The term “rational” originates from the word “ratio,” indicating a connection to proportions or comparisons. Rational numbers are a fundamental concept in mathematics, closely tied to the notion of ratios. A rational number can be written as p⁄q, where p and q are integers and q is not equal to zero. However, an irrational number cannot be expressed using simple fractions. √2 is an irrational number, whereas 2⁄3 is an example of a rational number.
In this article, we have covered definitions, examples, and properties of rational and irrational numbers, and others in detail.
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What are Rational Numbers?
Numbers that may be represented as fractions, as well as positive, negative, and zero, are known as rational numbers. It may be expressed as p/q, where q is not zero. The term “rational” comes from the word “ratio,” which refers to a comparison of two or more values or integer numbers, and is also known as a fraction. It is the ratio of two numbers in simple terms. All whole numbers, natural numbers, fractions of integers, integers, and terminating decimals are rational numbers.
Examples of Rational Numbers
Examples of rational numbers include integers such as 3, 4, 5, etc., which can be expressed in fractional form as 3/1, 4/1, and 5/1, respectively. Additionally, the number “0” is rational, as it can be represented in various fraction forms like 0/1, 0/2, 0/3, and so forth.
What are Irrational Numbers?
Contrary to rational numbers, irrational numbers cannot be expressed as fractions. They are characterized by an infinite sequence of non-repeating digits in their decimal representations. Examples of irrational numbers include square roots of non-perfect squares (√2, √3, √5), transcendental numbers like Euler’s number and π, and numbers with non-repeating decimal expansions.
Examples of Irrational Numbers
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Irrational numbers such as √2, √3, √5, Euler’s number, Golden Ratio, and π cannot be represented as fractions. Their decimal expansions continue indefinitely without repeating patterns, distinguishing them from rational numbers.
Is 22/7 a Rational or Irrational Number?
Solution:
Given the fraction 22/7, let’s analyze its properties to determine whether it qualifies as a rational number.
Rational Number Criteria:
- Integer Numerator(p): The numerator of 22 is an integer.
- Integer Denominator(q): The denominator of 7 is also an integer.
- Non-Zero Denominator(q ≠0): The denominator, 7, is not zero.
Verification:
Since 22/7 satisfies all the criteria for a rational number, we can confidently conclude that 22/7 is indeed a rational number.
Therefore, based on the properties of rational numbers, the fraction 22/7 qualifies as a rational number.
Similar Problems
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Problem 1: Determine whether √3 is a rational number.
Solution:
A rational number is a sort of real number that has the form p/q where q≠0. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal. Here, the given number, √3 cannot be expressed in the form of p/q. Hence, it is an irrational number.
Problem 2: Determine whether 1.232323…. is a rational number.
Solution:
A rational number is a sort of real number that has the form p/q where q≠0. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal. Here, the given number, 1.232323… has recurring digits. Hence, 1.232323…. is a rational number.
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