Basic math with infinity

Physicist: Several questions about doing basic math with infinity have been emailed over the years, so here’s a bunch of them!  (More can be added later)

Infinity comes in a lot of shapes and flavors.  However, the most straightforward infinity that makes the most intuitive sense (for most people) is probably the infinity that “sits at the end of the number line“.  \infty is defined as the “value” such that given any number, x, we always have x < \infty.  “Value” is in quotes there (‘ ” ” ‘) because infinity isn’t an actual value, it’s more of a place-holder for “bigger than anything else”.

As soon as the word “infinity” is dropped anywhere on the internet the tone suddenly becomes a little… philosophical.  So, just to be specific: \infty > x, for any number x, and \infty is assumed to be the unique thing that has that property (Nothing more or less).

\infty + 1 = ?

Like you’d expect, \infty + 1 = \infty.  This is because \infty > x-1, for any x, and therefore \infty + 1 > x for any x, and therefore (because \infty is defined to be the only thing with this property), \infty +1 = \infty.  This brings up the interesting fact that \infty + 1 is not bigger than \infty (no matter what any second grader might say).  They’re the same.

\infty + \infty = ?

\infty again.  Pick any number, x, and you find that \infty\infty\infty + x/2 > x/2 + x/2 = x.  So, \infty\infty\infty.  This one isn’t too surprising either.

\infty\infty = ?

This is a bit more nuanced.  Your first inclination might be “0”, but keep in mind that that would mean that \infty\infty = (\infty + 1) – \infty = 1.  The “not-a-number-ness” of infinity means that subtracting it from itself doesn’t make sense, or at the very least, doesn’t have a definitive result.

\infty/\infty = ?

This is just as nuanced.  You may think “1”, because you’re probably more reasonable than not.  But consider, if that were the case, then: \infty/\infty = (\infty + \infty)/\infty = 2.  So, again, there’s no definitive result.

Is \infty even or odd?

Something is even if, when divided by two, the result is an integer.  But is \infty/2 an integer?  \infty is generally considered to not be an integer (or rational, or even irrational), so \infty isn’t generally considered to be either even or odd.


On a case-by-case basis you can sometimes have “disagreeing infinities” and figure out what they equal.  For example, 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}\cdots involves a positive infinity (1+\frac{1}{3}+\frac{1}{5}\cdots) and a negative infinity (-\frac{1}{2}-\frac{1}{4}-\frac{1}{6}\cdots), but if you add up the sum one terms at a time you find that it equals ln(2) = 0.6931…

But in general, the operations we freely use with ordinary numbers (addition, subtraction, ) need to be considered very, very carefully before they’re applied to infinities (or even zeros).  In fact, mathematicians almost never “plug in \infty“.  Instead, they “sneak up” on it using by limits.  For example, if you want to figure out what 1/\infty is, you say “what is the limit as the number x gets arbitrarily large of the function 1/x?”.  In this case you can reasonably say that 1/\infty = 0, but without actually plugging in weird stuff that doesn’t have an actual, numerical, value.

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16 Responses to Basic math with infinity

  1. Anonymous says:

    There seems to be a lack of lip service regarding negative infinity. And how about divisions by infinity? Is it safe to always consider these equal to 0 when the numerator exists? Are there any interesting or counterintuitive results arising from exponentiation by either signed infinity?

  2. Edward says:

    Did this article get cut off at the end there?

  3. The Physicist The Physicist says:

    Crap! It really did.

  4. Flavian Popa says:

    Limits and calculus are awesome, and were it not for Aristotelian view on infinity, zero, vacuum, Europe would have embraced the figure “0” and would have started working with “infinities” long before it actually did. Charles Seife has a wonderful book about how hard it was for “zero” to be accepted in Europe.

  5. Locutus says:

    All real numbers are hyperreal numbers, and all hyperreal numbers are surreal numbers. Infinity is a hyperreal and surreal number, but not a real number. Same for an infinitesimal.

  6. The Cool Dude says:

    I can see why ∞ would not be even, odd, or rational, but I don’t see why it can’t be irrational.
    An irrational number is simply anything that isn’t a rational number, and a rational number times an irrational number is another irrational number, and ∞√(2)=∞, so if infinity were rational, it would have to be irrational simultaneously, which defies the definition of being irrational, so it can’t be rational, and therefor must be irrational.

    How does infinity deal with the complex plane? If I add √(-1) to infinity, is it still infinity, or is it some kind of slightly different, barely imaginary infinity?

  7. Igor says:

    @The Cool Dude: The way to answer your questions is to replace the infinity symbol with X and ask, does the answer converge as X grows? For example, “X + 1” becomes arbitrarily large as X grows, so you can sort of say that “infinity + 1” is infinity. Similarly, “-1 / X” becomes arbitrarily close to 0 as X grows, so you can say that “-1 / infinity” is 0.

    Now, to your specific questions. You ask whether sqrt(2) * infinity is rational or irrational. Let’s transform the question: is “sqrt(2) * X” rational or irrational for large X? Well, there are large X values that make “sqrt(2) * X” rational, but there are also large X values that make it irrational. So, the answer does not converge as X grows, which is why we’d have to say that the answer is undefined.

    You also ask what is “sqrt(-1) + infinity”. It is just a complex number of the form “i + X”, where X is a large real number. If you are very careful, you can do certain calculations with “sqrt(-1) + infinity”, again by replacing “infinity” with “large real value”. For example, the absolute value of “sqrt(-1) + infinity” is infinity.

    BIG DISCLAIMER: This blog post uses one particular concept of infinity. There are different concepts of infinity in mathematics (e.g., see the “shapes and flavors” link early in this post), and for those, some of the answers are different. For example, “2^infinity = infinity” is true for the infinity as defined in this blog post, but false for other infinities. For more details, see my blog post from a couple of years ago.

  8. slick rick says:

    well i have always thought infinity to be zero. well because if you have a set of positive numbers and negative numbers and added them together for example 1+(-1)=0, 2+(-2)=0 and so on. As you tended to positive and negative infinity you would get a infinite number of zeros. Therefore are infinity and zero equal to each other and if this is not the case what is infinity multiplied by zero?

  9. Curious says:

    Was hoping to find this answer: Are there roughly twice as many integers as there are even integers? Is the amount of even integers half of the amount of all integers? Or are both == infinity, and therefore equal. That is, the number of even integers == the number of integers == infinity (?)

  10. The Physicist The Physicist says:

    There’s an old post here that answers that. Or at least tries to.

  11. Stan says:

    For me the word “infinity” is very similar to the word “all”. If we have a set of numbers, for example the set of even numbers we can add nothing more to it unless we take another different set of numbers (odd numbers). Then we can talk about infinity+1 or infinity+infinity. If we have only one set of numbers adding to infinity and multiplicating it (with number>1) are not allowed. If we have a set including all of the even numbers it is simply impossible to add another even number. We can add an odd number but it comes from a different set.

    About infinity-infinity my personal opinion is that it is undefined if we have to sets. If we take the sets of odd and even numbers for example again, we have three possibilities: they are equal; we have one even number more; we have one odd number more. But if we have only one set of numbers, I think, infinity-infinity=0. It is just the same thing we subtract from itself.

  12. Jai says:

    A very useful website, thanks. !

  13. Pingback: Infinity

  14. Shaun (rhymes with yawn) Cena says:

    An odd number can be written as (odd #)/(2n+1)=1 where n is 1/2 odd# rounded down, for example: 15/2•7+1=15/15=1
    1/2 of infinity is still infinity, so infinity divided by infinity plus one should equal one if infinity were too be odd. However, this is clearly not the case.
    Another example of an odd # is where an odd number divided by two results in a modulo of one. Infinity/2=infinity, and with this there is a modulo of zero.
    An even number can be written as (even #)/2 results in a modulo of zero. We have already determined that infinity divided by two results in a modulo of zero.
    Therefore, infinity is even.

  15. ENGLISH BOB says:

    Why do mathematicians make life so complicated. And who decided what you can and cannot do ? I know this is a mathematical question, but it does seem a bit over the top. As a layman , infinity means never ending. so of course you can’t add 1 or divide by 2. But when you start dealing with number, not maths, you can always add 1 or divide by 2. Daft or wot .

  16. Ángel says:

    @slick rick:

    Infinity is not zero. Your explanation is absurd. 1 + (-1) = 0, but that does not mean 1 = 0. Also, note that INFINITY – INFINITY is an indeterminate expression: it can be equal to any value and every value. The reason is that infinity has simultaneously multiple sizes. Zero times INFINITY is also an indeterminate expression.


    Infinity does not mean all, it means ever growing or never ending.

    @Shaun Cena:

    Your analysis is wrong. You ignore that infinity has multiple sizes, and thus it yields indeterminate expressions. You claimed that, if infinity were odd, then INFINITY/INFINITY + 1 = 1, and that this is expression is not true. However, INFINITY/INFINITY is an indeterminate expression and can thus be equal to any value, so you cannot make a conclusion. The reason is that, because infinity can simultaneously have multiple sizes, the numerator may be bigger than denominator, or vice versa, and that would depend on what limit one is evaluating such that, when direct substitution is executed, one obtains the expression we analyze. The same applies with INFINITY/2 = INFINITY. INFINITY is not a real finite number either, so the concept of evenness or oddness regarding infinity is not logical in arithmetic. INFINITY can be implemented as an extension to the real numbers, but not as a real number itself.


    Mathematicians make life simpler by solving problems. And, no one decides what can we do and not do. Logic and the axioms it underlies decide it.

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