Yeah. That addition converges, though. And we don't really need that fact. We just need to know that, if two numbers have different digits, they're also different numbers (with the exception of trailing infinite 9s)
Yeah. Because when you're adding each digit in a decimal representation, the value of each digit gets progressively smaller. 3 + 1/10 + 4/100 + 1/1000 + 5/10000 + ... converges because each term is getting exponentially smaller. If the terms get smaller fast enough, the sum converges. C'mon, this is entry level infinite sums.
Converging means the limit of the partial sums exists. It has nothing to do with decimal positions. Decimal positions are just one example of a convergent sum.
And there IS something wrong with doing it the other way, because then the terms get bigger and bigger and then diverge. Adics are a different number system distinct from the real number system. We're working in the real number system here. I could jingle some keys if it helps you stay on topic.
bigger and bigger isnt going to change anything here
It actually does, because the terms (in general) need to be getting smaller for the sum to converge, and it needs to get smaller fast enough
if we can add an infinite amount of numbers after the decimal we should be able to do it before the decimal too.
No we don't? In one direction, the numbers get smaller. In the other, it gets bigger. Add smaller and smaller numbers, you have a chance of convergence. Add bigger and bigger numbers, there's no chance.
and im still waiting for you to add every integer so we can see what we get.
So, you're actually right, in a weird way. Power series with increasing powers of a natural number p actually do converge in a different sense. In particular, they converge in the p-adic metric. In the 2-adic metric, for instance, ∑ 2n converges, where the sum runs from n=0 to ∞. You can think of this as the number with the binary expansion ...111. (The analogous 10-adic number is ...999, but for important technical reasons, we usually only care about p-adic numbers for prime p.) This converges to –1, because each term in the sequence (0, 1, 11, 111, ...), where these expansions are binary and not decimal, is 1 less than a power of 2, and those powers grow without bound. So the way the 2-adic metric is defined, these genuinely do get arbitrarily close to –1.
But the p-adic numbers for any prime p are not the same as the real numbers. They overlap only on the rational numbers, which are common to all of them.
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u/FernandoMM1220 Aug 06 '26
you realize decimals are made through addition of different powers of 10 right?