√2 is not a fraction
Somewhere between 1.4 and 1.5 is a number whose square is exactly 2. Try to write it as a fraction, watch a search fail, then see a picture that shows why every search must. About five minutes.
1. A square of area 2
Put four unit squares together. The tilted square inside them is made of four half-squares, so its area is exactly 2. Its side is the diagonal of one unit square. So that diagonal, times itself, is 2. Its length is what we call √2.
Is there a fraction, one whole number over another, whose square is exactly 2?
2. Search for it
If √2 were a fraction p/q, then p × p would equal exactly 2 × q × q. So check every denominator q from 1 to 10,000, take the nearest whole-number p each time, and measure the miss: p² − 2q². A miss of 0 would be a hit.
No hits in 10,000 denominators. The fractions get astonishingly close: 8119/5741 matches √2 to seven decimal places. But the miss p² − 2q² is a whole number, so it can't creep towards 0. It's either 0 or at least 1 away, and every near miss in the list is off by 1 or 2.
Still, a search can only fail to find. It can't show there's nothing to find: the answer could be at q = 10,001. For that, you need a proof.
3. Why no search can succeed
Suppose some whole numbers did work: p² = 2q². Then a square of side p has exactly the same area as two squares of side q. Put the two smaller squares inside the big one, in opposite corners.
They overlap in the middle, and they leave two corners uncovered. The areas are equal, so the doubly covered middle must exactly make up for the two uncovered corners. That means a square of side 2q − p has the same area as two squares of side p − q. That's a new, smaller solution in whole numbers.
Do it again with the new, smaller squares, and you get a smaller solution still, and again, forever. But whole numbers can't get smaller forever: counting down, you reach 1 in finitely many steps. So the first solution can't have existed. No fraction squares to 2.
p² = 2q² ⟹ (2q − p)² = 2(p − q)²
Two small facts hold this up, and it's fair to name them. Because √2 is between 1 and 2, the new numbers are positive and smaller than the old ones. And a list of whole numbers that keeps getting smaller must stop. That second fact is where the proof really rests.
4. Something new
A friend runs the same search for √3, up to a million denominators. No fraction squares to exactly 3. They announce that they've proved √3 is not a fraction.
Has their search proved that √3 is not a fraction?
√3 isn't a fraction, but the search didn't prove it. A search checks the cases it reaches and says nothing about the rest. A proof like the squares argument covers every whole number at once.
The same kind of argument works for √3, √5, and the square root of any whole number that isn't itself a perfect square.