So You Think You Drew √2 Back In School ?

Remember you were taught back in school that √2 has non-terminating, non-repeating decimal expansion, like every irrational number — which is to say, after the decimal point, its digits never end nor follow any pattern. They just flow, they just keep going on forever without any end.It has value like 1.414213562...........

But in the next year, subsequently, or in the same year, you are taught how to draw it on paper. Hallelujah.

The teacher simply draws a right-angled triangle with two sides of length one unit, and boom! — the hypotenuse is √2, guaranteed by the Pythagorean theorem. And you can't complain it's not √2; it has been guaranteed by the Pythagorean theorem itself — which is the OG theorem in math, with the most proofs in math, proven in 300 different ways! infact the great Einstein also came with one & its valid in higher dimensions too ,which is to say its some solid truth in math.

Anyway, you take your ruler, compass, and do as told by your teacher — and this is it. You just drew √2.

But wait a minute —you just drew impossible! if you recall, in the first place, you were taught that √2 has a non-determinate decimal expansion, no? So how did you draw it on paper as a fixed line? or else if you spotting it on number line ,how its a well settled measure? what you did is no less than capturing a djinn in a bottle,i mean how you did it ? How does your teacher do it?

Let me explain.

You have been lied in school, in a way. Hiding half-truth is still a whole lie. You can't truly draw it on paper. What you are claiming is √2 is not √2 in reality. You see it as √2 to be consistent with the Pythagorean theorem. Otherwise, math breaks.

If you remember in the same class, or in the next class, you were taught little Euclid's geometry. Euclid kicks in and gives his four, five postulates, which seem utter nonsense but which govern all of the geometry besides non-euclidean geometry — and one postulate is: a line has no thickness — it has only length, no breadth. And if you just think — when you draw something with your pen or pencil, no matter how sharp — you are always making room for thickness,the error is inevitable.

Which concludes it's not a true math line. It's just a thick physical drawing pretending to be a line.And the number line itself doesn't exist in real sense,it's all smoke and mirrors .

Your teacher never taught you to connect these dots, I bet, might be said you represent it but you're not made conscious about the word represent .

What you have drawn is just a representation of √2, not the actual number. You can't call it an approximation either — by calling it an approximation, you're full-fledgedly dissing the Pythagoreans, whose work guarantee that side is √2. It has a measure of √2, not an approximation. And you can’t break math here.

It is a representation of √2. It may measure like 1.4, or 1.41, or 1.4142 — whatever, depending on your tools. But it's not those values. It is √2, although not √2 numerically. And the blame is on your tools.And you can never achieve that infinite precision sadly.

Moreover, when you draw rational numbers like 1, 2, or even fractions like 1/2, the error margin introduced by your tools is usually so small that we ignore it. The inaccuracy gets absorbed — it doesn’t shake your belief in what you’ve drawn. But the moment you try to draw irrational numbers — things get tricky. The error is no longer just an inconvenience; it becomes a fundamental barrier. You can't claim you've captured √2, π, or e or any other number with infinite decimal expansion, on paper — only a shadow of them.

And if you’ve studied a little physics, you know there is a concept called Planck length — around 1.66 × 10^-35 meters,below that scale, space and time do not exist. And if your tool reaches that concept — which it can't, in practicality — the universe itself does not recognize it. Can’t reach it. That’s for another day.

To draw something in mathematics is like drawing something in your mind. And to draw something in real life are two completely different things. In reality, you can't perfectly draw any number — not even 1, 2, 3 — or any real number you feel in your surroundings. They don’t exist.

Yes, they simplify your life. But they only exist as ideas. They don’t have any form. Like God — they just exist as ideas for now. Maybe they might exist given some more dimensions. I don’t know.

Next time when you pick something to draw, remember: you are not drawing the real thing.Its just a rough shadow of unreachable math truth.

And i dont mean making simple things complex i just mean telling you this if you care.idk.