In the CBSE Class 10 Real Numbers chapter, understanding the distinction between rational and irrational numbers is essential. A rational number can be expressed as a fraction p/q where both p and q are integers and q ≠ 0. An irrational number cannot be expressed in this form — its decimal expansion is non-terminating and non-repeating.
A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. The decimal expansion of a rational number either terminates (e.g., 3/4 = 0.75) or eventually repeats in a pattern (e.g., 1/3 = 0.333…). All integers, whole numbers, and fractions are rational numbers.
An irrational number cannot be expressed as p/q for any integers p and q. The decimal representation of an irrational number is non-terminating and non-repeating. Common examples include √2 ≈ 1.41421356…, √3 ≈ 1.73205080…, π ≈ 3.14159265…, and e ≈ 2.71828182…
Terminating decimals (like 0.5, 0.125) and recurring decimals (like 0.333…, 0.142857142857…) are always rational. A decimal that goes on forever without any repeating block is irrational. This distinction is a key test for classifying numbers on the number line.
The sum or difference of a rational number and an irrational number is always irrational. The product of a non-zero rational number and an irrational number is always irrational. However, the sum of two irrational numbers may be rational (e.g., √2 + (−√2) = 0) or irrational (e.g., √2 + √3).
To check if a number is rational: try expressing it as p/q. For square roots, if the number under the root is not a perfect square, the result is irrational. For decimal numbers, check whether the digits eventually form a repeating pattern or not.
Zero is rational. It can be expressed as 0/1 or 0/n for any non-zero integer n.
π is irrational. Although we often approximate it as 22/7 or 3.14, its actual decimal expansion never terminates or repeats.
Yes. For example, −√2 is an irrational number. The negative of any irrational number is also irrational.
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