ย 

Irrational Numbers

Numbers that never end, never repeat

๐Ÿ“– Definition
๐Ÿ“–
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 โ‰  0.

๐Ÿ‘‰ In simple words: An irrational number cannot be expressed as a simple fraction. Its decimal form goes on forever without repeating!
๐Ÿ”ข
Famous Example โ€” Pi (ฯ€)
Pi (ฯ€) in action
ฯ€ = 3.14159265358979323846264338327950288โ€ฆ
โ†‘ Digits continue forever and never repeat in any pattern.
๐Ÿ’ก
More Examples
โˆš2
Square root of 2
โ‰ˆ 1.41421356237โ€ฆ never ends, never repeats
e
Euler's Number
โ‰ˆ 2.71828182845โ€ฆ never ends, never repeats
ฯ†
Golden Ratio
โ‰ˆ 1.61803398874โ€ฆ never ends, never repeats
๐Ÿ”น Properties
๐Ÿ”น
Key Properties of Irrational Numbers
1
Non-Terminating
The decimal expansion never ends โ€” 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.
โš™๏ธ
Operations with Irrationals
Sum / Difference
Rational + Irrational = Irrational  |  Irrational + Irrational = may be rational or irrational
Product
Rational ร— Irrational = Irrational (if rational โ‰  0)  |  โˆš2 ร— โˆš2 = 2 (Rational!)
Comparison
Irrational numbers can still be compared and placed on a number line.
โœจ Examples
โˆž
Irrational Numbers
โˆš2
โˆš3
โˆš5
โˆš7
ฯ€
e
ฯ†
โˆ›2
โˆ›5
log 2
โˆš11
2ฯ€
eยฒ
โˆœ3
โˆš2
โˆš2 (Root 2)
โ‰ˆ 1.41421356237โ€ฆ โ€” proof: assume pยฒ/qยฒ=2, leads to contradiction
ฯ€
Pi (ฯ€)
โ‰ˆ 3.14159265358โ€ฆ โ€” ratio of circumference to diameter
e
Euler's Number
โ‰ˆ 2.71828182845โ€ฆ โ€” base of natural logarithm
ฯ†
Golden Ratio (ฯ†)
โ‰ˆ 1.61803398874โ€ฆ โ€” found in nature, art, architecture
โœ“
Rational Numbers (NOT Irrational)
ยฝ
ยพ
0.5
โˆš9
22/7
0.333โ€ฆ
7
โˆ’6
0
โˆš4
1.25
โˆš9
โˆš9 = 3
Perfect square โ†’ whole number โ†’ Rational
22/7
22/7 (approx. of ฯ€)
= 3.142857142857โ€ฆ repeating โ€” NOT equal to ฯ€, is Rational
0.3ฬ„
0.333โ€ฆ (repeating)
= 1/3 โ€” repeating decimal โ†’ Rational
๐ŸŽฎ Sort: Rational vs Irrational
Round 1 / 10
๐ŸŽ‰
Round 1 of 10
All Sorted!
You correctly placed all numbers!
๐Ÿ‘† Drag each number into the correct column below
โœ“ Rational
p/q ยท repeating decimals
Drop here
VS
โˆž Irrational
never-ending ยท never-repeating
Drop here