To 'Infinity' And Beyond! 2

You also have to remember that you can have different densities of infinity, so could never know if the infinities in the fraction have the same density. The only answer therefore is undefined.

infinity isn't a defined number so there isnt a number for you to equate it to. I definitely see what you are saying but seeing as infinity is concept and not a number you cannot define it BY a number

Infinity can actually be a lot of different values. That doesn't seem to make sense, but think about this. There are an infinite amount of positive integers, and there are an infinite amount of integers. If you subtracted the sum of all the positive integers from the sum of all the integers, you just have the set of negative integers. (that isn't 100% correct, but for now just picture that) You can see that infinity - infinity here is not 0 or infinity, it is actually -infinity. You could think of a lot of values of infinity where if you take their sums which equate to infinity, and subtract the two, you get weird values. if we thought about it enough, we could get infinite values subtracted from others to get an real number.

You may add or subtract a finite number from an infinite number and still have infinity.

There are degrees of infinity, which is why an operation containing two infinites is undefined (unless the compositions of the infinite sets are given).

I'll explain it without going into a mathematical proof, though it could easily be done.

Consider the set $\left\{1,2,3,4 \ldots \right\}$ (the set of natural numbers, regardless of whether you consider 0 to be in this set).

I think we can agree that this is an infinite set. Now let's add one more number to this set, 0, to make it $\left\{0,1,2,3,4 \ldots \right\}$ . This operation would be $\infty + 1$ . I would think that we can agree that the result is likewise infinite. Hence, $\infty + 1 = \infty$

However, suppose we take two infinite sets and subtract them ( $\infty - \infty$ ). Unless we define the composition of these infinite sets, we have no way of knowing what the result would be.

For example:

$\infty - \infty$ by removing all values within one set from another, with the two sets being identical $\left\{1,2,3,4 \ldots \right\}$ . This would yield $\infty - \infty = 0$ in that the resultant set would be empty.

$\infty - \infty$ as given by the first set being all integers $\left\{\ldots -3,-2,-1,0,1,2,3 \ldots \right\}$ and the second set being all naturals $\left\{1,2,3,4 \ldots \right\}$ . The result would be another infinite set (all negative integers with the addition of zero). Thus, $\infty - \infty = \infty$

So by now, hopefully you can understand that one infinity may be greater than another.

So:

$\infty - \infty = -\infty$ is a possibility too. As is $\infty - \infty = \dfrac{1}{2}$ , or any other value.

This is why we say equations with two infinites are undefined. Because the symbol infinity tells us nothing of the actual size or composition of that infinite number.

I believe no definitive proof has been published to support that equality.

If I were you, I would simply not attempt that examination... $\ddot\smile$

If someone asks the area of a right triangle with sides (3, 4, 5), and 6 is not in options, what would you do?

You're assuming a value for infinity. There are many different degrees of infinity, but infinity itself is undefined. For instance, if we take the set of all natural numbers [1, 2, 3, 4...] , which is infinite, and divide that by itself, the answer would indeed be 1. However, to divide it by the set of all real numbers (anything that be shown on the number line), the answer would be less than 1, seeing as real numbers encompass all naturals, integers, rationals, irrationals and transcendentals. Dividing 1 by this fraction (thus switching the numerator and denominaor), whilst still using the naturals and reals, would still be infinity/infinity, but the answer would this time be greater than 1. Because of the massive amount of possible answers (one could argue it is infinite, since there is theoretically an infinite number of infinite sets), the answer is simply undefined.

According to my calculations infinity-infinity=0, why is everyone so confused? this is a really simple question