Theoretical Math: Accept the Absurdity or Break Reality (And I Hate It!)

The 0.999... "Thing"

I recently found myself distressed by a seemingly simple math problem (which turned out to be more complex than I thought): Is 0.9999... less than or equal to 1?

The question seems simple at first, even a bit foolish. The answer appears obvious at first glance: of course, 0.9999... would be less than 1.

Let’s imagine a number line, with smaller numbers positioned to the left and larger numbers to the right.

Now, if we were to place the number 0.9999... on this number line alongside the number 1, it would seem straightforward. We’d place 0.9999... just to the left of 1 on the number line.

So Calum, what is the problem here?

The problem here is, but how far apart?

The number 0.9999... is INFINITELY close to 1.

But mathematically, 0.9999... is, in fact, equal to 1.

Many brilliant mathmaticians have found ways to make this arguments a sound one. Let's take a look at an algebraic demonstration to this argument.

Let's begin with X = 0.9999...

Now, let's multiply both sides by 10 to get the second equations 10X = 9.9999...

Let's subtract the first equation from the new equation 10X - X = 9.9999... - 0.9999...

That will leave us with 9x = 9

You see where this is heading!

Divide both sides by 9

9X/9 = 9/9 and we have X = 1

Therefore, we have established that:

(P1) x = 0.9999...

(P2) x = 1

∴ (C) 0.9999... = 1

This elegant proof demonstrates one of mathematics' most fascinating properties - that 0.9999... and 1 are actually the same number.

While this might seem counterintuitive at first glance, the algebraic approach shown above provides clear and irrefutable evidence of their equality.

This isn't just a mathematical trick; it's a fundamental truth that challenges our intuitive understanding of numbers and reveals the beautiful precision of mathematical logic.