Both a/0 and 0/0

[mathdad]

Both a/0 (a is not 0) and 0/0 are undefined, but for different reasons. Explain the different reasons.

Let me see.

I know that a/0 indicates division by zero, which is undefined. I forgot the mathematical reason why 0/0 is undefined. I recall that 0/0 is indeterminate. Does calculus answer the 0/0 mystery? To be undefined is not the same as to be indeterminate, right?

 

[tkhunny]

Both a/0 (a is not 0) and 0/0 are undefined, but for different reasons. Explain the different reasons.

Let me see.

I know that a/0 indicates division by zero, which is undefined. I forgot the mathematical reason why 0/0 is undefined. I recall that 0/0 is indeterminate. Does calculus answer the 0/0 mystery? To be undefined is not the same as to be indeterminate, right?

There is no mystery. In all cases, "a/0" and "0/0" are not, and cannot be, Real Numbers. If you refer to these sorts of expressions as FORMS, and not as actual results or values, then there is some possible discussion.

If we disregard what I just said, there is one, most elementary discussion:

6/3 = 2 because 2+2+2 = 6 -- It takes three (3) '2's to make 6.

6/0 is undefined since you cannot possibly add up enough '0's to make 6.
0+0 < 6,
0 + 0 + 0 + ... + 0 < 6

0/0 is indeterminate because:
0 + 0 = 0 -- Maybe it is 2.
0 + 0 + 0 = 0 -- Maybe it is 3.
0 + 0 + 0 + ... + 0 = 0 -- Maybe it is 9,473,841.
0*0 = 0 -- Maybe it's 0.
0+0+0+... = 0 -- Maybe it's just a REALLY BIG number.
Add up ANY number of '0's and you STILL get 0.

Apologies in advance for real analysis folks who already cringed at least three times during this brief discussion.

Okay, now let's get back to actual studies in elementary algebra.

 

[mathdad]

There is no mystery. In all cases, "a/0" and "0/0" are not, and cannot be, Real Numbers. If you refer to these sorts of expressions as FORMS, and not as actual results or values, then there is some possible discussion.

If we disregard what I just said, there is one, most elementary discussion:

6/3 = 2 because 2+2+2 = 6 -- It takes three (3) '2's to make 6.

6/0 is undefined since you cannot possibly add up enough '0's to make 6.
0+0 < 6,
0 + 0 + 0 + ... + 0 < 6

0/0 is indeterminate because:
0 + 0 = 0 -- Maybe it is 2.
0 + 0 + 0 = 0 -- Maybe it is 3.
0 + 0 + 0 + ... + 0 = 0 -- Maybe it is 9,473,841.
0*0 = 0 -- Maybe it's 0.
0+0+0+... = 0 -- Maybe it's just a REALLY BIG number.
Add up ANY number of '0's and you STILL get 0.

Apologies in advance for real analysis folks who already cringed at least three times during this brief discussion.

Okay, now let's get back to actual studies in elementary algebra.

Does calculus explain 0/0 mathematically?

 

[tkhunny]

As a FORM, it is studied.

 

[mathdad]

As a FORM, it is studied.

In calculus it is studied as a FORM.