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

What Are Irrational Numbers?

WHY this matters: Not all numbers fit neatly into fractions. The ancient Greeks discovered this with √2 (side vs diagonal of a square), shattering their belief that all numbers were ratios. This discovery opened up the real number line beyond just rationals.

Contrast with rationals:

Rational: $\frac{1}{2} = 0.5$ , $\frac{1}{3} = 0.333...$ , $\frac{22}{7} = 3.\overline{142857}$ (repeating)
Irrational: $\sqrt{2} = 1.41421356...$ (never repeats), $\pi = 3.14159265...$ (never repeats)

The Big Three Irrational Numbers

√2 — Geometric origin: diagonal of a unit square. Length = $\sqrt{1^2 + 1^2} = \sqrt{2}$ .
π (pi) — Ratio of circumference to diameter of ANY circle. $\pi = 3.14159..$
e (Euler's number) — Base of natural logarithms. $e = 2.71828...$ , appears in growth/decay.

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 √2 IS rational, derive an impossible conclusion, therefore √2 must be irrational.

Formula Proof that

$\sqrt{2}$ is Irrational Step 1: Assume (for contradiction) that $\sqrt{2}$ is rational.

Then $\sqrt{2} = \frac{p}{q}$ where $p, q \in \mathbb{Z}$ , $q \neq 0$ , and $\frac{p}{q}$ is in lowest terms (i.e., $\gcd(p, q) = 1$ , no common factors).

WHY lowest terms? Any fraction can be reduced. If they share a factor, cancel it first. This ensures $p$ and $q$ are not both even.

Step 2: Square both sides. $\left(\sqrt{2}\right)^2 = \left(\frac{p}{q}\right)^2$ $2 = \frac{p^2}{q^2}$ WHY? To eliminate the square root and work with integers.

Step 3: Multiply both sides by $q^2$ . $2q^2 = p^2$

WHY? Now we have an equation relating integers only.

Step 4: Conclude that $p^2$ is even. Since $p^2 = 2q^2$ , $p^2$ is divisible by 2, hence even.

WHY? Any multiple of 2 is even by definition.

Step 5: Conclude that $p$ itself is even.

If $p^2$ is even, then $p$ must be even. (If $p$ were odd, $p = 2k+1$ , then $p^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$ , which is odd — contradiction.)

So $p = 2m$ for some integer $m$ .

WHY this step? Squaring preserves parity. Odd squared is odd, even squared is even.

Step 6: Substitute $p = 2m$ back into $2q^2 = p^2$ . $2q^2 = (2m)^2 = 4m^2$ $q^2 = 2m^2$

WHY? Divide both sides by 2.

Step 7: Conclude that $q^2$ is even, hence $q$ is even.

By the same reasoning as Step 5, $q = 2n$ for some integer $n$ .

WHY? $q^2 = 2m^2$ means $q^2$ is even, so $q$ is even.

Step 8: CONTRADICTION!

We've shown $p$ and $q$ are both even, meaning they share a common factor of 2. But we started by assuming $\frac{p}{q}$ was in lowest terms ( $\gcd(p,q) = 1$ ).

This is impossible. Our assumption must be false.

Therefore, $\sqrt{2}$ cannot be rational. $\sqrt{2}$ is irrational. $\square$

Worked example Worked Example 2: Is √4 Irrational?

No! $\sqrt{4} = 2 = \frac{2}{1}$ , a ratio of integers, so it is rational.

Where does the √2 proof break down for √4? Assume $\sqrt{4} = \frac{p}{q}$ in lowest terms. Squaring gives $4 = \frac{p^2}{q^2}$ , i.e. $p^2 = 4q^2$ . This says $p^2$ is divisible by 4, so $p$ is divisible by 2: write $p = 2m$ , giving $4m^2 = 4q^2$ , hence $m^2 = q^2$ and $m = q$ . Then $p = 2q$ , so $\frac{p}{q} = 2$ — a perfectly valid integer! No contradiction arises.

WHY? The √2 proof relied on $2$ being prime and appearing an ODD number of times when you square (it can't). For $\sqrt{4}$ , $4$ is a perfect square, so the parity argument collapses to a consistent answer. The proof only forces a contradiction when the number under the root is not a perfect square.

Why This Proof Technique Works

The proof by contradiction uses:

Assumption of lowest terms — forces $p, q$ to be coprime (no shared factors).
Parity argument — even/odd properties of squares cascade: $p^2$ even $\Rightarrow$ $p$ even $\Rightarrow$ $q^2$ even $\Rightarrow$ $q$ even.
Infinite descent — if $p, q$ share factor 2, we could divide them out, get a smaller fraction, and repeat forever. Impossible for positive integers.

HOW to remember: "If √2 were a fraction, I could simplify it forever — but integers can't shrink forever. Boom."

Recall Feynman Technique: Explain to a 12-Year-Old

Imagine you're measuring the diagonal of a square tile with a ruler marked in centimeters. If the side is exactly 1 cm, the diagonal is √2 cm.

Now, someone says "√2 is just a fraction — like 14/10 or 1414/1000." But when you square 14/10, you get 196/100 = 1.96, not 2. When you square 1414/1000, you get 1.999396, still not 2.

No matter how precise your fraction, it never squares to exactly 2. Why?

The proof shows that IF √2 were a fraction p/q (simplified so p and q share no factors), then BOTH p and q would have to be even numbers (divisible by 2). But if both are even, they DO share a factor (2)! That's impossible if we already simplified the fraction.

So √2 can't be a fraction at all. It's a number that goes on forever without repeating, slipping between all the fractions on the number line.

Properties of √2, π, e

Number	Value (approx)	Why Irrational	Famous For
√2	1.41421356...	Proved above (parity/contradiction); algebraic	Diagonal of unit square
π	3.14159265...	Irrationality proved by Lambert (c. 1768); transcendence proved by Lindemann (1882)	Circumference/diameter ratio
e	2.71828182...	Irrationality proved by Euler (1737); transcendence proved by Hermite (1873)	Natural growth/decay, $e^x$

Transcendental means not the root of any polynomial with integer coefficients. (√2 satisfies $x^2 - 2 = 0$ , so it's algebraic, not transcendental. π and e are deeper — they are transcendental, and transcendence is a strictly stronger property than mere irrationality.)

Connections

Proof by Contradiction — core technique used here
Greatest Common Divisor (GCD) — lowest terms means gcd(p,q) = 1
Pythagorean Theorem — geometric origin of √2
Rational Numbers — what irrationals are NOT
Real Number Line — rationals + irrationals = reals
Decimal Expansions — non-terminating, non-repeating for irrationals
Prime Factorization — even numbers have factor of 2
Algebraic vs Transcendental Numbers — √2 vs π, e

#flashcards/maths

What is an irrational number? :: A real number that cannot be expressed as a fraction p/q where p, q are integers and q ≠ 0. Its decimal expansion is non-terminating and non-repeating.

Name three famous irrational numbers.

√2, π (pi), and e (Euler's number).

State the key assumption in the proof that √2 is irrational.

Assume √2 = p/q where p, q are integers in lowest terms (gcd(p,q) = 1), then derive a contradiction.

In the √2 irationality proof, after assuming √2 = p/q and squaring, what equation do you get?

2 = p²/q², which rearranges to p² = 2q².

Why does p² = 2q² imply p is even?

p² is even (divisible by 2). If p were odd, p² would be odd. So p must be even, meaning p = 2m for some integer m.

What is the final contradiction in the √2 irrationality proof?

Both p and q are even (share factor 2), contradicting the assumption that p/q was in lowest terms with gcd(p,q) = 1.

Is √4 irrational? Why or why not?

No, √4 = 2 = 2/1, which is a ratio of integers, so it is rational. The √2-style proof fails because 4 is a perfect square, so no contradiction arises.

What does "proof by contradiction" mean?

Assume the opposite of what you want to prove, show that leads to an impossible conclusion, therefore the original statement must be true.

Why can't √2 be approximated "close enough" by a fraction like 1414/1000?

Approximation ≠ equality. (1414/1000)² = 1.999396 ≠ 2. No fraction, no matter how close, squares to exactly 2.

Who proved π is irrational, and who proved it is transcendental?

Lambert proved π irrational (c. 1768); Lindemann proved π transcendental (1882).

Who proved e is irrational, and who proved it is transcendental?

Euler proved e irrational (1737); Hermite proved e transcendental (1873).

What is the difference between algebraic and transcendental irrational numbers?

Algebraic irrationals (like √2) are roots of polynomial equations with integer coefficients. Transcendental irrationals (like π, e) are not roots of any such polynomial; transcendence is strictly stronger than irrationality.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dosto, irrational numbers woh numbers hain jo fraction ke form mein kabhi nahi likh sakte, chahe kitna bhi try karo. Matlab √2, π, aur e — ye sab decimals mein likho toh forever chalte rahenge, aur repeat bhi nahi honge. School mein sabse pehli baar √2 ke sath yeh concept ata hai.

Classical proof bahut clever hai: assume karo ki √2 rational hai, matlab √2 = p/q (lowest terms mein, matlab common factor nahi hai). Phir squaring karne pe dikhta hai ki p aur q dono even hone chahiye (factor 2 common hai), but yeh toh lowest-terms assumption ke against hai! Iska matlab assumption hi galat thi — √2 rational ho hi nahi sakta. Yeh proof-by-contradiction technique bahut powerful hai number theory mein. Dhyan rakho: yeh proof sirf tab kaam karta hai jab number ek perfect square nahi hai — √4 ke liye contradiction nahi milta kyunki 4 perfect square hai.

Real-world mein yeh concept tab aya jab Greeks ne square ka diagonal measure kiya — side 1 unit thi toh diagonal √2 aayi, aur unhone realize kiya ki koi bhi fraction exactly woh length nahi de sakta. Ek important baat: π aur e sirf irrational nahi, balki transcendental bhi hain (Lindemann ne π 1882 mein, Hermite ne e 1873 mein prove kiya). Transcendence ek stronger property hai — matlab yeh kisi bhi integer-coefficient polynomial ki root nahi hain. √2 sirf algebraic hai (x² − 2 = 0 ka root).

Yad rakho: rational numbers dense hain number line pe (har do rational ke bech infinite rationals), but uske bech-bech mein infinite irrational bhi hain. In fact, irrationals zyada hain rationals se (uncountably infinite vs countably infinite)!

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

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Number Theory (Intermediate)

Connections