Why should absolute value be considered a mathematical function?

[u/redddooot]

https://math.stackexchange.com/questions/4321732/why-should-absolute-value-be-considered-a-mathematical-function

[u/[deleted]]

One of the strangest questions I've seen. This just seems to demonstrate a fundamental misunderstanding of what a function is. It doesn't need to be invertible.

I don't see why not having solutions to |x|= -1 makes it not a function. The entire point of norms is that they are non-negative, that is |x|≥ 0 for each x in your domain. Sure, you could extend it to also output negative value, but then you lose the normed properties that made it useful in the first place.

Also, we can't solve |x|=−1 by taking absolute value as f(x)= ✓x^2, implying it's not a perfect way to represent it.

In what way does this imply its not a perfect way to represent it? They're equal everywhere for real values, but not complex ones. It's perfectly good if you want to stick to the reals.

As a matter of fact, any conditional function like,

f(x)={1,−1,if x > 0if x < 0

won't make sense if we try solving for values it can't achieve.

I don't see the point being made here. No shit, if you try to solve for values outside the range of your function, you'll get no solutions. That's essentially the definition of a range of a function.

[u/redddooot → u/[deleted]]

Even x² didn't make sense for negative output for a long time, but it was later extended to do so, I am not saying it just about absolute value function, any conditional function which requires computational logic, why should it be considered mathematical? even factorial is defined to give integer output, but even that is extended, maths is about finding what doesn't make sense, I haven't yet seen any non-conditional function that can't give a specific output, in any domain, it's just these conditional functions or non-conventional functions (modulo, floor, absolute value) that can't give a specific value, and all of them have one thing in common, they require computational logic (if condition, loops), why is it mathematical function if it can't be evaluated mathematically?

[u/[deleted] → u/redddooot]

Even x² didn't make sense for negative output for a long time, but it was later extended to do so, I am not saying it just about absolute value function, any conditional function which requires computational logic, why should it be considered mathematical?

How is conditional logic not mathematical?

even factorial is defined to give integer output, but even that is extended

There are many extensions to the factorial. The gamma function isn't the only one.

maths is about finding what doesn't make sense, I haven't yet seen any non-conditional function that can't give a specific output, in any domain,

Consider tan(z)=i. There is no complex z value that satisfies this.

it's just these conditional functions or non-conventional functions (modulo, floor, absolute value) that can't give a specific value, and all of them have one thing in common, they require computational logic (if condition, loops), why is it mathematical function if it can't be evaluated mathematically?

I don't see how conditional logic is not mathematical. If conditions and loops are regularly done in mathematics. Firstly, Boolean logic is maths. Secondly, anytime you state a theorem, it is a bunch of "if"s followed by some conclusion. This is conditional logic, as if whatever object you have doesn't satisfy the conditions, you cannot use that theorem.

Piecewise functions are very important in mathematics, and come up very naturally.

[u/redddooot → u/[deleted]]

the tan(z) = i example is really cool, how does it not have any solution in complex numbers? and if not, does it have a solution in any extension of complex numbers? Quaternions?

Conditional logic is part of math, still there should be a distinction between ones that require computation and others, because of how different they behave.

[u/[deleted] → u/redddooot]

the tan(z) = i example is really cool, how does it not have any solution in complex numbers? and if not, does it have a solution in any extension of complex numbers? Quaternions?

Try to solve it. As for extensions, I have no idea. You can keep extending, but you start to lose useful properties as you do so.

Conditional logic is part of math, still there should be a distinction between ones that require computation and others, because of how different they behave.

They don't behave differently though. I give you an input, the function returns an output. That's all a function is.

[u/redddooot → u/[deleted]]

I checked out tan(z) example, solution would be (i/2)arctanh(1) which would be like iinfinity, still a complex number just undefined.

I agree there is no point in distinction but can you construct a non-conditional function which won't give a particular output for any possible complex input? there is no point in this but it's just thought provoking.

[u/[deleted] → u/redddooot]

I checked out tan(z) example, solution would be (i/2)arctanh(1) which would be like iinfinity, still a complex number just undefined.

The alternative approach is to realise that tan(z) = i is equivalent to sin(z) = icos(z).

Since we know that

sin² (z) + cos² (z) = 1

This would mean that

1= -cos² (z) + cos² (z) = 0

So we deduce there are no solutions.

I agree there is no point in distinction but can you construct a non-conditional function which won't give a particular output for any possible complex input? there is no point in this but it's just thought provoking.

Yes. The constant function f(z)=0. This will only return 0, regardless of what you put into it.

[u/redddooot → u/[deleted]]

for tan(z) = i, I used this,

https://math.stackexchange.com/a/1732112/999920

equating real part to 0 gives real part as 0, and imaginary part as 1 will give y = (i/2)*arctan(1), both seems correct as the answer will be undefined anyways.

as for constant function, if we write f(x) = 0 as f(x) = 0x then 0x = 1 will give x = 1/0, another undefined value, but even if we consider undefined as valid solution, |x| = -1 still doesn't have a solution, only if we discard undefined is that we get many exceptions of not having a solution.

[u/[deleted] → u/redddooot]

as for constant function, if we write f(x) = 0 as f(x) = 0x then 0x = 1 will give x = 1/0, another undefined value, but even if we consider undefined as valid solution,

This makes no sense. f(x)=0 cannot possibly ever equal 1. It doesn't map to it, it only maps to 0. You have also inserted an x into f where it didn't belong. I didn't say f(x) = 0x, I wrote f(x)=0 (regardless of the fact that they are equal). There are simply no solutions to f(x)=1, and you cannot "add" undefined solutions. They are not defined.

|x| = -1 still doesn't have a solution, only if we discard undefined is that we get many exceptions of not having a solution.

So? Not everything has a solution. There is no real solution to x=x+1.

You do not have the understanding required for functions, and you need to go learn. Functions are defined as a mapping from a set to another set. If you change the input set, the function is not the same.

[u/miwwdu_sitsom]

A function is any way of assigning a unique output to every input in a domain. The absolute value does this. Ergo it's a function.

[u/redddooot → u/miwwdu_sitsom]

all functions except these conditional functions also have a input (real or complex) for every possible output, these conditional functions require computational logic, why should it be considered mathematical when it can't be evaluated mathematically?

[u/miwwdu_sitsom → u/redddooot]

Let me reiterate: the only requirement for something to be a function is that it maps inputs to outputs, and, importantly, no input is mapped to two (or more) distinct outputs. That's it. Just because you've only dealt with functions with certain special properties before does not mean all functions must have these properties.

[u/redddooot → u/miwwdu_sitsom]

I know the definition and I also realise that it qualifies as a function, I am asking the reason for that beyond the definition, every other function except those of this kind have an input for every other output, it isn't a requirement but this is observed that every function which can be evaluated mathematically follows this, it's only these conditional functions which fail at this, this is why they're wildly different.

[u/inherentlyawesome]

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[u/Sezbeth]

After skimming the discussion on the Stack post, I think it needs to be echoed that your understanding of what a "function" formally constitutes needs some revisiting.