Dividing by 0, undefined?

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AvgStudent

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What exactly does it mean when we say "undefined" when dividing by 0?

My attempt to answer the question is as follows:
I started with the definition of division which repeated subtraction. For example, 15/3= 5 because 15-3-3-3-3-3=0. So what is 1/0?
By the division definition, 1/0 => 1-0-0-0-....... so infinity?

The second attempt was to define division as a ratio of two numbers. r=a/b=> a=r*b. So if b=0, then a=r*0. However, r*0=0, then r can be any number, i.e. infinity?

So far, I've come up with the conclusion that it should be infinity, so why do we say "undefined"? Is there a difference?
 
Prior to Georg Cantor, people would say that inifinity is not a number because it does not exist in the physical world.

Most mathematicians after Cantor would agree that infinity does exist, at least as a logically consistent concept in the mind of mathematicians.

There is a version of mathematics that agrees with you that any non-zero number divided by 0 is defined and equals infinity. (One point compactification of the reals.) Furthermore, as happens with most extensions of the idea of numbers, we need a revised arithmetic.

For those who are committed to compactification, they should not say “division by zero is undefined.” Rather they should say “division of a non-zero number by zero is defined but not finite.”

HOWEVER, that means that any process requiring division by zero can have no application in the physical universe. Infinity is a mental concept that has no physical exemplification. We teach people mathematics so that they can use it for practical purposes. That means that division by zero must be avoided.

Personally, I think it makes little difference if we say “don’t divide by zero because it is not defined [in standard arithmetic]”
or “don’t divide by zero because you will get incorrect or physically meaningless results.”

Within standard arithmetic

[math]a * b = c \iff a = \dfrac{c}{b}.[/math]
If we do not exclude b = 0 from that rule, we get weirdness like

[math]5 * 0 = 0 \text { and } - 100 * 0 = 0 \implies 5 = \dfrac{0}{0} \text { and } -100 = \dfrac{0}{0} \implies 5 = -100.[/math]
If you allow division by zero, you must throw out standard arithmetic and accept practically unrealizable answers.

So the actual rule in standard arithmetic is

[math]b \ne 0 \implies \{ a = \dfrac{c}{b} \iff a * b = c\}.[/math]
EDIT: I do not think that Dr. Peterson and I have much if any real disagreement. He says you cannot define the result of division by zero as a real number. Even you agree with that. Infinity is not a real number.

[math]r \text { is a real number} \implies r \ne r + 1.\\ \infty = \infty + 1.[/math]
 
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AvgStudent said:
What exactly does it mean when we say "undefined" when dividing by 0?

My attempt to answer the question is as follows:
I started with the definition of division which repeated subtraction. For example, 15/3= 5 because 15-3-3-3-3-3=0. So what is 1/0?
By the division definition, 1/0 => 1-0-0-0-....... so infinity?

The second attempt was to define division as a ratio of two numbers. r=a/b=> a=r*b. So if b=0, then a=r*0. However, r*0=0, then r can be any number, i.e. infinity?

So far, I've come up with the conclusion that it should be infinity, so why do we say "undefined"? Is there a difference?
Click to expand...
We say the division is undefined, because we can't define it -- there is no real number that works as an answer. The result of an operation on real numbers has to be a real number; infinity isn't a number, because it doesn't behave like a number.

Your statement that r can be any number is valid only if a=0; otherwise, a=r*0 means a=0, which is false. There is no number you can multiply by 0 and get a non-zero value a. (And when a=0, the fact that there are infinitely many values of r that work doesn't mean that the answer is infinity.)

In calculus, of course, we can treat infinity as a limit. That's a different thing.
 
@JeffM Thank you for the history, and I've realized something through your example. If I were to make the same argument for 2/0, I would also get infinity. So if 1/0=infinity and 2/0=infinity, then 1=2, which cause math to break.
@Dr.Peterson Speaking of limits (which is something I recently learned), what if I try to find the lim(x->0) of 1/x? By looking at the graph, I do see a vertical asymptote at x=0, does that mean it should be 0? But 0 contradicts what I said earlier, which is infinity.
 
AvgStudent said:
@JeffM Thank you for the history, and I've realized something through your example. If I were to make the same argument for 2/0, I would also get infinity. So if 1/0=infinity and 2/0=infinity, then 1=2, which cause math to break.
@Dr.Peterson Speaking of limits (which is something I recently learned), what if I try to find the lim(x->0) of 1/x? By looking at the graph, I do see a vertical asymptote at x=0, does that mean it should be 0? But 0 contradicts what I said earlier, which is infinity.
Click to expand...
How do you get limit 0 from a vertical asymptote?
 
lev888 said:
How do you get limit 0 from a vertical asymptote?
Click to expand...
My mistake, I was thinking about lim(x->infinity) instead. The limit from the left is negative infinity, while the limit from the right is positive infinity, so the two-sided limit does not exist. Is this what "undefined" means, it's "undefinable".
 
lim(x->0+) (1/x) = +infinity
lim(x->0-) (1/x) = -infinity
Hence lim(x->0)(1/x) DNE (does not exist).

You say that 1/0 is infinity but then -1/0 is -infinity. So dividing by 0 is not very well defined even if we allow infinity and -infinity to be real numbers. Of course, we can't allow them to be real numbers.
 
Jomo said:
lim(x->0+) (1/x) = +infinity
lim(x->0-) (1/x) = -infinity
Hence lim(x->0)(1/x) DNE (does not exist).

You say that 1/0 is infinity but then -1/0 is -infinity. So dividing by 0 is not very well defined even if we allow infinity and -infinity to be real numbers. Of course, we can't allow them to be real numbers.
Click to expand...
Thank you for confirming my thought. Discovering a lot of things that my math teachers didn't tell me ☹️
 
Yes, you have given a reasonable approach to the answer.
The fact that every effort at a definition leads to a contradiction is why the result of a division by zero is undefined.
r *0 = 0
There is no solution to the equation unless a=0 for all integers r.
You might now conclude that r=infinity satisfies. That's one way of stating it, but what exactly is infinity? It's not a figure! What's to stop you? We'd end up with paradoxes if we regarded it as a number. Consider what happens when we multiply an integer by infinity. Infinite plus any number is still infinity, according to popular belief. If that's the case,
infinity = infinity+1 = infinity + 2
If infinity were a number, it would mean that 1 equaled 2. That would mean, for example, that all integers are equal, and our entire number system would collapse.
Dividing 0
0 /a = 0

Any integer multiplied by zero equals a zero. When you multiply or divide any integer by zero, nothing changes.
a/ 0 is undefined

Finally, and maybe most importantly, remember:
a/ 0 is not defined
A number cannot be divided by zero!
Division by zero is a division with zero as the divisor (denominator) in mathematics. The expression has no significance in simple arithmetic since there is no number that when multiplied by 0 provides.
 
\(\displaystyle \frac{0}{1} × \frac{1}{0}\)

\(\displaystyle 0^{1} × 0^{-1}\)

\(\displaystyle 0^{1 - 1}\)

\(\displaystyle 0^{0}\)

\(\displaystyle \text{?}\)

?

[imath]\;[/imath]
 
See. I am going to say the reason very intuitively. The numerator of the fraction says what we have and the denominator says the total numbers of it. For example a pizza is cut into 4 pieces and I have 2 pieces of it. So,it can be denoted as 2/4 in fraction. But if we divide something by 0 that means the denominator is 0So,n/0 makes no sense as we cannot have something out of nothing.

Mathematically speaking,division by 0 is undefined. It can have many value. But 1/0 is also sometimes referred as infinity.
 
Piyush Ranjan Dash said:
Mathematically speaking,division by 0 is undefined. It can have many value. But 1/0 is also sometimes referred as infinity.
Click to expand...

"It can have many value [sic]."

If the numerator is non-zero, it cannot have any value, as in it is impossible.
If the numerator is zero, it cannot be assigned a particular value, because any real number would work. Please note a few of the posts above that already discuss this.

I have not seen "1/0" referred to as "infinity."
 
lookagain said:
"It can have many value [sic]."

If the numerator is non-zero, it cannot have any value, as in it is impossible.
If the numerator is zero, it cannot be assigned a particular value, because any real number would work. Please note a few of the posts above that already discuss this.

I have not seen "1/0" referred to as "infinity."
Click to expand...
It is referred as infinity. tan 90=sin 90/cos 90=1/0 which is considered as infinity. You may not have noticed but in Physics also the same is considered in the famous Theory of Relativity.
 
Piyush Ranjan Dash said:
It is referred as infinity. tan 90=sin 90/cos 90=1/0 which is considered as infinity.
Click to expand...

You mean you claim "tan(90 degrees) = sin(90 degrees)/cos(90 degrees) = 1/0, which
is considered as infinity."
At x = pi/2, there is a vertical asymptote for y = tan(x). The limit does not exist as x approaches pi/2. On the left side, the graph is going toward +oo, but on the right side,
the graph is going toward -oo.

You cannot call tan(90 degrees) "infinity." "Infinity" has the idea that you are going in
the direction of increasing, that is, +oo.
 
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Piyush Ranjan Dash said:
Yes 1/0 can be referred to as any value which would cause the entire arithmetic to crumble. So,division by 0 makes no sense
Click to expand...

No, 1/0 cannot be referred to as any value. Therefore, 1/0 makes no sense.

Suppose 1/0 could equal a real number, n.

Then, by rules of arithmetic, we would multiply each side by the denominator,
which is 0:

1 = 0*n

Then, you ask is there a real number which when you multiply it by 0, it can equal 1.

The answer is "no."

There are no values.

----------------‐---------------------‐

And now I must sleep after these posts.
 
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