Rational Numbers

What is a Rational Number? When someone asks you about your age, you may say you are 15 years old. The date, the number of pages in a book, the fingers on your hand. What numbers are these? These numbers are something known as rational numbers. Let us study in detail about rational numbers and their properties.

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What is a Rational Number?

We already know about some types of numbers. The numbers that we are familiar with are natural numbers, whole numbers, and integers. Natural numbers are the ones that begin with 1 and goes on endlessly up to plus infinity. If we include 0 in these sets of numbers, then these numbers become whole numbers.

Now in these sets, if we also include the negative numbers, then we call it as integers. So all the numbers that we see collectively on the number line are called integers. But what is a rational number?

A rational number is a number that can be written in the form of a numerator upon a denominator. Here the denominator should not be equal to 0. The numerator and the denominator will be integers. A rational number is of the form

\( \frac{p}{q} \)

p = numerator, q= denominator, where p and q are integers and q ≠0

Examples: \( \frac{3}{5} \), \( \frac{-3}{10} \), \( \frac{11}{-15} \). Here we can see that all the numerators and denominators are integers and even the denominators should be non-zero.

Browse more Topics under Number Systems

• Irrational Numbers
• Irrational Numbers Between two Numbers
• Operations on Irrational Numbers
• Laws for Exponents for Real Numbers

Positive and Negative Rational Numbers

Any rational number can be called as the positive rational number if both the numerator and denominator have like signs. A rational number which has either the numerator negative or the denominator negative is called the negative rational number.

Identify the Rational Numbers

• \( \frac{2}{7} \): Here 2 is an integer, 7 is an integer so yes it is a rational number.
• \( \frac{0}{0} \): Here there is 0 in the denominator too. So it is not a rational number.
• -9: Here -9 can be written \( \frac{-9}{1} \). So it is a rational number.
• 0: 0 is a  rational number.

Properties of Rational Number

1. A rational number remains unchanged when a non zero integer m is multiplied to both numerator and denominator.

\( \frac{p × m }{q ×m } \)

Suppose we take the number \( \frac{2}{5} \) and multiply both numerator and denominator by 3 then, \( \frac{2 × 3 }{5 ×3 } \) the result that we get is \( \frac{6}{15} \). Now this \( \frac{6}{15} \) is the standard form. If we express it in its simplest form we get it as \( \frac{2}{5} \).

2. A rational number remains unchanged when a non zero same integer m is divided to both numerator and denominator.

\( \frac{p ÷ m }{q ÷ m } \)

Suppose we take the number \( \frac{6}{15} \) and divide both numerator and denominator by 3 then, \( \frac{6 ÷ 3 }{15 ÷ 3 } \) the result that we get is \( \frac{2}{5} \).

Standard Form of Rational Number

Now \( \frac{24}{36} \) is a rational number. But when this number is expressed in its simplest form, it is \( \frac{2}{3} \). A rational number is in its standard form if it has no common factors other than 1 between the numerator and denominator and the denominator is positive.

Solved Examples for You

Question 1:  What fraction lies exactly halfway between \( \frac{2}{3} \) and \( \frac{3}{4} \) ?

1. \( \frac{3}{5} \)
2. \( \frac{5}{6} \)
3. \( \frac{7}{12} \)
4. \( \frac{9}{16} \)
5. \( \frac{17}{4} \)

Answer : The correct option is “E”. Comsider 3 × 4 = 12. So,
\( \frac{2}{3} \) = \( \frac{8}{12} \)
\( \frac{3}{4} \) = \( \frac{9}{12} \)
Multiplying the numerator and denominatoe by 2
\( \frac{16}{24} \) = \( \frac{18}{24} \)
The midpint is \( \frac{17}{24} \)

Question 2: If we divide a positive integer by another positive integer, what is the resulting number?

1. Always a natural number
2. Always an integer
3. A rational number
4. An irrational number

Answer : The correct option is “C”. If we divide a positive integer by another positive integer, the resulting number is always a rational number. Though it can be a natural number and an integer only if the denominator is 1.

Question 3: Give a simple definition of rational number?

Answer: A rational number refers to a number that one can write as a fraction. Rational numbers happen to be real numbers. Moreover, these numbers can be positive or negative.

Question 4: How can you identify a rational number?

Answer: A rational number is a number whose writing can take place as a ratio. That means it is possible to write it as a fraction such that both the numerator and the denominator are whole numbers. The number 8 is a rational number because one can write it as the fraction f8/1.

Question 5: Can we say that 3.5 is a rational number?

Answer: Yes, we can say that 3.5 is a rational number. Furthermore, 3.5 has a decimal because of which it is not a whole number. A rational number refers to any value which has equivalence to the ratio two integers. 3.5 is equivalent to the ratio of 7 and 2, thus it is a rational number.

Question 6: Can we say that 0.25 is a rational number?

Answer: Decimal 0.25 is certainly a rational number. This is because it shows the ratio or fraction 25/100 and both 25 and 100 are integers.