Trying to understand the property of absolute value inequality

[PeroK]

You are saying that we cannot prove the property by this definition of distance $|x-y|$ but there are other definitions by which we can.

I don't understand what you mean by that statement. The number line isn't a physical thing you can measure. We're not talking about physical distance here. There are no physical units involve, like inches or centimeters. So, you have to define the "distance" between $x$ and $y$ in some abstract mathematical way. The usual definition is $|x - y|$.

This makes some physical sense, because if you have a physical length of something like a rope and you measure $x$ units from the end and $y$ units from the end, then the distance you measure between these two points is $|x - y|$. That can't be proved either, in the sense that it's an experiment you must do.

 

[mark2142]

I

I don't understand what you mean by that statement. The number line isn't a physical thing you can measure. We're not talking about physical distance here. There are no physical units involve, like inches or centimeters. So, you have to define the "distance" between $x$ and $y$ in some abstract mathematical way. The usual definition is $|x - y|$.

This makes some physical sense, because if you have a physical length of something like a rope and you measure $x$ units from the end and $y$ units from the end, then the distance you measure between these two points is $|x - y|$. That can't be proved either, in the sense that it's an experiment you must do.

There is some miss understanding. Maybe my English.

it is taken as an axiom/definition. You can prove properties of this axiom, ie., triangle inequality and other stuff. But one does not prove an axiom.

Do you mean ‘$|x|<c$ is equivalent to $-c<x<c$’ as axiom?

 

[PeroK]

Do you mean ‘$|x|<c$ is equivalent to $-c<x<c$’ as axiom?

There's a difference between an axiom and a definition. Axioms are more fundamental. The modulus function must have a definition. That's been given above: $|x| = \pm x$, depending on whether $x$ is positive or negative. From that definition, you can prove that $|x|<c$ is equivalent to $-c<x<c$, where $c$ is positive..

What you can't do in mathematics is introduce some undefined notion of "distance". You must define what you mean by distance. In this case, we define the distance between $x$ and $y$ as $|x - y|$.

 

[mark2142]

it is taken as an axiom/definition. You can prove properties of this axiom, ie., triangle inequality and other stuff. But one does not prove an axiom.

Translation: Axioms are not that difficult to understand.
But I don’t know why this one doesn’t feel obvious.
By ‘it’ you mean property of absolute value inequality just to be clear.

 

[mark2142]

There's a difference between an axiom and a definition. Axioms are more fundamental. The modulus function must have a definition. That's been given above: |x|=±x, depending on whether x is positive or negative. From that definition, you can prove that |x|<c is equivalent to −c<x<c, where c is positive..

Ok. We define things and combine them to make fundamental statements which can be proved by the definition itself like here.
But I am not satisfied that distance definition doesn’t work to prove axiom.

 

[PeroK]

But I am not satisfied that distance definition doesn’t work to prove axiom.

Then, perhaps, mathematics isn't for you? Perhaps you should study something else instead?

 

[mark2142]

Then, perhaps, mathematics isn't for you? Perhaps you should study something else instead?

You sound like I can’t learn maths.

 

[PeroK]

You sound like I can’t learn maths.

The modulus function is simple, algebraically and graphically. You don't have time to spend days trying to understand it. You have to find some way to move on. This is a practical matter. You don't have an unlimited amount of time. This thread, IMO, is now a waste of your time. You have to move on.

 

[MidgetDwarf]

I would watch the 3 videos.