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Rational and irrational numbers on the number line
01/31/2√2/21√2φ2π

Blue: rational numbers (exact fractions). Red: irrational numbers (non-repeating decimals). Between any two rationals lies an irrational, and vice versa.

Geometric proof: √2 is irrational
Unit square diagonal = √2. Assume √2 = p/q (lowest terms).
Then 2 = p²/q², so p² = 2q² — p² is even, so p is even. Write p = 2k.
Then 4k² = 2q², so q² = 2k² — q is also even. Contradicts p/q in lowest terms. ∎
Decimal expansions: how to spot the difference

Comparison table of rational numbers with repeating or terminating decimals versus irrational numbers with non-repeating non-terminating decimals

RASIONAL: berakhir atau berulangIRASIONAL: tidak pernah berulang
1/4 = 0,25000...√2 = 1,4142135...
endetkein Muster, niemals
1/3 = 0,3333...π = 3,1415926...
periodischer Block: {3}kein Muster, niemals
22/7 = 3,142857...e = 2,7182818...
periodischer Block: {142857}kein Muster, niemals
5/11 = 0,454545...φ = 1,6180339...
periodischer Block: {45}kein Muster, niemals
How many irrationals are there compared to rationals?
REAL NUMBERS R (uncountable) Rationals Q (countable) 1/2, 3/7, -5, 0... Irrationals (uncountably more numerous) sqrt(2), pi, e, phi... Cantor (1874): |Irrationals| is strictly and infinitely larger than |Rationals|

The rational numbers, despite being infinitely numerous, can be listed (they are countable). The irrationals cannot be listed. If you picked a real number at random, the probability of it being rational is exactly zero.

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