there are infinite rational numbers.
- Not only fractional numbers are rational numbers but any number that can be expressed in a fraction is a rational number.
- A real number which is not considered to be a rational number is an irrational number.
Rational Numbers- Solved Questions
Q1: In a restaurant, there are 18 chefs and 12 waiters. What fractions of the total number of staff are waiters?
Answer: Let, the number of waiters be ‘p’= 12
And the total number of staff in the restaurant will be ‘q’ = 18 chefs + 12 waiters.
So, the fraction of waiters in the restaurant = number of waiter ÷ total number of staff in the restaurant.p/q
= 12/ (18 + 12)
= 12/ 30.
So, the fraction of waiter in the restaurant is 12/30 or ⅖.
Q2: Write any five rational numbers less than 5
Answer: A rational number is an integer with zero. So, -2, -1, 0, 1, and 2 are the five rational numbers less than 5.
Q3: Find out the rational numbers from the following:
√9, -8/10, 8/–9, √5, √5/2, 0.25, 0.333… 2.41421254…..
Answer: By evaluating each given number:
√9=3, This can be written as 3, which is an integer so, it is a rational number.
-8/10, 8/–9, are rational numbers as it is in the form pq, where q≠0.
√5, √5/2 These two when simplified are non-terminating with non-repeating decimal form so, they are not rational numbers.
0.25, is a terminating decimal and can also be written as ¼, so it is a rational number.
0.333… is a non-terminating with repeating decimal form so, they are rational numbers.
2.41421254… again it is a non-terminating with non-repeating decimal form so, they are not rational numbers.
Hence, √9, -8/10, 8/–9, 0.25, 0.333… are rational numbers from the given numbers.
Q4: Find out the sum of the given rational numbers?
8/9+6/5 and 9/4+8/2
Answer: To find the sum of two rational numbers if have to make the denominator of both the rational number the same that can be written in the form (py + qx)qy.
Let, the first set of numbers can be written as p/q=8/9 and x/y=6/5, putting into the equation, (py + qx)qy = (8×5) + (9×6)9×5 = 40 + 5445=94/45
Now, again the second set of numbers can be written as p/q=9/4 and x/y=8/2, putting into the equation, (py + qx)qy = (9×2) + (4×8)4×2 = 18 + 328= 50/8
Q5: Find out the difference between the given rational numbers?
3/2-5/4 and 6/2-7/5
Answer: The difference between two rational numbers can be found easily but before that, we have to make the denominator of both the rational number the same, which can be written in the form (py – qx)qy.
Let, the first set of numbers can be written as p/q=3/2 and x/y=5/4, putting into the equation, (py – qx)qy = (3×4) – (2×5)2×4 = 12 – 108=2/8=1/4
Now, again the second set of numbers can be written as p/q= 6/2 and x/y=7/5, putting into the equation, (py – qx)qy = (6×5) – (2×7)2×5 = 30 – 1410 = 16/10 this can be written as 1.6
Q6: The given fraction is a mixed fraction 5½. Find out if it is a rational number or not?
Answer: The given form is first converted to a simple fraction then 5½ is 11/2
Now, check whether the given form is in the p/q where q≠0.
Then p=11 and q= 2 an integer number and also, q≠0
So, Yes, 5½ is a rational number.
Q7: Check whether the following given numbers are rational or not?
0.50, 0.111.., 0.6, 2.789, 4.234
Answer: As we can see these all are decimal numbers, so, to find out whether they are rational or not we just have to check that the given number is terminating or non-terminating with repeating.
If we consider these numbers 0.50, 0.6, 2.789, and 4.234 are terminating decimals while 0.111 is a non-terminating repeating decimal,
Hence all the given numbers are rational numbers.
Q8: Find out at least 4 rational numbers between 4 and 5?
Answer: As 4 and 5 both are integers then we can convert them into equivalent fractions. So 4 can be written as 4/1 and 5 can be written as 5/1, now the equivalent fraction can be 8/2 and 10/2, respectively.
Now, we have the numbers are 8/2 and 10/2,
So the required rational number can be obtained in between these two equivalent numbers.
Hence, the four rational numbers between 4 and 5 are 9/2, 17/4, 18/4 and 19/4.
Q9: What is the rational number that is equivalent to 8/7?
Answer: To convert 8/7 into an equivalent fraction. Then multiply both numerator and denominator by the same number that is 4, so the equivalent rational number is 32/28.
Q10: Check whether 3.14 is a rational number?
Answer: To check whether 3.14 is a rational number or not firstly consider it as a decimal and check whether it is a terminating or non-terminating repeating decimal. Here, 3.14 is a terminating decimal. In spite of this 3.14 is not a rational number because the exact value of 3.14 is non-terminating and non-repeating i.e 3.141592653589…. As it is π. So, 3.14 is not a rational number.
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