An irrational number is a real number that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. Irrational numbers have decimal expansions that are non-terminating (never end) and non-repeating (no repeating pattern). Numbers like √2, √3, π, and e are irrational. A common MCQ trap: √4 = 2 (rational), √9 = 3 (rational), √16 = 4 (rational) — even though they have a square root sign, the result is a whole number, making them rational, not irrational.
Irrational numbers: cannot be expressed as p/q; non-terminating, non-repeating decimals.
Rational numbers: can be expressed as p/q; includes terminating and repeating decimals.
√2 = 1.41421... (irrational), but √4 = 2 (rational — perfect square).
π = 3.14159... is irrational; 22/7 is rational (only an approximation of π).
e = 2.71828... is irrational.
All perfect square roots are rational: √1=1, √4=2, √9=3, √16=4, √25=5...
All integers are rational (e.g., 5 = 5/1).
Rational + Irrational numbers = Real numbers.
Rational numbers: • Can be expressed as p/q where p and q are integers, q ≠ 0 • Decimal form: either terminating (e.g., 0.5 = 1/2) or repeating (e.g., 0.333... = 1/3) • Examples: 2, -3, 0, 1/2, 3/4, 0.25, 0.666..., √4 = 2, √9 = 3
Irrational numbers: • Cannot be expressed as p/q for any integers p, q • Decimal form: non-terminating and non-repeating • Examples: √2 = 1.41421356..., √3 = 1.73205..., π = 3.14159265..., e = 2.71828..., √5, √7, √11
Key point: Every integer is a rational number (e.g., 5 = 5/1, -3 = -3/1). Every terminating or repeating decimal is rational.
Together, rational and irrational numbers form the set of real numbers.
Watch out for perfect squares and cubes under a root sign — their roots are rational:
Number | Result | Rational or Irrational? √4 | 2 | Rational ✓ (NOT irrational) √9 | 3 | Rational ✓ (NOT irrational) √16 | 4 | Rational ✓ (NOT irrational) √25 | 5 | Rational ✓ (NOT irrational) √36 | 6 | Rational ✓ (NOT irrational) √49 | 7 | Rational ✓ (NOT irrational) √100 | 10 | Rational ✓ (NOT irrational) √0.25 | 0.5 | Rational ✓ (NOT irrational) √(1/4) | 1/2 | Rational ✓ (NOT irrational) ∛8 | 2 | Rational ✓ (NOT irrational) ∛27 | 3 | Rational ✓ (NOT irrational)
These ARE irrational: √2, √3, √5, √6, √7, √8, √10, √11, √12, √13, √14, √15, π, e
Note: 22/7 is a rational number (fraction of two integers). It is only an approximation of π, not equal to π.
Common irrational numbers:
√2 = 1.41421356237... → Proved irrational by ancient Greeks (Pythagoras). The hypotenuse of a right triangle with both legs = 1.
√3 = 1.73205080757... → The ratio of the diagonal to the side of a regular hexagon.
π (pi) = 3.14159265358979... → Ratio of circumference to diameter of a circle. Proved irrational in 1761 by Johann Heinrich Lambert. → Note: 22/7 ≈ 3.14285... is only an approximation of π.
e (Euler's number) = 2.71828182845... → Base of natural logarithm. Proved irrational in 1737.
√5 = 2.23606797...
√7 = 2.64575131...
√11 = 3.31662479...
Golden ratio φ (phi) = 1.61803398... = (1+√5)/2
To check if a number is irrational:
If it is a non-terminating, non-repeating decimal → irrational • 1.41421356... → irrational • 3.14159265... → irrational • 0.333... → rational (= 1/3) • 0.142857142857... → rational (= 1/7, repeating)
If it is √n, check if n is a perfect square: • n = 4, 9, 16, 25, 36, 49... → √n is rational • Any other positive integer under √ → irrational
π, e, and most named mathematical constants → irrational (unless proved otherwise)
MCQ strategy: When asked 'Which is NOT an irrational number?', look for: • Square roots of perfect squares: √4, √9, √16, √25... • Cube roots of perfect cubes: ∛8, ∛27, ∛64... • Simple fractions: 2/3, 7/5, -1/4 • Any terminating decimal: 0.5, 0.75, 2.5
√4 is not an irrational number. √4 = 2, which is an integer and therefore a rational number. √2, √3, and √5 are irrational because 2, 3, and 5 are not perfect squares — their square roots are non-terminating, non-repeating decimals.
An irrational number is a real number that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. Its decimal expansion is non-terminating and non-repeating. Examples: √2 = 1.41421356..., √3 = 1.73205..., π = 3.14159265..., e = 2.71828..., √5, √7.
No. 22/7 is a rational number — it is a fraction of two integers. It is commonly used as an approximation for π, but it is NOT equal to π. π itself is irrational (non-terminating, non-repeating). 22/7 = 3.142857142857... (repeating) → rational. π = 3.14159265... (non-repeating) → irrational.
No. √9 = 3, which is an integer and therefore a rational number (3 = 3/1). √9 is NOT irrational. Only square roots of non-perfect-square integers are irrational (like √2, √3, √5, √7).
Rational numbers can be expressed as p/q (integers p and q, q ≠ 0). Their decimals are terminating (0.5) or repeating (0.333...). Irrational numbers cannot be expressed as p/q. Their decimals are non-terminating and non-repeating. Examples: rational — 2, -3, 1/2, 0.75, √4; irrational — √2, √3, π, e.
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