Irrational Numbers, denoted by P, are numbers that cannot be written in the form
where:
In simple words, Irrational numbers cannot be expressed as fractions and their decimal expansion is non-terminating and non-repeating.
Set Representation
P⊂R
Key Features of Irrational Numbers
⭐ 5 Examples of Irrational Numbers (with explanation)
📝Explanation: √3 is not a perfect square → decimal is non-terminating and non-repeating → irrational.
📝 Description: Irrational numbers cannot be expressed as any fraction p/q where both p & q are integers.
📝 Description: √2 is non-terminating & non-repeating. √4 = 2 and √9 = 3 are rational.
📝 Description: π = 3.141592653… (decimal goes on forever without pattern).
📝 Description: √16 = 4 → rational (integer).
📝 Description: 1.41421… is the approximate value of √2 → non-terminating → irrational.
📝 Description: This is the formal defining property of irrational numbers.
📝 Description: Digits have no repeating pattern → irrational.
📝 Description: Rational + irrational = irrational (unless rational = 0, still irrational).
📝 Description: √50 = 5√2 (contains √2 → irrational).
📝 Description: π is a fundamental irrational number.
22/7 & 3.14 are rational approximations.
📝 Description: √45 = √(9×5) = √9 · √5 = 3√5 → still irrational because √5 is irrational.
📝 Description: √3 & π are irrational; others are rational..
📝 Description: Irrational number decimals → non-terminating & non-recurring.
📝 Description: √2 = 1.414213… (approx).
📝 Description: Irrational numbers are dense → infinitely many between any two real numbers.
📝 Description:
2 + √2 = irrational
√2 + √2 = 2√2 also irrational, but not in options? Wait: option a is also irrational.
But b is the MOST standard answer because rational + irrational always = irrational.
📝 Description: √12 = 2√3 → irrational.
📝 Description: Repeating decimal → rational.
📝 Description: Correct definition → non-terminating, non-repeating decimals.
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