Cannot divide by zero!

We cannot divide by 0, see it by using long division.

[Pining_for_the_fjords]

I convinced myself in my childhood that you cannot divide by zero by asking "If you have 10 sweets and zero people, how many sweets do they get each?" The fact that there are no people to share the sweets among makes the question nonsensical, just like asking "If I don't have a dog, how old is my dog?"

[Faladrin]

1/0 = 0 remainder 1

MIND BLOWN.

j/k

[solstakao]

This is my 'proof'
1*0=0
2*0=0
Therefore 1*0=2*0
Now divide by 0
1=2

[alejandrosegovia]

"Crazy as using purple pen" lol

[Bignic]

The coolest part, to me, is that division by 0 is undefined purely because of the property that x + 0 = x for any number x whatsoever. From this, one can show that

x*0=x*(0+0), by the above property
-> x*0=x*0+x*0, by distributing
-> 0=x*0, subtracting x*0 from both sides

Hence x*0=0 for any number x only makes use of the fact that 0+0=0 (and the cancellation property of subtraction). Next, suppose 1/0 has a real number solution, so that 1/0=x for x a real number. Then:

1=x*0, multiplying by 0 on both sides
-> 1=0, by the property we just proved

Which is obviously nonsense. But hold on! Maybe you can accept that some weird number system exists where 1=0. Can we work with that? Well, remember the fundamental property of 1, that 1*x=x for any number x. If we allow 1=0, then:

1=0
-> 1*x=0*x, multiplying by x on both sides
-> x=0*x, using the property above
-> x=0, using the fact we already proved

But remember that x stands for any real number whatsoever! So if 1=0, then every other number is also 0, and so our number system is quite sparse indeed.

This is all in the area of abstract algebra, personally my favourite area of math. In a subject in algebra called ring theory, one can show that if we have any abstract "number system", which can actually be functions, matrices, vectors, sequences... almost anything you've encountered in math can be investigated in ring theory, and in this system we have objects 0 and 1 so that x+0=x, and 1*x=x, where 0 and 1 aren't the familiar numbers but rather any two objects of the same type, and + and * are any operations satisfying familiar properties like distribution, then one can never divide by 0.

[alexatg]

a÷b=c is actually a=b×c
If b=0,
a=0×c
But we know that 0×any number is zero,
Therefore a÷0 ≠ c assuming a ≠ 0
If a=0 on the other hand, 0=0×c where c can be any number, 0÷0=any number which is also not, well, logical. Therefore any number÷0 is not defined.

[GhostyOcean]

Purple pen?!?! UNSUB, DISLIKE, REPORT!

[Crojach]

"This is just as crazy as that I am using a purple pen" :D

[infinitymfg]

Proof by contradiction. I learned a new term, I never knew what this method was called.

[Mat_Rix]

For example I presented graph of the function f(x) =1/x ... without discontinuity point ! :) It can be presented for every x, and I'm also explaining why our traditional (wrong) graph has discontinuity at 0.
If you really want to understand it ... you need to read it and understand all presented examples.
Enjoy :)

[snakespeak]

we need more math professors like this

[danuttall]

The technique I use with my students is to consider that division is the reversing operation of multiplication.
So 12 divided by 4 is the same as asking, "What number, when you multiply by 4 gives you 12?" The answer in this case is 3. Let's try dividing by 0. 12 / 0 is the same as asking, "What number, when you multiply by 0 gives you 12?" Well, anything times 0 is 0, so we have no answers. Let's try another way: 0/0. What number, when you multiply by 0, gives you 0? This time any and all numbers qualify. We have gone from famine to feast, from no answers to any number as the answer.

I then explain we have a technique we will cover in grade 12 Calculus (limits) which can help us determine which 0/0 is the correct one for our context, but we won't look at this right now (unless this is at the start of the calculus class). For now, we will call 0/0 indefinite and anything else over 0 undefined.

[Shadow]

If I didn't know that it was true, I could argue that probably the method of long division is flawed or it might work for any divisor that is not 0. While it doesn't find the answer to a division by 0, it might still exist.

[angelcaru]

Another way to convince:
Supose we can divde by zero
0=0
1(0)=2(0)
1(0)/0=2(0)/0
We cancel and we get
1=2

[RashadSaleh]

Nobody on the Internet seems to understand this. The real answer is because *division by zero is not defined.* Dividing by zero is like abruptly changing the topic in a conversation: You start talking about division, which is well defined for non-zero, then suddenly a new 'division' appears that _looks_ like real division to the eye, but actually exist outside the scope of real division. The concept or 'sense' of division, which underlies division operations, was not generalized to include division by zero the way, say, addition was generalized to include negative operands.

But if this causes disagreement, division in the comments won't be by zero I hope.

[tomraj]

I wouldn't call this very convincing. People for a long time said, "Of course you cannot take the square root of a negative number. Let me show you why: we start by asking 'what is the square root of four?' Well, what number squared gives four? Two does! (We could use Newton-Raphson as an algorithm). Now what number squared gives negative four? See, it doesn't make sense!" Lo and behold, we later introduced the imaginary unit and complex numbers, and these became very important in many parts of mathematics, and it all, in the end, makes sense! Why couldn't there be a system of numbers we could introduce, where they'd be defined as a division by zero? A good answer to that would, in my opinion, be much more convincing!

[martinepstein]

Here's a proof that works in any nontrivial division ring (which is what the 4th graders REALLY want)
1/0 = x . . . cross multiply
1 = x*0 = x*(0 + 0) = x*0 + x*0 . . . by the distributive law
x*0 = x*0 + x*0 . . . subtract x*0 from both sides
x*0 - x*0 = x*0 + x*0 - x*0 . . . simplify
0 = x*0 = 1
0 = 1

[fgme]

Black pen
Red pen
Blue pen
Purple pen
Pink pen
Orange pen
Yellow pen
Green pen

[MushroomManToad]

I like this. This guy takes something math teachers just tell you that you can't do, and uses math that a 4th grader could do to prove it in a fun way. You actually made me chuckle at how obvious this is to prove without using infinities and such. Nicely done!

[shacharh]

It's easy to prove that 0a = 0 for all a (except 0; rightside multiplication by 0 is ill-defined) using ordinal arithmetic.

Consider: for all a, a + 0 := a, a*1 := a
For a consecutive ordinal b: a*b := a*(b-1) + a, for a limit ordinal b: a*b := sup {a*x | x<b}.

So 0 *1 = 0, next you can inductive show for all positive integrers using regular induction or for all ordinals using generalized induction: suppose for all x<b 0x = 0 then ob = 0(b-1) + 0 = 0 + 0 = 0 from the definition. If b is alimit ordinal then 0b = sup{0x | x<b} = sup {0} = 0.