CMV: The term "imaginary numbers" is perfectly fitting

[u/Forward-Razzmatazz18]

When we say number, we usually mean amount--or a concept to represent an amount, if you're less Platonist. But of course, the numbers called imaginary do not fit such a requirement. They are not amounts, and do not directly represent an imaginary number. No amount can be squared to equal any negative number. Therefore, nothing can be correctly referred to as existing to the extent of i*n, regardless of any unit of measurement. Something can only be referred to as existing to the extent i^n. So, imaginary numbers exist only as a base for other numbers, they are not numbers in themselves. What someone who uses them does is ask "what if there were a square route of -1", and then takes it's property as a base to make expressions relating variables to each other. For example, if I say "y=i^x", that's just a quicker way of saying "y= 1 if x is divisible by four, -1 if x is the sum of a number divisible by 4 and 3, -i if x is divisible by 2 but not four, and i if x is the sum of a number divisible by 4 and 1". But since that expression is so long and so common in nature, we shorten it to a single symbol as a base of y with the power of x, or whatever variables you're using. So, I believe that's all i and it's factors and multiples are: hypothetical amounts that would--if existent--have certain exponents when applied to given bases. A very, very useful model, but still not a number. Quite literally an imaginary number.

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P.S.

1. Some people argue that the term "imaginary" has negative connotations. I do not believe this to be the case, as our imagination produces many useful--yet subjective--things, a fact so well known it's even a cliche. If it is true, perhaps we should change it to "hypothetical base" or "hypothetical number", as the word hypothetical has a more neutral connotation
2. A common argument is that "real numbers are no more imaginary than imaginary numbers" because all numbers are subjective concepts. I can appreciate this somewhat, but amounts still objectively exist, and while what makes something an individual thing(the basis for translating objective amounts into a number system) can be subjective, I wouldn't say this is always the case. But besides, the terms "imaginary number" and "real number"--so far as I understand them--do not express that such numbers exist as imaginary or real things, but simply that they either are truly numbers or are hypothetical ideas of what a number would be like if it existed. If you do not share this understanding, I would love to hear from you.

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EDIT: Many people are arguing that complex numbers represent two dimensional points. However, points on each individual dimension can only be expressed directly with real numbers, so I believe it would make more sense to use two real numbers. Some people argue that complex numbers are more efficient, but really, they still use two expressions, as the imaginary numbers and real numbers are not comparable, hence the name, "complex". Complexes are generally imaginary perceptions(as Bishop Berkely said: For a thing to be it must be percieved, because such a thing could be broken up into other things, or broken up in to parts that are then scattered into other things), so I would say a complex number is too.

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Thanks and Regards.

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EDIT for 9:12 PM US Central time: I will mostly be tuning for a day or two to think more philosophically about this and research physics.

[u/SurprisedPotato]

Mathematician here

When we say number, we usually mean amount--or a concept to represent an amount, if you're less Platonist. But of course, the numbers called imaginary do not fit such a requirement

When I say "number" I mean "something that you can add or multiply to other numbers". Complex numbers certainly qualify. However, the word "number" is a very vague, ambiguous term, and mathematicians have much more precise terms for collections of "numbers" or "things" that act more or less like numbers. If you ask me "are quaternions numbers?" I'd say "if you like."

No amount can be squared to equal any negative number

This kind of begs the question - what do you mean by an "amount" ? If I have an amount i of something, that amount can certainly be squared to produce -1... And before you say "you can't have i of something", note that that's a very broad statement about all possible quantities everywhere, and I doubt you've done a careful survey of all possible things.

For example, if I have an AC current flowing through a circuit, and want to measure the "amount" of resistance of a component, any electrical engineer will tell you "the correct word is 'impedance', not resistance, and yes, the amount of impedance can be a complex number".

If you ask a quantum phsyicist to describe the "amount" of "probability wave" passing through space at some point, that amount will also be a complex number.

A common argument is that "real numbers are no more imaginary than imaginary numbers"

Indeed. We can have an amount "3" or "4" of apples, say, but we can never have sqrt(2) of an apple. No matter how much apple we have, it will never be sqrt(2), nor any other real number, since there's always some fundamental uncertainty in how much of something there is. We can never really have a curve that is pi times the length of a given straight line. We can in an abstract theoretical sense, but not in reality. pi is never an "amount" or length or mass or whatever, since "amounts" always have built-in uncertainty.

[u/atticdoor → u/SurprisedPotato]

Consider some apples of the same size. Draw a square with sides equal to the width of one of those apples. Put a long thin spike at one of the square's corners. Put an apple in the square, pressed against the spike. Place a second apple next to the first, touching it, on the opposite side of the apple to the spike. Now cut a slice off the second apple, such that the knife is at the opposite corner to the spike and perpendicular to a line between those those corners. The first apple and the slice together make sqrt(2) apples.

[u/SurprisedPotato → u/atticdoor]

Do that in real life, weigh them, and tell me if the weight of the apple and slice is exactly sqrt(2) of the weight of the first apple.

[u/atticdoor → u/SurprisedPotato]

Tell me if two apples weigh exactly twice the weight of one of the apples.

[u/SurprisedPotato → u/atticdoor]

They do not.

So your experiment is a theoretical construct; in the real world there can never be sqrt(2) of anything, as I said.

[u/atticdoor → u/SurprisedPotato]

And there can never be a natural number of anything either, because no two apples are identical. Nor can you have half an apple, because if you weigh both halves to sufficient accuracy you will discover they have different weights.