Ask ChatGPT why 0.999⋯ is equal to 1, and it will give you two proofs: one with algebraic manipulation, and the other using infinite series.
First common proof: Multiplying 0.999⋯ by ten and then subtracting 0.999⋯ from it gives just 9. This is supposed to be nine (=10–1) times the original value. If nine lots of something is 9, one lot of it is obviously 1, thereby proving 0.999⋯ is actually 1. While this may be mathematically sound, it honestly feels like one-time special-case trickery, albeit a clever one. A solution with zero intuitiveness, a dead end to any learning path.
The series-based proof is also equally unintuitive where I am told — okay, reminded — that the sum of 0.9+0.09+0.009+ and so on, equals a/(1-r) where a is the first term (=0.9) and r is 10% of each previous term (=0.1). Solving for the sum returns 1. Understanding this requires rummaging through high-school maths textbooks or their digital equivalents. Again, far from an intuition-based approach.
Intuition, in this case, is based on the notion of inequality. How can a number that starts with a zero ever be equal to 1? It simply can’t, right?
Here is my non-mathematician, intuition-based approach to this problem, one that anyone can follow through. Let’s suppose 0.999⋯ is indeed strictly less than 1. This implies that there is a positive non-zero k representing the gap between 0.999⋯ and 1.
Next, extend the number line slightly and look on the other side of the fence. There must be a number along this line that is strictly greater than 1 where the gap is equal to k. This number, 1+k, is the mirror image of 0.999⋯, with 1 placed exactly halfway in between. As such, 0.999⋯ can also be written as 1-k (obviously).
Now, let’s attempt to work out what k is from the right side of 1. Is 1+k equal to 1.1? No way; it’s much smaller than that. Is it equal to 1.01? Nope. 1.001? 1.0001? 1.00001? No, no, and no. In other words:
k < 0.1
k < 0.01
k < 0.001
k < 0.0001 … and so on.
What is the smallest positive non-zero number that can be compared to k, then? Whatever it is, it must have a 1 after some finite number of 0’s repeating; for example: 0.0000000000000001. Yet, k is still smaller than that number because the string of 0’s in k continues into infinity just as the string of 9’s in 0.999⋯ continues into infinity. This can only mean that k is less than any positive non-zero number along the line of real numbers. By implication, the only possible value of k representing the gap is zero.
If the supposed gap between 1 and 1+k is zero, then the gap between 1 and 1-k is also zero. In other words, there is zero gap between 1 and 0.999⋯.
This proof began by assuming that there is a positive non-zero gap between 0.999⋯ and 1. That assumption is now evidently contradicted. Therefore, with the absence of any gap, 0.999⋯ is equal to 1.
I can follow this through better than what ChatGPT spits out.
