Irrational Numbers

Numbers that never end, never repeat

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What is an Irrational Number?

An irrational number is a number that cannot be written in the form p q where p and q are integers and q ≠ 0.

👉 In simple words: An irrational number cannot be expressed as a simple fraction. Its decimal form goes on forever without repeating!
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Famous Example — Pi (π)
Pi (π) in action
π= 3.14159265358979323846264338327950288…
↑ Digits continue forever and never repeat in any pattern.
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More Examples
√2
Square root of 2
≈ 1.41421356237… never ends, never repeats
e
Euler's Number
≈ 2.71828182845… never ends, never repeats
φ
Golden Ratio
≈ 1.61803398874… never ends, never repeats
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Key Properties of Irrational Numbers
1
Non-Terminating
The decimal expansion never ends — digits go on infinitely.
2
Non-Repeating
Digits never repeat in any fixed or predictable pattern.
3
Not a Fraction
Cannot be written as p/q where p, q are integers.
4
Real Number
All irrational numbers are real numbers on the number line.
5
No Integer Ratio
Cannot be expressed as a ratio of two integers.
6
Dense on Line
Irrational numbers are densely packed on the real number line.
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Operations with Irrationals
Sum / Difference
Rational + Irrational = Irrational  |  Irrational + Irrational = may be rational or irrational
Product
Rational × Irrational = Irrational (if rational ≠ 0)  |  √2 × √2 = 2 (Rational!)
Comparison
Irrational numbers can still be compared and placed on a number line.
∞
Irrational Numbers
√2
√3
√5
√7
π
e
φ
∛2
∛5
log 2
√11
2π
e²
∜3
√2
√2 (Root 2)
≈ 1.41421356237… — proof: assume p²/q²=2, leads to contradiction
π
Pi (π)
≈ 3.14159265358… — ratio of circumference to diameter
e
Euler's Number
≈ 2.71828182845… — base of natural logarithm
φ
Golden Ratio (φ)
≈ 1.61803398874… — found in nature, art, architecture
✓
Rational Numbers (NOT Irrational)
½
¾
0.5
√9
22/7
0.333…
7
−6
0
√4
1.25
√9
√9 = 3
Perfect square → whole number → Rational
22/7
22/7 (approx. of π)
= 3.142857142857… repeating — NOT equal to π, is Rational
0.3̄
0.333… (repeating)
= 1/3 — repeating decimal → Rational
Round 1 / 10
⭐ 0
0 / 10 sorted
⭐⭐⭐
🎉
All Sorted!
You correctly placed all numbers!
+100 pts
Drag each number to the correct column
✓
Rational
p/q · repeats · terminates
0
Drop here
VS
∞
Irrational
never-ending · never-repeating
0
Drop here