does infinity minus infinty always equal zero? I've been told yes and no with plenty of proof for either side. so which is it?
say you have an x and y coordinate plane, where y=5. If you take everything above that line, and subtract it from the bottom, you still have 5 left over. but both portions are technically infinty, because a coordinate plain extends infinitely.
\(\infty-\infty\) is undefined, meaning \(\infty-\infty\) can take any value - even infinity! The problem is that infinity is not a number so the usual operations like addition and multiplication don't work.
"The problem is that infinity is not a number" Exactly. So it can mean different things. If we are talking about simple numbers .9999 to infinity is understood. If we are talking about things that have no beginning and no end it is harder to apply the idea in mathematic terms.
Consider: \((\infty + 5) - (\infty + 3) = ?\) If infinity take infinity is assumed to be zero always, then the answer to the above problem is \((\infty - \infty) + (5 - 3) = 0 + 2 = 2\) Right? But consider. We are also told that \(\infty + 5 =\infty\) and \(\infty + 3 = \infty\) That is, any finite number added to infinity gives infinity. So, the initial problem above reduces to \((\infty + 5) - (\infty + 3) = \infty - \infty = 0\) accoring to our original assumption that infinity take infinity is always zero. So, using the assumption that infinity minus infinity is always zero has led us to two different answers to the problem stated here. And this is simple arithmetic. Something must be wrong! What's wrong must have to do with one of our assumptions about infinity. The short answer to your initial question is that infinity take infinity is NOT always zero. In fact, the "answer" to infinity minus infinity is undefined in mathematics.
Like asking what happens in Cluedo (Clue) when you pass Go. Like asking what happens in Football (Soccer) when you have a 3 of a kind. ...
Thats not quite right. \(\sum_{n=0}^\infty \frac{1}{2^n}\) has infinite values and is perfectly well defined (it's 2). The problem is when treating an object that is infinitely large, that is when the object is larger than any real number you can write down and is usually called \(\infty\). There are systems to deal with infinite numbers - the hyperreals and the surreals are both number sets that include infinite numbers and infinitesimals, but I'm afraid that is where my knowledge on them ends.
The pretext or assumption is a misnomer. Infinite is not a number to be addded, subtracted, divided or multiplied. Convince me otherwise.:shrug:
How about: "infinity is assumed to have indefinite value, which is always outside any real value". Infinity is kind of the idea of value itself, since it's always beyond the 'valuation process'
"Infinity is larger than any real number," is just fine, because you don't have to start getting into words like "assumed" and "indefinite" with this definition.
"any real number" suggests an indefinite number, though. A number that can be any value, then one that's always "beyond" any such value, which has to be an assumption because you can't ever get to it to see what it is. It's existence is assumed because numbers are "real". Infinity is real, then.
You are mistaking the mathematicians' definition of 'real' with the usual definition of 'real'. No it doesn't. The sentence "Infinity is larger than any real number" means that given a real number, z, infinity is larger than it, \(\infty > z\). 5? Infinity is bigger. 20? Infinity is bigger. \(10^{10^{10^{10}}}\)? Infinity is bigger. There is no \(x \in \mathbb{R}\) such that \(x>\infty\). Nothing vague about that.
Yes it bloody does. "Any value", presumes the actual value is indefinite but real. So if there is a value "beyond any value", then it's indefinite too. We connect to the idea of an infinite value (which can never exist in reality), to the idea of counting beyond a value. This gives us "infinite" value, or a value which is logically a value but real only in the sense you really count. Infinity is logically real, but not actually, like an actual real value is. Since we can imagine numbers we can imagine a logically infinite number. You're making the common mistake of also imagining that imagination is "real". When what imagination is is really imaginary.
No, it doesn't. "Larger than any real number" is similar to the well defined analysis concept of "|x|<d for all d<0". This means that |x|=0. Larger than any real number means that infinity is not a member of the reals. If it were then infinity+1 would be a member of the reals, since the reals are a field, and you'd have a real larger than infinity. You continue to think of the mathematicians use of 'real' and 'imaginary' as the same as the everyday use of those words. They aren't. If a number is real then it means that it belongs to the set which is defined as the Cauchy completion of the rationals. If a number is imaginary then it means it's a member of the set produced by multiplying each and every element of the set of reals by i, where i*i = -1. The complex numbers are not 'complicated' but the set of numbers formed by linear combinations of elements from the set of reals and the set of imaginaries. When I say "5 is real" I don't mean there's a planet somewhere which is populated by 5's, I mean that it satisfies the property that it's the limit of a Cauchy convergent sequence of rational numbers. It's the same as saying "3 is odd". I don't mean 3 is a bit weird and I don't want to hang around with it but I mean it can be represented in the form 2n+1 for an integer n. You are making the common mistake of thinking that because you've never opened a book on analysis in your life and you keep reading people using words you think you know the meaning of then your ignorant assumptions about end results are true. Or put another way, you're being a crank.
There's no 'processes' when defining a set. If I write down the definition of the set of integers, I don't have to wait a few seconds for the integer 100000000000000000000000000000000000000000000 to be included in that set or to become well defined. Saying "Infinity is never reached" is a vague and thus poorly defined definition.
