Please explain to me why this simple problem is so

[Dickfore]

Unfortunately I cannot find a 4D drawing package!

Your previous diagram was not three dimensional either.

 

[The Chaz]

chambershex, how come x + y \ne x \, y? I know that, for example, 2 + 3 = 5 and 2 \cdot 3 = 6 and 5 \ne 6, but what's the reason behind this?

Why are you being so insulting? You and H must be on a mission to demoralize and deter people from using PF. Sure, the OP doesn't think like us or know as much, but who here has explained WHY?!? I suspect the reason that no one has (explained why) is that you don't know. The first response was close... Something is not true unless it is ALWAYS true.

As for your question, they are different operations, and have different outputs.

 

[alice22]

Your previous diagram was not three dimensional either.

No it was a 2 dimensional representation of the vertices of a 3D shape to be precise, or at least my rather poor attempt at it.

There is a video on the web claiming to draw a representation of a 4D cube but I think I will be banned if I post a link to it so I won't, as I expect it may break forum rules, or something like that!

 

[Dickfore]

Why are you being so insulting? You and H must be on a mission to demoralize and deter people from using PF. Sure, the OP doesn't think like us or know as much, but who here has explained WHY?!? I suspect the reason that no one has (explained why) is that you don't know. The first response was close... Something is not true unless it is ALWAYS true.

As for your question, they are different operations, and have different outputs.

But, for example, when x = y = 2 or x = y = 0 I get the same answer.

 

[chambershex]

chambershex, how come x + y \ne x \, y? I know that, for example, 2 + 3 = 5 and 2 \cdot 3 = 6 and 5 \ne 6, but what's the reason behind this?

I know your being nasty here, but I can answer that. The operation + is fundamental and cannot be broken down into anything more simple - what it means is self-evident. The operation x is derived from +, being defined as Y lots of X. Or alternatively X + X + X..., Y times. However, I'm sure someone could provide a better formal description.

 

[Dickfore]

I know your being nasty here, but I can answer that. The operation + is fundamental and cannot be broken down into anything more simple - what it means is self-evident. The operation x is derived from +, being defined as Y lots of X. Or alternatively X + X + X..., Y times. However, I'm sure someone could provide a better formal description.

Oooh! Sounds too complicated ;) What does operation mean?

 

[chambershex]

Why are you being so insulting? You and H must be on a mission to demoralize and deter people from using PF. Sure, the OP doesn't think like us or know as much, but who here has explained WHY?!? I suspect the reason that no one has (explained why) is that you don't know. The first response was close... Something is not true unless it is ALWAYS true.

As for your question, they are different operations, and have different outputs.

Thank-you. I now understand why the first question I posed is true, however I'm still yet to really see a good, fundamental explanation of why the second question is true (the one referring to the roots).

The overarching reason I am asking these questions is because I have always had a problem with completing algebraic/arithmetical questions using explanations which are just accepted as truth. I am content enough once an explanation can get down to the level of the axiom, because I can accept certain fundamental, propositions are self-evident. "I suspect the reason that no one has (explained why) is that you don't know" - I think this statement is actually not far from the truth; on philosophical grounds much of the foundations of mathematics is problematic.

 

[The Chaz]

But, for example, when x = y = 2 or x = y = 0 I get the same answer.

No, 4 is not the same as 0. Surprised you didn't know that.
/trollfeeding

 

[Dickfore]

Thank-you. I now understand why the first question I posed is true, however I'm still yet to really see a good, fundamental explanation of why the second question is true (the one referring to the roots).

The overarching reason I am asking these questions is because I have always had a problem with completing algebraic/arithmetical questions using explanations which are just accepted as truth. I am content enough once an explanation can get down to the level of the axiom, because I can accept certain fundamental, propositions are self-evident. "I suspect the reason that no one has (explained why) is that you don't know" - I think this statement is actually not far from the truth; on philosophical grounds much of the foundations of mathematics is problematic.

Let us say that the second statement you were asking about was correct, i.e.

<br /> \sqrt{a + b} = \sqrt{a} + \sqrt{b}<br />

was correct. Then, take a = x^{2} and b = y^{2} for x, y &gt; 0. Our 'equality' becomes:

<br /> \sqrt{x^{2} + y^{2}} = x + y<br />

Then, square the above 'equality':

<br /> x^{2} + y^{2} = (x + y)^{2}<br />

But, you this is in contradiction with your first statement. Therefore, if you accept that the last equality does not hold, then neither can the second one.