Teaching Kids Programming – Two Ways to Prove Square Root of Two is Irrational (proof by contradiction and geometric infinite descent)

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How to Prove √2 Is Irrational — Two Methods (Proof by Contradiction & Geometric Infinite Descent)

Introduction

The statement “√2 is irrational” means there are no integers and with such that .

Method 1 — Proof by Contradiction

Assume, for contradiction, that is rational. Then there exist integers and with and such that

Squaring both sides gives:

From we see that is even, so must be even. Write for some integer .

Substitute back:

Thus is even, so is even.

Therefore both and are even, contradicting the assumption that . This contradiction shows the assumption was false; hence is irrational.

Method 2 — Geometric Infinite Descent (Median / Perpendicular Construction) geometric construction and argument

Consider an isosceles right triangle with right angle at , legs and hypotenuse , so .

Let be the midpoint of , and draw the line through perpendicular to the hypotenuse , meeting at .

Then is an isosceles right triangle similar to . Denote the small triangle’s hypotenuse and leg by and .

, confirming .

The crucial point is similarity (a fixed scale), not literal subtraction of whole sides. The small triangle is a constant fraction of the original.

From the , takes the middle point , and draw a perpendicular line to the hypotenuse . So the is similar to . We use the repeative subtraction to compute the . because is equal to , Thus .

Then . Thus: which is where is and is . And we can continue this process infinity as the is similar to .

Hence every time you repeat the same construction (take the midpoint of a triangle leg and draw the perpendicular to the hypotenuse), you obtain a new isosceles right triangle similar to the original with all side lengths multiplied by a factor . Thus the hypotenuse and legs shrink geometrically by a fixed factor at each step.

This produces the infinite-descent contradiction in the integer setting: if you started with integer side lengths satisfying , the construction (or the equivalent integer-preserving midpoint-of-hypotenuse → midpoint-of- construction that yields a strictly smaller positive integer solution of the same Diophantine equation. Repeating indefinitely would produce an infinite strictly decreasing sequence of positive integers, which is impossible. Therefore no such integer solution exists and is irrational.

Conclusion

Both methods prove that cannot be written as a ratio of integers.

• Method 1 (proof by contradiction) uses parity to show any assumed fraction in lowest terms leads to both numerator and denominator being even.
• Method 2 (geometric infinite descent) uses a similarity construction inside an isosceles right triangle to produce a smaller integer solution, contradicting minimality.

Either argument gives a clear, rigorous proof that is irrational.

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