Why is ?(infinite) not considered a number?
I mean, if zero is considered a number, it sounds reasonable to consider infinite a number too.
Why is infinite technically not a number.
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It depends on what field of mathematics you're talking about. There is a field that works with infinity as a "number." Considering the Arabic basis of numbers, a system of powers of ten placeholders, zero indicates that the number represented has no component of the power of ten at the position the zero occupies. I.e., 1023 = 1*1000 + (nothing)*100 + 2*10 + 3*1. In that context, a sequence of multiples of powers of ten, infinity is of no use. It would not have meaning as a power of ten in a specific position in an Arabic base number.
In Roman numerals, there is no concept of zero as it is not a placeholder-type system.
You pays your money and you takes your choice.
In Roman numerals, there is no concept of zero as it is not a placeholder-type system.
You pays your money and you takes your choice.
Infinity is a concept and doesn't work like a number. If you add 1 to infinity you still have infinity. On the other hand if you add 1 to 0 you get 1.
Look at it a different way - zero has a specific place on a number line. Where is infinity? You would say the number of points between zero and one is infinite and not a specific point.
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For example, in set theory,
M??=?
Set M union universal discourse equals universal discourse.
Author
Infinity is a concept and doesn't work like a number. If you add 1 to infinity you still have infinity. On the other hand if you add 1 to 0 you get 1.
Look at it a different way - zero has a specific place on a number line. Where is infinity? You would say the number of points between zero and one is infinite and not a specific point.
Actually, if you add one to infinity you get ?+1=?
Well, answer this questions and you might get the answer:
1 - What integer number comes before 0? And what integer number comes before infinity?
2 - What integer number comes after 0? And what integer number comes after infinity?
Basically, infinity means "a number bigger that any number you could think of", so you can't name ONE integer that comes exactly before or after.
When you say "why isn't it a number if zero is a number?" it doesn't make sense. How does zero being a number gives you enough information to conclude that infinity is a number too? Do you know about the mathematical concepts that define 0 and infinity? You can find hundreds of results in google about why 0 IS a number and why infinity isn't, if you want to suggest that infinity read some results and think again if that makes sense.
Maybe you did the basic operations with integers and infinity and tought "hey, this could be set as a rule: ANYTHING + infinity = infinity, ANYTHING - infinity = infinity", but it doesn't mean you've discovered something new. When you work with infinity you're really working with limits, and those are the rules that are beign applied, you can't do basic operations with it.
Infinity is a concept and doesn't work like a number. If you add 1 to infinity you still have infinity. On the other hand if you add 1 to 0 you get 1.
Look at it a different way - zero has a specific place on a number line. Where is infinity? You would say the number of points between zero and one is infinite and not a specific point.
Actually, if you add one to infinity you get ?+1
So... you get a "the next number after infinity"? a number bigger than infinity? Wouldn't that also be infinity? What dfinition of infinity are you using? Maybe that explains it all...
Zero compared to Zero is equal. There is only one value for 0, and that is 0.
Infinity compared to Infinity can be equal, greater, or less. "x²" grows at much higher rater than "x", and as such the infinity of the limit x² is "greater" (in a sort of speak) than that of "x".
In this case, if we try to subtract Inf - Inf; we could end up with x - x² which has a limit at negative infinity, or with x² - x; which has a limit at positive infinity. Although we're looking at the same symbol ? on both sides, turns out ? - ? is not the same as ? - ?
Infinity is a concept, not a number.
In a certain sense, Infinity can't be *a* number, because it is all numbers at once. There is no single value for infinity, There are also multiple infinities, and some infinities are demonstrably larger than other infinities. Eg, take the set of all odd integers and the set of all even integers. Clearly they are equal in size, but if you take the set of all integers, it is the sum of the odd integers set and even integers set, and is thus larger than either of them.
"Infinity" means several different things, some of which allows you to think of it as a number:
* If you are measuring how big sets are, you define two sets to be as big as each other if there is a bijection between them, and you say a set is infinite if it is as big as one of its proper subsets (in other words, if you add one element that wasn't in the set, you get a set exactly as large as the original). But thinking of the cardinality of an infinite set as being simply "infinity" doesn't really cut it, because there is a large collection of different sizes of infinite sets.
* If you are thinking of the behavior of sequences of real numbers in the limit, it is convenient to extend the real line to include +infinity and -infinity, so you can say that a sequence has limit +infinity or -infinity, meaning for any bound you propose, there is a point from which all the elements of the sequence are beyond your proposed bound.
* If you are dealing with the slope of a straight line, it makes sense to think of it as a real number or infinity (a single infinity, in contrast with the previous bullet point). In this case we are dealing with the projective real line. There is also an analogous "projective complex line", which consists of the complex numbers plus a number called infinity (this is also known as the Riemann sphere).
* Any topological space can be extended into a compact topological space by adding a single point called "infinity" (Alexandroff one-point compactification).
* The surreal numbers contain a number "omega" that you could call "infinity" instead (they call it "omega" because it is one very specific type of infinity: the first infinite ordinal). However, these surreal numbers are kind of weird, and they contain notions like "infinity minus 5", "infinity times 3", "infinity squared", and even "square root of infinity".
I would say the notions of infinity from the second, third and fifth bullet points fit well into considering it as a number. Notice how the floating-point representations of numbers on a computer typically include +infinity and -infinity, which leads me to believe they are based around the notion of infinity from the second bullet point.
* If you are measuring how big sets are, you define two sets to be as big as each other if there is a bijection between them, and you say a set is infinite if it is as big as one of its proper subsets (in other words, if you add one element that wasn't in the set, you get a set exactly as large as the original). But thinking of the cardinality of an infinite set as being simply "infinity" doesn't really cut it, because there is a large collection of different sizes of infinite sets.
* If you are thinking of the behavior of sequences of real numbers in the limit, it is convenient to extend the real line to include +infinity and -infinity, so you can say that a sequence has limit +infinity or -infinity, meaning for any bound you propose, there is a point from which all the elements of the sequence are beyond your proposed bound.
* If you are dealing with the slope of a straight line, it makes sense to think of it as a real number or infinity (a single infinity, in contrast with the previous bullet point). In this case we are dealing with the projective real line. There is also an analogous "projective complex line", which consists of the complex numbers plus a number called infinity (this is also known as the Riemann sphere).
* Any topological space can be extended into a compact topological space by adding a single point called "infinity" (Alexandroff one-point compactification).
* The surreal numbers contain a number "omega" that you could call "infinity" instead (they call it "omega" because it is one very specific type of infinity: the first infinite ordinal). However, these surreal numbers are kind of weird, and they contain notions like "infinity minus 5", "infinity times 3", "infinity squared", and even "square root of infinity".
I would say the notions of infinity from the second, third and fifth bullet points fit well into considering it as a number. Notice how the floating-point representations of numbers on a computer typically include +infinity and -infinity, which leads me to believe they are based around the notion of infinity from the second bullet point.
If you think like a mathematician, you can make your own definitions (as long as you take the consequences). So you can define infinity as a number if you like (e.g. as the sum of a sufficient number of ones so that it becomes larger than any finite number). And you'll end up with the surreal or hyperreal numbers. You may ask, are those numbers real? Well, are the real numbers real? And are the imaginary numbers just imaginary? Hmmm...
[edit]
As for the original question:
Assume x is a positive integer that is greater than any other integer. Let y = x + 1. Then y is greater than x (by rules of addition). But x was supposed to be greater than y. Contradiction. Assumption was false. So there can't be such an integer x.
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