Understanding 0: How Maths Rules Changed Through Time

[Stephanus]

Dear PF Forum,
I just read Math FAQ

The goal of this FAQ is to clear up the concept of 0 and specifically the operations that are allowed with 0.[..]Our concept of "zero" as a both a placeholder and a number originated in India in the 9th century. But mathematicians were quite unsure about how to work with zero. For example, some rules involving zero were

• A number when divided by 0 is a fraction with 0 in the denominator.
• Zero divided by zero is zero.

As we shall soon see, these rules are not in practice today.

Please confirm.
These rules are not int practice. Is this true?
Because 30 years ago in high school, I was taught.
x/zero is infinity, error in computer.
zero/zero is undefined, not zero. Also error in computer.

So zero/zero is undefined or zero?
Thanks for anyhelp confirming my doubt.
[Add: is not in practice today, implying that in the old days some calculation still use zero/zero = zero.
When was it's not in practice?]

 

[micromass]

x/zero is infinity

This is wrong, something divided by 0 is undefined. Always.

So zero/zero is undefined or zero?

Undefined.

 

[Stephanus]

Please confirm one more thing.
A. $\lim h{\to 0} \text{ } \frac{6}{h} \rightarrow \text{ infinity}$

B. $h = 0; \frac{6}{h} \rightarrow \text{ undefined}$

C. $\lim h{\to 0} \text{ } \frac{6h}{h} \rightarrow 6$

D. $h = 0; \frac{6h}{h} \rightarrow \text{ undefined}$
Could you please tell me which ones are wrong?
Thanks.

 

[Mark44]

Please confirm one more thing.
A. $\lim h{\to 0} \text{ } \frac{6}{h} \rightarrow \text{ infinity}$

No.
The limit doesn't exist. If you don't understand why, read up on left-hand and right-hand limits.

B. $h = 0; \frac{x}{h} \rightarrow \text{ undefined}$

Correct

C. $\lim h{\to 0} \text{ } \frac{6h}{3h} \rightarrow 2$

Correct, although it would normally be written as $\lim_{h \to 0} \frac{6h}{3h} = 0$. In other words, without the last arrow, which indicates "approaches."

Edit: I miswrote 0 instead of 2 as the limit. It should be $\lim_{h \to 0} \frac{6h}{3h} = 2$

D. $h = 0; \frac{6h}{3h} \rightarrow \text{ undefined}$

Yes

Could you please tell me which ones are wrong?
Thanks.

 

[Stephanus]

Correct, although it would normally be written as $\lim_{h \to 0} \frac{6h}{3h} = 0$. In other words, without the last arrow, which indicates "approaches."

You mean $\lim_{h \to 0} \frac{6h}{3h} = 2$?
The "last arrow" I think is ambiguous in math.
I should have said $\lim_{h \to 0} \frac{6h}{3h}$ is equal/match to 2.

 

[micromass]

Please confirm one more thing.
A. $\lim h{\to 0} \text{ } \frac{6}{h} \rightarrow \text{ infinity}$

B. $h = 0; \frac{6}{h} \rightarrow \text{ undefined}$

C. $\lim h{\to 0} \text{ } \frac{6h}{h} \rightarrow 6$

D. $h = 0; \frac{6h}{h} \rightarrow \text{ undefined}$
Could you please tell me which ones are wrong?
Thanks.

I want to be a bit more careful than what Mark says. Although in principle I agree with him, you seem to use arrows $\rightarrow$ everywhere, and it is not clear to me why you write this arrow. So the following are correct:

A. $\lim_{h\rightarrow 0} \frac{6}{h}$ is undefined.
B. If $h=0$, then $\frac{6}{h}$ is undefined.
C. $\lim_{h\rightarrow 0}\frac{6h}{h} = 6$.
D. If $h=0$, then $\frac{6h}{h}$ is undefined.

 

[micromass]

You mean $\lim_{h \to 0} \frac{6h}{3h} = 2$?

Yes.

The "last arrow" I think is ambiguous in math.

It's not ambiguous, it's just wrong to use it in that context.

 

[Stephanus]

It's not ambiguous, it's just wrong to use it in that context.

I should have asked directly to Mark44, but your post is the last one.

A. $\lim h{\to 0} \text{ } \frac{6}{h} \rightarrow \text{ infinity}$

No.The limit doesn't exist. If you don't understand why, read up on left-hand and right-hand limits.

Actually this

A. $\lim h{\to 0} \text{ } \frac{6}{h} \rightarrow \text{ infinity}$

doesn't have anything to do with "left-hand and right-hand limits", because it's the last arrow that I put wrongly? I just read left-hand, right-hand arrow, http://www.millersville.edu/~bikenaga/calculus/limlr/limlr.html
but it doesn't explain why there's no limit.

 

[micromass]

The point of the left-hand and right-hand limits is the following:
While
$$\lim_{h\rightarrow 0} \frac{6}{h}$$
is undefined. It is true that the left-hand limit
$$\lim_{h\rightarrow 0-} \frac{6}{h} = -\infty$$
and the right-hand limit is
$$\lim_{h\rightarrow 0+}\frac{6}{h} = +\infty$$

 

[Stephanus]

The point of the left-hand and right-hand limits is the following:
While
$$\lim_{h\rightarrow 0} \frac{6}{h}$$
is undefined. It is true that the left-hand limit
$$\lim_{h\rightarrow 0-} \frac{6}{h} = -\infty$$
and the right-hand limit is
$$\lim_{h\rightarrow 0+}\frac{6}{h} = +\infty$$

Ahh, so if the left hand and the right hand lmits differs very big, than we can say "There is no limit"?
[Add: differ is the wrong word I think. So, the left-hand and the right-hand limit must match/exact?]
[Add: $\lim_{h \to 0+} \frac{6h}{h}$ is equal to $\lim_{h \to 0-} \frac{6h}{h}$, so we can say that there is a "limit" there?]

 

[micromass]

Yes. A limit exists if and only if the left-hand and right-hand limits exist and are equal.

 

[Stephanus]

Yes. A limit exists if and only if the left-hand and right-hand limits exist and are equal.

Thanks! A new concept. This helps me much in understanding math (at least about this "limit" thing . Left hand should match right hand)

 

[HallsofIvy]

Part of the problem is that you are trying to interpret "infinity" as if it were a number- it isn't. Saying that a limit "is equal to infinity" is the the same as saying the limit "does not exist", just in a particular way.

 

[Mark44]

You mean $\lim_{h \to 0} \frac{6h}{3h} = 2$?

Yes, that's what I should have written. I have edited my earlier post to indicate this.