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There is a popular illustration called the Hilbert’s paradox of the grand hotel.

Suppose Hilbert’s hotel has an infinite number of rooms and infinite number of guests are booked into the hotel. By common sense, it seems like the hotel is fully booked right? Wrong. Infinite sets just defy logic. Suppose there was another guest who wanted to book into the hotel, all the hotel staff have to do is just shift guest in room number 1 to the next, the guest in room number two to the third and so on… So by this logic

∞ + 1 = ∞ ” id=”MathJax-Element-3-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}+1=\mathrm{\infty}$Similarily

∞ − 1 = ∞ ” id=”MathJax-Element-4-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}-1=\mathrm{\infty}$. Just remove the guest from room number 1 and shift the remaining guests to the predecessor of their room numbers. You still have an infinite number of guests.

Let’s apply the logic to your question. Seemingly

∞ − ∞ = 0 ” id=”MathJax-Element-5-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}-\mathrm{\infty}=0$.

But suppose we remove the guests which are present in the rooms having an odd number(1,3,5…..) we still have infinite number of guests. So we get

∞ − ∞ = ∞ ” id=”MathJax-Element-6-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}-\mathrm{\infty}=\mathrm{\infty}$.

Let’s remove all the guests except the ones present in the first 50 rooms. So

∞ − ∞ = 50 ” id=”MathJax-Element-7-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}-\mathrm{\infty}=50$. You see where I’m going with this? Simply that

∞ − ∞ ” id=”MathJax-Element-8-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}-\mathrm{\infty}$is indeterminable.

Usually,

1 ∞ ” id=”MathJax-Element-14-Frame” role=”presentation” tabindex=”0″>

$\frac{1}{\mathrm{\infty}}$is nonsensical, because

∞ ” id=”MathJax-Element-15-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}$is not a number; it’s a symbol used in limits to mean “without limit”, and doesn’t really have meaning outside the concept of limits.

However, there is the case of the Riemann sphere, which extends the Complex Numbers by adding a multiplicative inverse to zero called

∞ ” id=”MathJax-Element-16-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}$:

With the Riemann Sphere,

∞ ” id=”MathJax-Element-17-Frame” role=”presentation” tabindex=”0″>

$\mathrm{\infty}$is a number, and

1 ∞ = 0 ” id=”MathJax-Element-18-Frame” role=”presentation” tabindex=”0″>

$\frac{1}{\mathrm{\infty}}=0$.

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See this

1 1000 = 0.001 ” id=”MathJax-Element-19-Frame” role=”presentation” tabindex=”0″>

$\frac{1}{1000}=0.001$1 100000 = 0.00001 ” id=”MathJax-Element-20-Frame” role=”presentation” tabindex=”0″>

$\frac{1}{100000}=0.00001$Now taking an extremely large number

1 1000000000000 = 0.000000000001 ≈ 0 ” id=”MathJax-Element-21-Frame” role=”presentation” tabindex=”0″>

$\frac{1}{1000000000000}=0.000000000001\approx 0$So we see that as large the number gets in the denominater, closer the fraction gets to

0 ” id=”MathJax-Element-22-Frame” role=”presentation” tabindex=”0″>

$0$So if we keep on increasing the denominater, we will end up getting closer and closer to

0 ” id=”MathJax-Element-23-Frame” role=”presentation” tabindex=”0″>

$0$.

Thus

1 ∞ = 0 ” id=”MathJax-Element-24-Frame” role=”presentation” tabindex=”0″>

$\frac{1}{\mathrm{\infty}}}=0$This can be better put up by saying

lim x → ∞ 1 x = 0 ” id=”MathJax-Element-25-Frame” role=”presentation” tabindex=”0″>

$\underset{x\to \mathrm{\infty}}{lim}{\displaystyle \frac{1}{x}}=0$This is read as, “As

x ” id=”MathJax-Element-26-Frame” role=”presentation” tabindex=”0″>

$x$tends to infinity,

1 x ” id=”MathJax-Element-27-Frame” role=”presentation” tabindex=”0″>

$\frac{1}{x}$equals 0.

According to the definition of limit if ,

We have to calculate the value of

Limit n tends to infinity of 1/n then it is equal to zero.

But here the value of 1/infinity is not equal to zero.(Because when a no. is divide by sufficient large no. then its value come out to be practically zero,but not actually).

I think the above reason is sufficient to explain the fact.

Infinity can be defined in many ways,

In our case say,

Infinity = A/0, except A=0.

Ie., Any number ( except 0 ) divided by 0 leads to infinite value. That’s y infinity = anything/0

Example,

“2/0” which means 2 is divided 0 times which clearly results in Infinite value.

Similarly it is applicable for all Numbers ( Real + Complex ) except 0.

Let’s see in vise versa case,

“1/infinity” which means 1 is divided infine times surely the result will approaches to Zero because, let’s see

1/1 = 1

1/2 = 0.5

1/4 = 0.25

1/8 = 0.125

…..

1/100 = 0.01

1/1000 = 0.001

1/10000 = 0.0001

As it continues..

As denominator becomes larger, the result approaches to 0.

So, 1/Infinite = 0

This not only for 1/infinity, it is applicable for all the numbers except 0.

Therefore we conclue that, Anything/infinite = 0, except anything=0.

First of You need to know the when we increase the denominator numerator remaining constant value decrease for example-

- 1/1=1
- 1/2=0.5
- 1/3=0.33
- 1/4=0.25
- 1/5=0.20

Now we should talk about infinity

infinity is extremely HUGE number (nobody can find it till date) being the largest number when divides 1 gives a extremely small number example

0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 UP TO INFINITY

this value is very very close to ZERO that why we take 1/infinity=ZERO

I hope I am clear !!!!

∞ does not belong to the Real Line, it belongs to the Extended Real Line (R+), where R+ is defined as R U {+∞, -∞}. Check Extended real number line – Wikipedia

So, if your question is : what is 1/∞ , by definition of ∞, it is 0.

But if your question is : what is Lim x->∞ 1/x, then it is still zero, because the limit is zero (since take any epsilon neighbourhood around zero, and after a certain value of x say M, all the 1/x will fall in that epsilon neighborhood. So, by definition of limits, the limit of 1/x is zero.)

No

Imagine having a napkin. Now, split that napkin in half. You now have 2 half-napkins, meaning each piece is 50% of the original napkin. Now, split each half-napkin in half. You now have 4 quarter-napkins, meaning each piece is 25% of the original napkin.

Okay, now imagine take a napkin and splitting it into infinity pieces. You just can’t. Infinity is not a true number, it is more of a concept. You can imagine splitting it into a million pieces, a billion pieces, etc. However, you cannot imagine splitting a napkin into infinite pieces.

No matter how many pieces you split it into, you can still split it smaller. That is infinity. Infinity is infinite. It cannot be added, subtracted, multiplied, or divided. It is not a number, it is a symbol of a concept.

As the division operator takes precedence over the subtraction operator, the expression simplifies to

1-∞.- ∞ is not a Real number, so if this is your domain, the expression is undefined.
- ∞ is an unsigned number in the projective extension of Real numbers,[1] and in this system, 1-∞=∞.
- -∞ and +∞ are distinct numbers in the affinely extended Real number system,[2] so here, 1-∞=-∞.

I hope this helps; you did not affix a Comment directly to the Question to give context as to the domain.

Footnotes

Most likely an infinite portion of 1.

Like 0.01 with infinite 0sThe first thing you have to understand is that infinity is not a number. Infinity is a concept of a number so large that we can’t conceptualize it. So because infinity is not a number it often does not behave like a number. For example we can’t determine what infinity divided by infinity is, and if you add any real number to infinity it will still be infinity. And if you think of multiplication as just a quicker version of addition you will realize that infinity times infinity is still equal to infinity.

In some higher maths they deal with varying levels of infinity. These specific infinities can have different rules than just a general “Infinity”. My personal favorite is the smallest of the infinite ordinals:

Aleph Null is the set of all natural numbers. Being a set it isn’t really a number but I like to think of it as the smallest of the impossibly large numbers. If you want a massive number that is actually a number you should look into Busy Beaver numbers. These numbers are so massive we haven’t even determined BB7 yet, much less BB1000 or BB1000000.

Here is a picture of infinity times infinity:

But really you can’t multiply infinity in the same way you multiply 1 times 1 or 5 times 10, because it isn’t truly a number. We just imagine it that way to help us grasp the concept.

*Edit: fixed a couple typos

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