Irrational numbers — √2, π, e — proof that √2 is irrational

2.5.4 · Maths › Number Theory (Intermediate)

What Are Irrational Numbers?

YEH kyun matter karta hai: Saare numbers neatly fractions mein fit nahi hote. Ancient Greeks ne yeh √2 ke saath discover kiya tha (ek square ki side vs diagonal), aur unka yeh belief toot gaya ki saare numbers ratios hote hain. Is discovery ne real number line ko sirf rationals se aage badha diya.

Rationals se comparison:

Rational: $\frac{1}{2} = 0.5$ , $\frac{1}{3} = 0.333...$ , $\frac{22}{7} = 3. \overline{142857}$ (repeating)
Irrational: $2 = 1.41421356...$ (kabhi repeat nahi karta), $π = 3.14159265...$ (kabhi repeat nahi karta)

The Big Three Irrational Numbers

√2 — Geometric origin: unit square ka diagonal. Length = $1^{2} + 1^{2} = 2$ .
π (pi) — KISI BHI circle ki circumference aur diameter ka ratio. $π = 3.14159..$
e (Euler's number) — Natural logarithms ka base. $e = 2.71828...$ , growth/decay mein aata hai.

Figure — Irrational numbers — √2, π, e — proof that √2 is irrational

The Classic Proof: √2 is Irrational

Strategy: Proof by contradiction (reductio ad absurdum). Assume karo ki √2 rational HAI, ek impossible conclusion nikalao, isliye √2 irrational hona chahiye.

Formula Proof that

$2$ is Irrational Step 1: Assume karo (contradiction ke liye) ki $2$ rational hai.

Tab $2 = \frac{p}{q}$ jahan $p, q \in Z$ , $q \neq = 0$ , aur $\frac{p}{q}$ lowest terms mein hai (yaani $g cd (p, q) = 1$ , koi common factors nahi).

YEH lowest terms kyun? Koi bhi fraction reduce kiya ja sakta hai. Agar unmein koi factor common ho, pehle cancel karo. Yeh ensure karta hai ki $p$ aur $q$ dono even nahi hain.

Step 2: Dono sides ko square karo. $(2)^{2} = (\frac{p}{q})^{2}$ $2 = \frac{p ^{2}}{q ^{2}}$ YEH kyun? Square root hataane ke liye aur integers ke saath kaam karne ke liye.

Step 3: Dono sides ko $q^{2}$ se multiply karo. $2 q^{2} = p^{2}$

YEH kyun? Ab hamare paas sirf integers wali equation hai.

Step 4: Conclude karo ki $p^{2}$ even hai. Kyunki $p^{2} = 2 q^{2}$ hai, $p^{2}$ 2 se divisible hai, isliye even hai.

YEH kyun? 2 ka koi bhi multiple definition se even hota hai.

Step 5: Conclude karo ki $p$ khud even hai.

Agar $p^{2}$ even hai, tab $p$ bhi even hona chahiye. (Agar $p$ odd hota, $p = 2 k + 1$ , tab $p^{2} = 4 k^{2} + 4 k + 1 = 2 (2 k^{2} + 2 k) + 1$ , jo odd hai — contradiction.)

Isliye $p = 2 m$ kisi integer $m$ ke liye.

YEH step kyun? Squaring parity preserve karti hai. Odd ka square odd hota hai, even ka square even hota hai.

Step 6: $p = 2 m$ ko $2 q^{2} = p^{2}$ mein substitute karo. $2 q^{2} = (2 m)^{2} = 4 m^{2}$ $q^{2} = 2 m^{2}$

YEH kyun? Dono sides ko 2 se divide karo.

Step 7: Conclude karo ki $q^{2}$ even hai, isliye $q$ bhi even hai.

Step 5 jaisi hi reasoning se, $q = 2 n$ kisi integer $n$ ke liye.

YEH kyun? $q^{2} = 2 m^{2}$ ka matlab hai $q^{2}$ even hai, isliye $q$ even hai.

Step 8: CONTRADICTION!

Humne dikhaya ki $p$ aur $q$ dono even hain, matlab unka 2 common factor hai. Lekin humne shuru mein assume kiya tha ki $\frac{p}{q}$ lowest terms mein tha ( $g cd (p, q) = 1$ ).

Yeh impossible hai. Humara assumption galat hona chahiye.

Isliye, $2$ rational nahi ho sakta. $2$ irrational hai. $□$

Worked example Worked Example 2: Kya √4 Irrational Hai?

Nahi! $4 = 2 = \frac{2}{1}$ , integers ka ratio hai, isliye yeh rational hai.

√4 ke liye √2 proof kahan fail karta hai? Assume karo $4 = \frac{p}{q}$ lowest terms mein. Square karne par $4 = \frac{p ^{2}}{q ^{2}}$ milta hai, yaani $p^{2} = 4 q^{2}$ . Isse pata chalta hai ki $p^{2}$ 4 se divisible hai, isliye $p$ 2 se divisible hai: likho $p = 2 m$ , jisse $4 m^{2} = 4 q^{2}$ milta hai, isliye $m^{2} = q^{2}$ aur $m = q$ . Tab $p = 2 q$ , isliye $\frac{p}{q} = 2$ — ek bilkul valid integer! Koi contradiction nahi aata.

YEH kyun? √2 proof is baat par depend karta tha ki $2$ prime hai aur square karne par ODD number of times aata hai (jo nahi ho sakta). $4$ ke liye, $4$ ek perfect square hai, isliye parity argument ek consistent answer de deta hai. Proof tab hi contradiction force karta hai jab root ke andar wala number perfect square nahi hota.

Why This Proof Technique Works

Proof by contradiction use karta hai:

Lowest terms ka assumption — force karta hai ki $p, q$ coprime hon (koi shared factors nahi).
Parity argument — squares ki even/odd properties cascade karti hain: $p^{2}$ even $\Rightarrow$ $p$ even $\Rightarrow$ $q^{2}$ even $\Rightarrow$ $q$ even.
Infinite descent — agar $p, q$ factor 2 share karte hain, hum unhe divide out kar sakte hain, ek chota fraction paate hain, aur yeh forever repeat ho sakta hai. Positive integers ke liye impossible hai.

YAAD kaise rakhen: "Agar √2 ek fraction hoti, main use forever simplify kar sakta tha — lekin integers forever shrink nahi ho sakte. Boom."

Recall Feynman Technique: Ek 12-Saal ke Bacche ko Explain Karo

Socho tum ek square tile ka diagonal ek ruler se measure kar rahe ho jo centimeters mein marked hai. Agar side exactly 1 cm hai, diagonal √2 cm hai.

Ab, koi kehta hai "√2 sirf ek fraction hai — jaise 14/10 ya 1414/1000." Lekin jab tum 14/10 ko square karte ho, 196/100 = 1.96 milta hai, 2 nahi. Jab tum 1414/1000 ko square karte ho, 1.999396 milta hai, phir bhi 2 nahi.

Chahe tera fraction kitna bhi precise ho, yeh kabhi exactly 2 tak square nahi hoga. Kyun?

Proof dikhata hai ki AGAR √2 ek fraction p/q hota (simplified taki p aur q koi factor share na karein), tab DONO p aur q ko even numbers (2 se divisible) hona padta. Lekin agar dono even hain, toh unka ek factor common HAI (2)! Yeh impossible hai agar humne already fraction simplify kar liya.

Isliye √2 bilkul bhi fraction nahi ho sakta. Yeh ek aisa number hai jo forever chalta rehta hai bina repeat kiye, number line par saare fractions ke beech se slip karta hua.

Properties of √2, π, e

Number	Value (approx)	Why Irrational	Famous For
√2	1.41421356...	Upar prove hua (parity/contradiction); algebraic	Unit square ka diagonal
π	3.14159265...	Irrationality Lambert ne prove ki (c. 1768); transcendence Lindemann ne prove ki (1882)	Circumference/diameter ratio
e	2.71828182...	Irrationality Euler ne prove ki (1737); transcendence Hermite ne prove ki (1873)	Natural growth/decay, $e^{x}$

Transcendental ka matlab hai ki integer coefficients wale kisi bhi polynomial ka root nahi. (√2, $x^{2} - 2 = 0$ satisfy karta hai, isliye yeh algebraic hai, transcendental nahi. π aur e zyada deep hain — yeh transcendental hain, aur transcendence mere irrationality se strictly stronger property hai.)

Connections

Proof by Contradiction — yahan use ki gayi core technique
Greatest Common Divisor (GCD) — lowest terms ka matlab gcd(p,q) = 1
Pythagorean Theorem — √2 ka geometric origin
Rational Numbers — irrationals jo NAHI hain
Real Number Line — rationals + irrationals = reals
Decimal Expansions — irrationals ke liye non-terminating, non-repeating
Prime Factorization — even numbers ka factor 2 hota hai
Algebraic vs Transcendental Numbers — √2 vs π, e

#flashcards/maths

Irrational number kya hota hai? :: Ek real number jo fraction p/q ke roop mein express nahi kiya ja sakta jahan p, q integers hon aur q ≠ 0. Uska decimal expansion non-terminating aur non-repeating hota hai.

Teen famous irrational numbers ke naam batao.

√2, π (pi), aur e (Euler's number).

Proof mein jo key assumption hai ki √2 irrational hai, woh state karo.

Assume karo ki √2 = p/q jahan p, q integers lowest terms mein hain (gcd(p,q) = 1), phir ek contradiction nikalo.

√2 irrationality proof mein, √2 = p/q assume karke aur square karke kaunsi equation milti hai?

2 = p²/q², jo rearrange hokar p² = 2q² banta hai.

p² = 2q² se p even kyun hota hai?

p² even hai (2 se divisible). Agar p odd hota, toh p² odd hota. Isliye p even hona chahiye, yaani p = 2m kisi integer m ke liye.

√2 irrationality proof mein final contradiction kya hai?

P aur q dono even hain (factor 2 share karte hain), jo assumption ko contradict karta hai ki p/q lowest terms mein tha jahan gcd(p,q) = 1.

Kya √4 irrational hai? Kyun ya kyun nahi?

Nahi, √4 = 2 = 2/1, jo integers ka ratio hai, isliye yeh rational hai. √2-style proof fail karta hai kyunki 4 ek perfect square hai, isliye koi contradiction nahi aata.

"Proof by contradiction" ka kya matlab hai?

Jo prove karna chahte ho uska opposite assume karo, dikhao ki isse ek impossible conclusion aata hai, isliye original statement true honi chahiye.

√2 ko 1414/1000 jaisi fraction se "close enough" approximate kyun nahi kiya ja sakta?

Approximation ≠ equality. (1414/1000)² = 1.999396 ≠ 2. Koi fraction, chahe kitna bhi close ho, exactly 2 tak square nahi hota.

π irrational kisne prove kiya, aur transcendental kisne prove kiya?

Lambert ne π irrational prove kiya (c. 1768); Lindemann ne π transcendental prove kiya (1882).

e irrational kisne prove kiya, aur transcendental kisne prove kiya?

Euler ne e irrational prove kiya (1737); Hermite ne e transcendental prove kiya (1873).

Algebraic aur transcendental irrational numbers mein kya fark hai?

Algebraic irrationals (jaise √2) integer coefficients wali polynomial equations ke roots hote hain. Transcendental irrationals (jaise π, e) kisi aisi polynomial ke root nahi hote; transcendence strictly stronger property hai mere irrationality se.

Irrational numbers — √2, π, e — proof that √2 is irrational

What Are Irrational Numbers?

The Big Three Irrational Numbers

The Classic Proof: √2 is Irrational

Why This Proof Technique Works

Properties of √2, π, e

Connections

Concept Map

Deep Dive