Today I am going to show you actual proof that 0 ÷ 0 has a solution!! Let's go!




To begin with, let's recall that any number divided by itself equals 1.


x ÷ x = 1.


So at first glance, you may think that 0 ÷ 0 = 1.


BUT WAIT!!!

We should also consider that any number multiplied by 0 equals 0.


1 * 0 = 0
2 * 0 = 0
0.25 * 0 = 0
-4 * 0 = 0


We can also fit in some algebra into this:




If a and b are factors of a multiplication sequence, and c is the product, then

a * b = c.

But if these variables were reversed and used in a division sentence, then c would be the dividend, since it was the product in the first place.

So,

c ÷ b = a &
c ÷ a = b


But now, let's say that b and c are 0, and a is any number from -∞ (negative infinity) to ∞ (infinity).


Using those rules:

1 * 0 = 0
50 * 0 = 0
-950,000 * 0 = 0

Every sentence here is true.


But do you remember c ÷ a = b? The divisor and quotient have been switched, and the division sentence is still true.


So,


0 ÷ 1 = 0
0 ÷ 50 = 0
0 ÷ 610 = 0

These are all true...

You should also consider the fact that 0 divided by any number will equal 0.


And remember:



c ÷ b = a &
c ÷ a = b


are both TRUE STATEMENTS, although they may not be equal...


10 ÷ 2 =5      and      10 ÷ 5 = 2 are both true statements...


50 ÷ -10 = -5         and      50 ÷ -5 = -10 are also true statements....

it all has to do with the position of the divisor and quotient.

But what about the division statements from earlier?



0 ÷ 1 = 0
0 ÷ 50 = 0
0 ÷ 610 = 0


What if we switched the divisor and quotient with these?


0 ÷ 0 = 1
0 ÷ 0 = 50
0 ÷ 0 = 610


And also, remember this:

If a * b = c, then c ÷ b = a! That's the inverse operation rule!!

So then...

1 * 0 = 0
50 * 0 = 0
610 * 0 = 0

These are true.


If the three other versions are true, then wouldn't the 0 ÷ 0 equations be true?

If that's the case, then....

CONCLUSION:

0 ÷ 0 has an infinite number of solutions.


So there you have it!



I hope you found it helpful!

;D