# Logic Puzzle: Is 0=1 When ∞=1/0?

• ojsimon
In summary, the conversation discusses the undefined nature of division by zero and the concept of infinity in arithmetic systems. It explains that infinity is not a number that can be used in arithmetic, and using it can lead to contradictions like 0=1. The conversation also mentions the existence of arithmetic structures that include infinity, but even in those, the product of 0 and infinity is undefined.
ojsimon
∞=1/0

so: 0* ∞=1

so: 0=1

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

Thanks

Division by zero is undefined.

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

ojsimon said:
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 ∞.

DaveC426913 said:
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 said:
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.

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

## 1. Is it possible for 0 to equal 1 when infinity equals 1 divided by 0?

No, it is not possible for 0 to equal 1 when infinity equals 1 divided by 0. This is because division by 0 is undefined, and infinity is not a number that can be used in regular arithmetic operations.

## 2. How can 0 equal 1 in a logical puzzle?

In a logical puzzle, the numbers 0 and 1 can be used as symbols to represent different values. Therefore, in this context, 0 could potentially be used to represent 1, but it does not mean that the actual values of 0 and 1 are equal in regular arithmetic.

## 3. Can this logic puzzle be solved using mathematical principles?

Yes, this logic puzzle can be solved using mathematical principles such as algebra and logical reasoning. However, the solution may not align with the traditional rules of arithmetic.

## 4. Is this puzzle meant to challenge the laws of mathematics?

No, this puzzle is not meant to challenge the laws of mathematics. It is a fun and creative way to think about numbers and their representations, but it does not change the fundamental principles of mathematics.

## 5. What is the significance of using infinity and division by 0 in this puzzle?

The use of infinity and division by 0 adds an element of complexity to the puzzle and challenges our traditional understanding of numbers and their operations. It allows for a unique and thought-provoking solution to the puzzle.

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