Logic Puzzle: Is 0=1 When ∞=1/0?

[ojsimon]

∞=1/0

so: 0* ∞=1

so: 0=1


Why is this, or have i made a mistake somewhere in my logic?

Thanks

 

[D H]

Division by zero is undefined.

 

[DaveC426913]

Also, infinity is not an entity upon which you can do arithmetic.

 

[Hurkyl]

Why is this, or have i made a mistake somewhere in my logic?

You probably have made a mistake. Have you learned any arithmetic systems that include a number called ∞? If not, then you made a mistake simply by writing ∞.

 

[Hurkyl]

Also, infinity is not an entity upon which you can do arithmetic.

If by "infinity" you mean "a vague and undefined notion that you don't really know anything about", then you're certainly correct.


But there are arithmetic structures that do include an object / some objects called infinity, and you can do arithmetic there. e.g. in elementary calculus you (implicitly) learn about one such number system: the extended reals.



Of course, in the extended reals, 1/0 is undefined, as is 0*(+∞).

And even in the projective reals where 1/0=∞, 0*∞ is still undefined.

 

[Hurkyl]

You probably have made a mistake. Have you learned any arithmetic systems that include a number called ∞? If not, then you made a mistake simply by writing ∞.

Just to expand on this...

In mathematics (and other disciplines) we might speculate -- we might consider "if we had a number system with something called ∞, how might that work out?"

And as your calculation shows, if we had such a number system with the properties that 1/0=∞, division has the property that a/b=c implies a=bc, and multiplication has the property that 0*x=0, then the number system couldn't be very useful because we could prove that 0=1.

But the calculation in this hypothetical number system says nothing about the real numbers (or the complex numbers, or the integers, or the extended real numbers or anything else). And it tells us that if we want a number system that contains ∞, we shouldn't insist on all of the properties we used in the above calculation.

 

[g_edgar]

There is a number system (the Riemann sphere) that has an element $$\infty$$.
But even there, the product $$0\cdot\infty$$ is undefined.