I’m with ya bullwinkle…
now can someone explain to me why, exactly, X^0=1?
Basically, it’s because 1 is the multiplicative identity. The intuitive idea is that x^0 means you have zero factors of x. An “empty product” (a product with no factors) is generally defined as the multiplicative identity 1, since it’s the “nothing” of multiplication–multiplying by 1 doesn’t change anything.
In addition to Cabbage’s good explanation, here’s one way I explain it to my algebra classes:
Consider:
10[sup]3[/sup] = 1000
10[sup]2[/sup] = 100
10[sup]1[/sup] = 10
10[sup]0[/sup] = 1
10[sup]–1[/sup] = 1/10
10[sup]–2[/sup] = 1/100
Each time the exponent goes up by 1, you multiply by another factor of 10.
Each time the exponent goes down by 1, you divide by 10. To continue the pattern, 10[sup]0[/sup] should be 10 divided by 10, which is 1.
(There’s nothing special about the number 10. It was chosen just as an example. Other numbers would work just as well.)
BJMoose, the problem with declaring that 1/0 = infinity, is then you have to ask what 2/0 is. Does 2/0 = 1/0? No matter how you answer that question, you end up breaking some of the “rules” of arithmetic. Which really just means that you have to invent new rules with exceptions built into them. It’s doable, but it usually ends up being more awkward than disallowing division by zero in the first place.
On the topic of exponentiation, another interesting question is 0[sup]0[/sup]. 0[sup]x[/sup] = 0 for any x other than 0, and x[sup]0[/sup] = 1 for any x other than 0. You can make arguments for 0[sup]0[/sup] to be any number between 0 and 1, and probably for it being any number at all. But it turns out that most of the arguments point to 0[sup]0[/sup] = 1, so that’s usually the convention used.
See, thats what always bugged me… x^0=1 to make it fit the pattern, but X/0=ERROR instead of infinity (which would fit the pattern…)
C’mon mathy guys, pick one!
How about this… according to my good ol’ TI-83+, 0^X=0, where X is positive. But 0^0 or 0^X, where X is negative gives an error. And the windows calculator says 0^0=1, but still gives and error for negative values of X. Who’s right?
In “Bullwinkle’s Axiom” above, substitute the word “divisor” everywhere you see the word “dividend”.
I knew I should have checked it at the time, but my ISP has the nasty habit of giving me the boot for “inactivity” (and indeed it started to do so just before I hit “submit”) and so I rushed things. (Waddya expect from a moronic moose?)
Chronos: It is, ultimately, a matter of definition, and I won’t quibble over that. Take, for example, the fraction 0/0 . If one follows the rule that any fraction with a numerator of zero equals zero, then 0/0=0. If one follows the rule that says any fraction with identical numerator and denominator equals one, then 0/0=1. If one follow’s Bullwinkle’s Axiom, where anything divided by zero equals unsigned infinity, then 0/0=unsigned infinity. Who says math ain’t fun?
As I think about your example, I see no problem. So what if 2/0=1/0? We’ve long been accustomed to the fact that 0/2=0/1. (And, I dare say, we see here the reciprocity I mentioned earlier: zero as a denominator gives us everything, zero as a numerator gives us nothing. Kinduva Zen balance thing.)
This is why you use limits, as, depending on the situation, it can tell you whether 0/0 is 0, 1, 2, or infinity. I don’t know why we can’t take division by zero to imply the limit of the function as x->0. Then again, I haven’t thought about it all that much.
Well, the larger price to pay would be that you look at something like -1a + 1a and have no ability to collect the terms, even though it’s a perfectly natural thing to do and has a perfectly natural interpretation. The system is just begging in all kinds of ways for 0*a to be defined as 0, which is why it usually is.
Generally, in a field, you do define multiplication for all pairs of elements, but you also speak of “the multiplicative group”, which refers to the nonzero elements and multiplication on them. Best of both worlds; you’ve got your natural multiplication by zero, and you also see the partial analogy between multiplicative structure and additive structure.
The trouble with x/0=infinity is that therefore 2x/0=infinity, which would seem to mean that 1=2. But then if you argue that infinity can’t be just plugged into algebra like that, that’s the same thing as arguing that infinity=ERROR, which is the same thing as arguing that x/0=ERROR.
Infinity is not a number; 1 is.
There’s a good amount of debate on 0^0. A lot of people think it should be undefined.
Because, if you’re just dividing two numbers, what’s the function? That is to say, if I ask you what 0/0 is, am I asking about x/x when x goes to 0, or am I asking about (x+x)/x as x goes to 0 or perhaps about x/sin(x) as x goes to 0, or maybe even just about the division of the constant function 0 by itself?
Besides, your solution doesn’t do anything improved to deal with division of nonzero numbers by zero numbers.
It does, however, show us a reasonable way to say things like “(x^2+1)/(x-1) is the function x+1” without having to qualify with “…except where x = 1”. To wit, we can take function equality to be modulo removable discontinuity at undefined points, so that the function which given by x+1 everywhere except undefined at 1 is considered the same as the function which is just given by x+1 everywhere.
Incidentally, BJMoose’s system exists and is well-studied (with, historically, the same slope of lines motivation that he gives, in fact!). It’s called the real projective line. You add an unsigned infinity to the real line as he says. Then the reciprocal operator is everywhere defined and continuous; however, there has to be a catch, as Chronos notes, and there is: multiplication becomes only partially defined, in the sense that 0unsigned infinity is undefined (thus preventing “1/0 = 2/0, therefore 1 = 2” style proofs; thus also preventing 0/0 from being defined). Thus, x(1/x) = 1 for all nonzero, noninfinite x.
It would seem from some of these posts that the true issue with “1/0 = infinity” isn’t with deviding by zero, but with the ambiguity in nailing down what, exactly, the number “infinity” is.
It’s not that difficult to conceptualize outside of the rigid definitions required by mathematics. It’s an idea that doesn’t behave itself when you do math stuff to it.
Oh, I should also point out that addition is only partially defined in the real projective line; infinity + infinity is undefined. (This is to avoid things like the following: 1+infinity = infinity, therefore (1 + infinity) - infinity = 0, therefore 1 + (infinity - infinity) = 0, therefore 1 + 0 = 0; since x + infinity = infinity for any finite x, there’s no good value to assign to infinity - infinity, which, since infinity is unsigned, is the same as infinity + infinity).
So, yeah, as Chronos says, you can make it work but you have to toss out some things. Reciprocation becomes everywhere defined, but in some of the new cases, addition and multiplication no longer are. But it’s a system which definitely has its uses.
The system has a natural correspondent for complex numbers, too; the Riemann sphere, which augments the complex numbers with a single point at infinity which is the reciprocal of 0 and is also the limit of all sequences of unboundedly increasing magnitude. Further conventional properties break (in the complex numbers, everything has a clear real part and imaginary part, but in the Riemann sphere, this is not so for the point at infinity; furthermore, the exponential function doesn’t extend smoothly to the point at infinity). But it’s pretty useful in some situations too. The real projective line is just a particular subspace of this.
If division by zero were allowed, wouldn’t the answer have to be “infinity”? And I don’t think infinity is a number. Euclid is often said to have proved that there were an infinite number of prime numbers, though what he actually showed was that no matter how big a prime you came up with, he could generate a bigger one.
I’m no mathemetician, but isn’t infinity more a concept than an actual number? And if not infinity, what would the quotient of division by zerio be.
My calculator says: “Arithmetic overflow/underflow.”
Infinity is a number in any system where you want it to be one. It’s not a number in any system where you don’t want it to be one. Some systems have more than one number in them which would be considered infinite, some systems have just one, some have none. It’s not like there’s just one static concept of number, is there? You’ve got your naturals, integers, rationals, reals, complex numbers. Which perhaps gives a sort of misleading picture of a strict linear hierarchy of containments. But you can come up with all sorts of other number concepts too, whatever suits your purpose at the moment. You’ve got things like your cardinal numbers (numbers that answer the question “How many things are in this bunch?”), ordinal numbers (numbers that answer the question “How long is this sequence?”), surreal numbers, dual numbers, quaternions, hyperreals, linear transformations, all kinds of things.
Maybe you want to think of board games as numbers, as was done by John Conway. You can define a notion of addition and multiplication of board games (for example, adding two games would be the new game where you play them in parallel), identify certain games with particular integers, and the whole thing has a lot of properties in common with more conventional “numbers”, but also a lot of unique properties of its own, and is very useful for some analyses. And you can impose a natural ordering on these games, and some of them will turn out to, in a natural sense, be infinite.
Maybe you want to think of yes/no statements about the world as numbers, as in the theory of Boolean rings. You can make addition into the XOR operator, multiplication into the AND operator, and equality into logical equivalence. You can impose an ordering where A <= B precisely when A logically implies B. You get associativity, commutativity, distributivity, inverses, all the usual goodies. You can talk about things like “All men are mortal” * “Socrates is a man” = “All men are mortal and Socrates is a man” <= “Socrates is a mortal”. This “number system” is useful for quite a lot of purposes too.
Movements of a Rubik’s cube are numbers of a sort. We can consider the addition of two moves to be doing them in sequence, the inverse of a move to be doing it in reverse. 0 will be the move where you do nothing. Plenty of ordinary number intuitions apply here.
Maybe you want to consider pieces music to be numbers. Addition of two pieces of music will be playing them in unison; multiplication will be playing the one after the other, say. Or perhaps you have some other, better definition of the operators, that’s more useful to you. Maybe this is a number system with no notion of multiplication, but you have some operation with a good claim to be the sine function.
There’s no point saying things are or aren’t numbers, with a concrete finality, as if there is just one single narrow particular concept identified with the term “number”. Things only are or aren’t numbers with respect to a particular number system, but you can make any mathematical system you want, and if it’s useful enough and has enough similarities/family resemblances to other systems, it may be useful to think of the things with in it as number of a sort. In some number systems, it will be useful to think of some numbers as Infinity or infinite; in others, it won’t be.
For the record, I think that 0^0 should be defined as 1 because of the binomial theorem. You would like to be able to expand (x + 0)^1 like you would any other binomial, and you can’t do that unless 0^0 = 1.
Yeah, I think in most contexts it should be 1, and definitely so when working with the natural numbers specifically. Not because of the binomial theorem specifically, but also on the grounds that, for example, the number of functions from a b-sized set to an a-sized set will always be a^b. Also, it’s just a cleaner recurrence relation; the base case of exponentiation becomes uniformly 1, no special cases.
With the reals and complex numbers, there’s more of an impulse to make things continuous, though. It all depends on the particular application at hand, like most things.
Well, now I’m just tickled pink because, so far as I can recall, I figured this all out myself. The idea of “infinite slope” occured to me in College Algebra when we were fiddling with Cartesian Graphs (thirty years ago). I came up with “Bullwinkle’s Axiom” a few years back in one of the threads resulting from Cecil’s column on whether 0.333~*3=1. I think this discussion is the first time I’ve used the term “unsigned infinity”, and I am simply flummoxed that that is, in fact, the correct technical term. Amazing what one can accomplish just by sitting around and thinking.
Incidentally, that cracking sound you hear is my arm breaking as I pat myself on the back. . . .