Alright. Would you agree that there is a first (positive) odd number, a second odd number, a third, and so forth? Indeed, for any n, could you not find the n-th odd number if they are arranged from least to greatest?
They are called “imaginary” for a reason
Imagine the whole natural numbers going to infinity, then the odd numbers next to them which of course skipping the even numbers. You are pulling the line of odd numbers and then saying they must be equal to the odd plus even. This is erroneous. Your argument that, for any odd number, you can find a whole natural number only shows that the odd numbers are a part of the whole natural number series. You can line up 3 with 2 instead of with 3 but there is still twice as many in the series of odd plus even. By your method there is no larger infinities, because every infinity goes on forever, and you can keep picking numbers from both infinities and putting them together, but one can still be larger because if I put 1 to 1, 3 to 2, 5 to 3, 7 to 4, and 9 to 5, I’ve ignored 5 numbers in the larger series
And what reason is that?
By the definitions of math, nothing times itself equals a negative number. Unless you are saying that rule is not based on anything
It is not a definition.
You didn’t answer my question. A simple “yes” or “no” would suffice. Can you or can you not associate with each positive integer n the n-th positive odd number, and conversely, can you not say that each positive odd number is the first, or the second, or third, etc?
By the definitions of mathematics, i * i = -1. That is how i is defined.
Are we supposed to be teaching high school math in this thread?
I systematically answered that in post 339. The odd numbers only line up with the odd or even numbers, not both. Bottom lineYou didn’t answer my question. A simple “yes” or “no” would suffice. Can you or can you not associate with each positive integer n the n-th positive odd number, and conversely, can you not say that each positive odd number is the first, or the second, or third, etc?
If you can, then we have a one-to-one correspondence. Now you can disagree with using the definition of “countable infinity” and say that it’s a stupid distinction to make. You can insist that nothing is gained by distinguishing various infinities in this way (I think you’d be wrong, but whatever). But if we’re using the definition that a set is countable if its members can be placed in one-to-one correspondence with the positive integers, then the odds are definitely countable if the above is admitted. It’s over, there’s nothing left to argue after that. Your disagreement lies with definitions, not the reasoning.
So is the principle “a negative times a negative is a positive” always TRUE or not
I am more revolutionary than you, willing to admit that something in math these days is wrong. Someday quantum physics, now accepted, may be refuted and brought back to classical physics. “Follow the truth wherever it leads you”
How would you explain the Stern Gerlach experiment or the quantum hall effect using classical physics?
It is true in the real numbers. It may or may not be true elsewhere. Hence, your “always” is incorrect.
There is no “lining up”. We are discussing one-to-one correspondences. Bijections, if you prefer. Am I speaking a foreign language?
=2n-1, where n can assume integer values, is f a bijection? It must be, as it is a linear function with non-zero slope. Its range is the set of odd numbers, so there is a one-to-one correspondence between the integers and the odd numbers.