Why is 0/0 indeterminate, but 1/0 is not?

Why do we say that some things are "indeterminate" instead of "undefined"?

EXERCISE FOR THE VIEWER: Suppose we define a new number "x" such that 0x = 1. Show that this number violates the distribitive property.

Music:
Blue Wednesday, All That Jazz

"Fireside"
Instrumental by Homage

Outro:
Lateralus, as performed by Sakis Strigas
Originally by Tool

My current equipment for making videos (affiliate links):

[moskthinks]

Suppose 0x=1, then by an axiom, 0=0+0, so
1=0x=(0+0)x
Hence, by the distributive property:
(0+0)x=0x+0x=1+1=2
Which leads to the contradiction 1=2.
Therefore, by defining that new element x, it violates the distributive property.

Nice exercise!

[AndrewDotsonvideos]

Um try again sweetie 1/0 is infinity check your privilege

[alkankondo]

This was THE BEST explanation I've seen of why 0/0 is "indeterminate"! Not only was it packed with content, but also it was very easy to follow and understand!

[toferg.]

I’m watching this, because i expect you to explain it so well — not because i need to learn it.

[MathManMcGreal]

this one is a sleeper hit. definitely saving this for "the talk" during my L'hopital's lesson!

[shawon]

Very clear and concise! Here’s what I would like to add: the term "undefined" here is referring to the definition of the division operator.

The division operator is literally not defined for 0. Like literally, division operation is defined as "the inverse operation of multiplication, i.e. a/b=c → a=b*c where b≠0". So, anything divided by 0 is undefined (e.g. 0/0, 1/0, -3/0 etc.). Out of these, 0/0 is both undefined and indeterminant (as mentioned in the video).

[VibingMath]

Thank you EMT! This is the most understandable explanation for the two things

[billywilliam]

I like the informal way you explain it - you don't mean to show something but to solve the problem. Thanks and cheers

[camdenwhite]

If you multiply 0 on both sides of 1/0 = n, you get 0 = 0, unless you are already assuming that 1/0 is the inverse of 0. If we assume that 1/0 = 0, we find NO CONTRADICTIONS (at least none that I have seen). This goes for any n/0 where n is any complex number (including zero). Furthermore, 0^n is also zero for all n (including negatives). Just keep in mind that zero times anything is zero!

[prestondebetaz]

I’m an incoming freshman into university and I am a math major so I will maybe one day finally understand all of your videos!! However, the calculus based ones (the few that I get now) are fantastic, keep up the great work!

[martinshoosterman]

The riemann sphere would like to have a word with you.

[mikinovak]

This video has had no dislikes until now and it better stay that way.

[ViewHog]

Consider x*(0+0). 0+0 = 0 thus x*(0+0)=x*(0) which equals 1 by definition. However, by the distributive property, x*(0+0) = x*0+x*0. This is equal to 1+1 or 2. 1 /= 2 so the number x with the property that 0x=1 consequently violates the distributive property.

[legomatiks]

Suppose 0x=1 and consider a in A (a not 0) where A is a set such that it satisfies distributive property i.e a(b+c)=ab+ac for all a, b, c in A. Then, 0= 0(a+x)=0a+0x=0+1=1 hence, we get the contradiction 0=1.

[perappelgren]

I would say the derivative of |x| at x=0 is slightly less indeterminate than hardcore 0/0. The possible values of the slope of the tangent go from -1 to +1. This restriction is what makes it a chicken 0/0 😎😎

[angelmendez-rivera]

A fun fact is that 1/0 and 0/0 are undefined in the field of real numbers, but there exists a wheel W with binary operators + and ·, additive and multiplicative identities 0 and 1, and the involution /, such that 1·/0 and 0·/0, abbreviated as 1/0 and 0/0 respectively, are well-defined elements of W. The wheel axioms generalize the field axioms, by generalizing the distributive property. Interestingly, addition and multiplication are still commutative and associative for all elements of W, so this is non-destructive generalization of the field.

[wiggles]

Undefined: used to describe the "value" of a function's output when the input is not sensible or suitable.


Indeterminant: used to describe the nature of a variable expression whose value could change depending on context.


That's all you really need to know.

[alanhere]

Maybe the full and empty solution sets could be useful here.

For example approaching from any direction the (0, 0) limit re N/M yields any solution.

[hydraslair]

I love your videos, although instead of saying "by multiplying by zero" and canceling out the zeros (wouldn't that mean you have established that 0/0 = 1, or that 1/0 is a unit and 1/0*0 = 1?) I would just have said that what we mean by fractions a/b is "the number q (in reality, equivalence class) such that a = qb" which makes a little bit more sense to me looking at the definition of field of fractions and all that.

[pipertripp]

Great vid... let me try my hand at the challenge at the end... no sure if this is any good, but here goes:

the distributive property states that a(b + c) = ab + ac
let 0x = 1 -- now, add 0 to both sides of the equation... this give us
0x + 0 = 1 + 0 -- now, factor out a zero on the lefthand side (this might be the bad part, but I'm doing it anyways dammit)
0(x + 0) = 1 + 0 -- use the additive identity to clean up the righthand side

0(x + 0) = 1 -- and now divide both sides by zero
(x + 0) = 1/0 -- apply additive identity to lefthand side
x = 1/0 which means that x is undefined.


Was that circular? I like the one where the guy showed that 1=2 better.