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Mental arithmetic is all little tricks and shortcuts. If the answer is right then there's no wrong way to do it, and maths is one of the few places where answers are right or wrong with no damn maybes!
Well, there are certainly wrong ways to arrive at the answer, e.g. calculating 2+2 by multiplying both numbers still gets you 4 but that is the wrong way to get there. That doesn't apply to any of the methods in the post though.
Unsolved problems do not all fall into binary outcomes. They can be independent of axioms (the set of assumptions used to construct a proof).
I like your funny words, mathemagic man
Unless you consider probabilities. That's a very strange field—you can't objectively verify it.
You can't objectively verify anything in mathematics. It's a formal system.
Once you start talking about objective verification, you're talking about science not math.
It is actually the opposite, since it is purely abstract everything in math is objective. There is literally no subjectivity possible in something that isn't in the real world.
That's also all common core is. Instead of teaching the line up method which requires paper and is generally impractical in the real world, they teach ways to do math in your head efficiently.
What is "common core" and what is the "line up method"?
Hmm, you seem to be completely discounting calculus, where a given problem may have 0, 1, 2, or infinite solutions. Or math involving quantum states.
In math, an answer is either right, wrong, or partially right (but incomplete).
Quantum states is physics, not math.
And mathematically a probabilistic theorem is still a theorem.
Yes, but physics is math with more variables.
But there's plenty of math related to quantum states that can make sense, such as if you know a given machine will give the right answer 51% of the time, and you want to know how many iterations you'll need to get a certain confidence that you are seeing the correct answer. That's basic statistics, which is also math, but it's relevant to quantum states in that you're evaluating a computing system based on qubits.
Those are quite far from mental arithmetic though
Calculus is generally pretty easy to do mental arithmetic on, especially when talking about real-world situations, like estimating the acceleration of a car or something. Those could have multiple answers, but one won't apply (i.e. cars are assumed to be going forward, so negative speed/acceleration doesn't make much sense, unless braking).
Math w/ quantum states is a bit less applicable, but doing some statics in your head for determining how many samples you need for a given confidence in a quantum calculation (essentially just some stats and an integral) could fit as mental math if it's your job to estimate costs. Quantum capacity is expensive, after all...