# If a/x*x = a then if x =0 a/0*0 = a

• ArielGenesis
In summary, division by zero is undefined in the real number system. This is because if one were to define division by zero, it would result in every number being equal to every other number, making the number system uninteresting. Infinity is not a defined number in the real number system, so any expression involving infinity is also undefined. In some cases, when taking the limit of a function, it may appear that division by zero results in infinity, but this is just a way of expressing that the function approaches infinity as the variable approaches zero. Overall, division by zero should be avoided in mathematics as it leads to undefined results.
ArielGenesis
if

a/x*x = a

then if x =0

a/0*0 = a

but they say that:

n *0 = 0

and when n =a/0

so that

a/0*0 = 0

ArielGenesis said:
and when n =a/0

so that

a/0*0 = 0

a/0 does not = 0

anything/0 is undefined

do you know that long ago when some one asked what 5-6 was, he was told undefined(ok not exactly that, but something that meant impossible)

sooner to present day, some was told that (-4)^.5 (that is square root of -4 without using square root sign) was impossible, until someone thought outside the box and cam up with imaginary numbers.

time to do that again

5*0 = 4*0

If you can divide by zero, then the zeros cancel and 5 = 4 and you end up with a number system with only 1 element. Its not so much impossible to come up with a system that allows you to divide by zero, its that it would be limiting rather that useful..

do you know that long ago when some one asked what 5-6 was, he was told undefined

And they were right.

sooner to present day, some was told that (-4)^.5 was impossible

And they were right.

It's not a matter of "thinking outside the box" -- it's a matter of definition. Long ago, people used a number system that consisted only of positive numbers. Thus, it is correct that 5-6 was undefined. And it's still undefined in that number system. The fact we invented negative numbers doesn't change that fact.

Similarly for your next example.

The problem with wanting to invent division by zero is this:

0 = 0*x - 0*x = (0 + 0)*x - 0*x = (0*x + 0*x) - 0*x = 0*x + (0*x - 0*X) = 0*x + 0 = 0*x

Each of the steps in this equation is something that is extremely desirable for a number system to have. Here, I've used:
Subtracting something from itself yields zero.
Adding zero to something leaves it unchanged.
Multiplication distributes over addition

Thus, anything that has these nice properties also has the property that 0*x = 0 for all x.

Thus, if we wanted to define division by zero, we must have:

0*x = 0 = 0*y
Therefore x = y. (Dividing by zero)

In other words, it would require every number to be equal to every other number. That's not a very interesting number system now, is it?

In order to have a useful division by zero, one has to give up at least one of the properties that makes a number system useful.

so hurky you mean that the number system where every number to be equal to every other number isn't interesting.

and by the way I'm confuse wether 1/0 = undefined or infinity

if i asked u that wether

1/0 < 2/0

if yes then 1/0 suppose to be a form of constant

It is just undefined(not infinity).

Let us say, 1/0 = infinity. As infinity still holds some mathematical value 0*infinity = 0, thus 1 = 0.

When a teacher was explaining his elementary school students about dividing a number with same number and always getting the value 1, he took an example of distributing 5 apples among 5 students, 4 apples among 4 students etc.

A boy stood up and asked that if there were no apples and they were distributed to nobody, still everybody would get 1 each?

The boy asking about division by zero was the great Srinivasa Ramanujam and he was 8 year old that time.

I am sorry. On a second thought, my explanation seems to be wrong for, 1/infinity = 0 and so if infinity*0 = 0 then 1 = 0.

You just consider 1/0 is undefinied.

so if we have a curve where y= 1/x and at a point where x=0, was taught that y = infinity instead of undefined. infinity point on cartesian plane is somewhere imaginary. while undefine should be nowhere or does not exist.

y = infinity means y is undefined.

Does someone know which are the different types of undefinitions in math?

and esclusively, what is zero times infinite?

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ArielGenesis said:
so if we have a curve where y= 1/x and at a point where x=0, was taught that y = infinity instead of undefined. infinity point on cartesian plane is somewhere imaginary. while undefine should be nowhere or does not exist.

when x approaches 0, y approaches +infinity or -infinity depending on direction... that's one of the reasons 1/0 is undefined.

there is no point at x= 0, it is untrue that "there is an imaginary point...", it does not exist.

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quark said:
I am sorry. On a second thought, my explanation seems to be wrong for, 1/infinity = 0 and so if infinity*0 = 0 then 1 = 0.

You just consider 1/0 is undefinied.

this doesn't work becaus infinity/infinity does not equal 1, it is undefined.

for <<<GUILLE>>>, 0*infinity = 0

nnnnnnnn said:
this doesn't work becaus infinity/infinity does not equal 1, it is undefined.

for <<<GUILLE>>>, 0*infinity = 0

thanks.

and zero/infinity?

and infinity/zero?

zero/infinity = 0 ...anything not (+,-) infinity is zero when divided by (+,-) infinity (this may actually be undefined, but its limit is 0)

infinity/zero DNE

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thanks again.

one more, what is -infinite+infinite?

There are not "different kinds" of "undefined"s. "Undefined" means exactly that: there is no definition for that combination of symbols- it makes no sense. As far as the real numbers are concerned, any formula involving "infinity" is "undefined" because infinity itself is undefined. Asking "what is 0/infinity" is exactly the same as asking "what is 0/green beans?".

isnt the more clear cut way of saying it like this: anything/0 is undefined over real numbers. however, when you get the limit thing, like lim 1/x as x -> 0, you get infinity bcause that is what it 'approaches'. but in general, in a pure math situation, if you come across 1/0 give up right there.

ofcourse, in most cases the 1/0 comes up in an 'applied' or semi applied problem, like some word problem or such, and when you solve the quadratic you come up with an imaginary number. that doesn't mean your equations are 'undefined', it means that relative to your specific example, you interpert a result of 'undefined' as the dog missed the train, or whatever. same idea with 1/0, if you have say an equation for profit an that's what you get, your profit is mathematically undefined, but if you think of it relative to that case it means your profit is 'infinite'.

so in a sense, i agree with Hurkyl, but i think that you're being a little over formal/general/precise. (I mean duh that's not quite what people mean when they write 5-6 :) )

for <<<GUILLE>>>, 0*infinity = 0

First off, there is no infinity in the reals, so this would be an undefined statement in the reals.

Secondly, in terms of the extended reals, it is still undefined.

Thirdly, in terms of limit forms, 0 * infinity is an indeterminate form. (Consider, for example the limits of x * 1/x, and x * 2/x)

1/0 < 2/0
do you agree...

We've ascertained 1/0 and 2/0 are both undefined...

Since "1/0 < 2/0" contains undefined expressions... what do you think my answer will be?

nnnnnnnn!

I was just showing that my earlier proof was wrong.

Guille!

The following are undefined to my knowledge.

n/0, 0/0, infinity/infinity, infinity/zero

thanks to all.

refering to up-over-their,: zero and infinity do exist in reality, you have two chickens and eat them and you have zero chickens.there is infinite time. where zero and infinite don't exist is in fisically. as in infinite dogs or zero space.

<<<GUILLE>>> said:
thanks to all.

refering to up-over-their,: zero and infinity do exist in reality, you have two chickens and eat them and you have zero chickens.there is infinite time. where zero and infinite don't exist is in fisically. as in infinite dogs or zero space.

You don't know that there is infinite time, it may be finite.

Alkatran said:
You don't know that there is infinite time, it may be finite.

true, I didn't think very much about it,

what I meant was that as much as there is motion, there will be time. so unless everything destroyes (energy and mass are supposed to be impossible to destroy) then there is only vacuum space, with absolute zero temp. Then there will be no motion adn thus, no time. it is the only way. but of course, you have to destroy energy, mass, rnd the temperature...quite dificult.

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primary student ?

ArielGenesis said:
primary student ?

what? who is a primary student? are you referring to me?

<<<GUILLE>>> said:
true, I didn't think very much about it,

what I meant was that as much as there is motion, there will be time. so unless everything destroyes (energy and mass are supposed to be impossible to destroy) then there is only vacuum space, with absolute zero temp. Then there will be no motion adn thus, no time. it is the only way. but of course, you have to destroy energy, mass, rnd the temperature...quite dificult.

What do you mean? I don't see energy and mass disappearing as a requirement for time to 'end'.

Alkatran said:
What do you mean? I don't see energy and mass disappearing as a requirement for time to 'end'.

time depends and is caoused directly by motion. it exists because motion exists. what has motion? matter and energy. you are matter, then you move, then you have motion, then time exists. then only way you and non of tyour particles will stop moving is with absolute zero. then, there is no time.

<<<GUILLE>>> said:
time depends and is caoused directly by motion. it exists because motion exists. what has motion? matter and energy. you are matter, then you move, then you have motion, then time exists. then only way you and non of tyour particles will stop moving is with absolute zero. then, there is no time.

How do you know time is caused by motion? That makes very little sense. It makes more sense to say motion is caused by the passage of time.

Hey Alkatran, you ever going to concede the point in eot-cat-cit?

Alkatran said:
How do you know time is caused by motion? That makes very little sense. It makes more sense to say motion is caused by the passage of time.

yes, sort of. Actually it is impossible for creaters which are under time and motion, to knwo which one is more fundamental. But if nothing moves then there is no time.

if time is relative to the observer, then it is possible for time not to "flow" at all. like if u observe a foton, it does not have any "flowing" time at all

ArielGenesis said:
if time is relative to the observer, then it is possible for time not to "flow" at all. like if u observe a foton, it does not have any "flowing" time at all

yes: time is relative. photons, for example, have infinite life, although can be created and destroyed.

This thread has completely digressed from the OP's question, which itself is not a brain teaser, but a question asking for clarification of the definition of the reals.

## 1. What is the meaning of the equation "If a/x*x = a then if x =0 a/0*0 = a"?

The equation is a mathematical statement that states if the quotient of a divided by x, multiplied by x, is equal to a, then if x is equal to 0, the quotient of a divided by 0, multiplied by 0, is also equal to a.

## 2. Why is the value of x set to 0 in the second part of the equation?

The value of x is set to 0 in the second part of the equation to test the validity of the statement and to see if it holds true even when x is equal to 0.

## 3. Is this equation always true for any value of a?

Yes, the equation is always true for any value of a. This is because any number multiplied by 0 is equal to 0, and division by 0 is undefined. Therefore, the second part of the equation becomes a/0, which is undefined, making the equation true for any value of a.

## 4. Can this equation be simplified or rewritten?

Yes, the equation can be simplified to a = a. This is because when x is equal to 0, the equation becomes a/0, which is undefined. Therefore, the equation can be simplified to a = a, which is always true.

## 5. What is the significance of this equation in mathematics?

This equation is significant in mathematics as it highlights the concept of undefined values and the importance of considering all possible scenarios when making mathematical statements. It also demonstrates the importance of understanding the properties of numbers, such as division by 0, in order to make accurate mathematical statements.

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