@user21820 would it be considered a basic math question to ask why can you never divide by zero?

@student You can ask any mathematical question here, basic or not. And yes I can answer that. =)

(in the set of real numbers, of course :)

The short answer is that the division operation on reals is only defined for certain inputs, namely the first input is a real and the second input is a non-negative real.

In other words, it's our choice to define the division operation that way, and that is that. However, one can then ask why that is the 'best' definition.

The answer is that it gives the neatest algebraic properties. It's not like we cannot do mathematics if x/0 is defined as 0. We can, but our theorems would have to be changed accordingly.

For example...

Sorry I don't know why I wrote "non-negative" above instead of "nonzero".

np, i've seen a two case argument

x=0 gives infinely many answers etc

in x/0

@student We cannot have infinitely many answers for an operation, because an operation must be a function.

Let's talk about non-functions later.

First consider this example:

1/(1/x) = x for every nonzero real x.

but functions come later?

Wait wait.

first comes arithmetic, no?

imo

:waiting:

:-)

2/(1/x) = x·2 for every nonzero real x.

sure, if you multiply the numerator and the denominator by x

that's just a fraction, right?

If you want to extend to allow x = 0 and yet still retain both of these facts, then you will be forced to define 1/0 = 0.

And 2/0 = 0.

This then breaks a third fact:

If a/b = c then a = b·c, for every reals a,b,c such that b ≠ 0.

0*x = 0

I'm not sure whether you are following me.

i'm not, sorry

What's your mathematical background? High-school? Undergraduate?

why ASSUME a/b = c

@user21820 middle school

Ah okay so I've to make things simpler. Let's start over again.

ok

We define arithmetical operations to capture some basic facts about concrete objects in the real world.

yes

For example, addition is defined to capture the notion of putting two collections of solid objects together and counting them together.

true

Similarly multiplication can be defined to capture replacing each object in one collection with a copy of the other collection. Alternatively, use a rectangular arrangement.

provided, they are "countable"

Yep.

That's why "solid objects" =)

:D

Okay so that's addition and multiplication. Division is not so 'natural' as those. For some kinds of objects, we simply cannot divide one into many pieces.

such as?

Well, humans? Single particles?

ok

I mean, the notion of counting does not extend automatically to half. Half should mean that two halves is the same as one.

same goes for subtraction

Yea at first people rejected negative numbers because you can't take away more than you already have.

there is no negative thing

In some sense that is correct, and we have to move away from counting solid blocks to go beyond arithmetic on natural numbers.

Natural numbers are 0,1,2,3,...

So, consider volume of some liquid like water.

please continue

And now define addition to be just pouring them together.

ok

We also have some specific unit volume (say the volume of a cup).

nonzero?

Yea. I don't know of any cup with zero volume. =)

hmm

If the amounts we are handling are multiples of cups, then we can use our original intuition for solid objects. For example, p·q is the volume of water in a rectangular arrangement of p times q cups, each of which is filled to the brim.

Makes sense so far?

yup

But now it also makes sense to ask whether there is a volume v such that two times of it gives 1.

sure 1/2 a cup

For water, it seems as if such a volume exists, and we call it 1/2.

Right.

And we can similarly extend this notion to get all fractions p/q where p and q are positive integers.

You can then observe (experimentally or conceptually) various facts about fractional volumes.

Such as (a+b)/c = a/c+b/c...

"such a volume exists" seems a bit guarded of a statement, no?

@student Haha I'm just being cautious, because with modern science we sort of know that water is made up of molecules of H2O, and if there are an odd number of them then we can't exactly get half.

Lol.

Just ignore that tiny problem. =P

:D

In any case, under this interpretation of fractions we already can see that there can be no such thing as 1/0, because it is asking for whether there is a volume v such that 0 times of it gives 1.

And since we assumed our cup has nonzero volume, that's just impossible.

right

If that is sufficiently satisfying to you, then you can go off happy. =)

But if you're curious about what happens if we want to develop mathematics with 1/0 being defined.

Just for fun.

Then I can try to explain.

So, what would you like?

@student: I thought the "what happens if" was your original question, but I'm not sure. =)

nope

why?

thanks for your time

@student I'm not sure what your question is then. My main point is that we define arithmetic operations to capture something about the real world, so that we can investigate it with predictive power.

So we won't arbitrarily define 1/0 unless we can interpret it to mean something in the real world that we are interested in.

The real world is the justification of our definition, correct?

Yes.

that makes sense now

For reference see this particular collection of axioms for facts about natural numbers that one could say were empirically observed.

Ignore the rest of the article for now.

ok

Do you know how to read the symbols in the axioms?

nope

"∀x,y,z∈N" means "for every x,y,z that are members of N", namely "for every natural numbers x,y,z".

So axioms (1) says "( Adding x to y, and then adding the result to z ) gives the same as ( adding y to z, and then adding x to the result )".

Which we observe when counting solid objects or cups of water.

right

so this doesn't apply to something like trying to count "clouds"

@student Yea. Clouds are nebulous. =)

And when they rain, they can decrease in size.

counting breaks down because of the objects?

@student It's just that it's not discrete.

And the objects should not be changing over time, because we take time to count.

the objects are not discrete

Yea that.

what then is "counting"?

That's another good question.

my teacher said "counting is what you can do on your fingers"

:-/

Hahahaha..

How do you get past 10?

how much more circular can you get than that?

Off-topic: can you figure out how to count from 0 to 1023 on your fingers?

change of base?

Yea try base 2. But don't start now; counting to 1023 would take some time.

Counting can be described as follows: You first have an unbounded supply of some standard blocks that you can arrange in a straight line. To count a collection of items, you place them one by one next to a block, such that no two are next to the same block, and if a block has an item next to it then the previous block in the line also has an item next to it.

This may take a while to digest, but I am basically trying to describe in a relatively non-circular way a process of counting.

one to one matching?

Yup.

i don't like the "unbounded" part

One-to-one matching to the smallest initial segment of the line of blocks (that has a starting block).

@student That's just too bad; we can't define counting without that.

:(

It's a valid philosophical concern though.

@student: Turing (the famous guy) tried to evade this issue by somewhat saying it's extendable rather than literally unbounded.

If you believe you can always make more blocks and lengthen the chain of blocks, then you're fine. If not, then there's no way around it.

hmm...

so, when we use "..." for "and so on" we mean forever?

@student Yes when mathematicians write 1,2,3,... they do mean all the natural numbers (unbounded).

like 0.999...

Yes that means an infinite sequence of digits that are following the pattern.

and there can be no number between 0.999... and 1.000...

Correct.

because there is no digit past 9

Basically yes.

afk

back

thanks, once again, for your time @user21820 let me think about this for awhile and come back with more questions

@student: Sure you're welcome and see you again! =)

:D