Today we’ll talk about some paradoxical things, like the logarithm of zero, and the maximum element of a set of real numbers that doesn’t contain any real numbers at all. More importantly, we’ll see how mathematicians try to wrap their heads around such enigmas.

All today’s logarithms will be base ten logarithms; so the logarithm of 100 is 2 (because 100 is 10^{2}) and the logarithm of 1/1000 is −3 (because 1/1000 is 10^{−3})). The logarithm of 0 would have to be an *x* that satisfies the equation 10^{x} = 0. Since there’s no such number, we could just say “log 0 is undefined” and walk away, with our consciences clear and our complacency unruffled.

**BUT WE AREN’T GOING TO DO THAT, ARE WE?**

Suppose we’re feeling willful and we decide that we absolutely want to have a solution to 10^{x} = 0, no matter what price we have to pay. (Think of Roald Dahl’s Veruca Salt, screaming that she *must* have an Oompa-Loompa.) When setting out in search of such an *x*, we ask (as one always does on a quest), what’s truly important to us? What will we hold onto during our quest, and what are we willing to give up? Our answers to these questions will determine what route we’ll take and what we’ll discover when we reach our destination.

On today’s quest, the thing we’ll guard is the formula log (*a* × *b*) = log *a* + log *b*. If this is going to be true, then setting *a* = 0 and *b* = 10 (say), we’ll find that log (0) = log (0) + 1. Upon realizing this, we might be inclined to just head home and give up the quest, because what kind of number stays the same when you add 1 to it?

But let’s stalwartly persevere, and imagine that there is such a creature, which we’ll call ℧. (A century ago electrical engineers used this symbol as a unit of conductance and called it “mho”, but since the engineers aren’t really using the symbol anymore I think they won’t mind if I borrow it for bit.) What price must we pay for admitting ℧ into the flock of ordinary numbers? We have ℧ + 0 = ℧ = ℧ + 1, so if we allow the ordinary rules of algebraic cancellation, we’ll end up with 0 = 1. And we certainly can’t have *that*! Fanciful numbers can join in our number games but not if they’re going to wreak havoc with their mundane numerical companions. So, no cancellation law for us, as far as ℧ is concerned. (Which also means we’re going to have to give up subtraction, or at least bar ℧ from its domain of applicability.)

So, we get a pseudo-number called ℧ with the property that ℧ + 1 = ℧. In fact, using our sacred formula log (*a* × *b*) = log *a* + log *b*, we can show that ℧ + *r* = ℧ for any real number *r* (see Endnote #1). And we also have ℧ + ℧ = ℧ (as a consequence of 0 × 0 = 0).

What else might be true about ℧? Let’s make use of the fact that when *a* < *b*, we have log *a* < log *b*. This fact holds when *a* and *b* are positive numbers, so let’s see what comes of assuming that it’s true when *a* is 0 and *b* is positive. Since for example 0 < 1/1000, we have ℧ = log 0 < log 1/1000 = −3. So ℧ < −3. In fact, ℧ is less than every real number, positive or negative. So we’ve created a new number system, consisting of the ordinary real numbers plus a new pseudo-number called ℧. We’ve decreed that ℧ is less than every real number, and that ℧ plus itself, or plus any real number, is just ℧ again. It’s a perfectly consistent, if initially strange, number system.

But what good is this system? Why try to define log 0 in the first place? It’s not as though taming the expression “log 0” will gives us some insight into how to take the log of a negative number (see Endnote #2 for more on this). Taken on its own, creating a value for the expression “log 0” isn’t an important thing to do; it’s just a way to illustrate the kind of conceptual move that pure mathematics allows us to make. Fortunately for ℧, there’s an area of math in which ℧ can find something more like an honest job: polynomial algebra.

**POLYNOMIALS AND THEIR DEGREES**

I’m going to assume that you’ve seen polynomials, so that you can recognize *x*^{3} + 2*x* + 1 as a polynomial of degree 3. It’s a sum of three terms: *x*^{3}; 2 times *x*^{1}; and *x*^{0} (better known as 1). The degree of the polynomial is the maximum of the exponents: the set of exponents is {3, 1, 0}, and the largest element of this set is 3.

But can you recognize 7 as a polynomial of degree 0? It certainly doesn’t look like a polynomial function of x; for one thing, there’s no *x* in it! But if I write it as 7*x*^{0}, you can see why we call it a polynomial of degree 0. (If you wonder why we want to call 7 a polynomial in some contexts, consider that *x*+3 and −*x*+4 are both polynomials; if we want it to be a general rule that you can add two polynomials and get a polynomial, we’re going to have to admit 0*x*+7, aka 7, into the polynomial club, even if we’re going to make it sit at a special table for all the really boring polynomials.)

But what about the polynomial 0*x*+0, or 0 for short? Here things get more complicated. Some textbooks teach that every constant polynomial has degree 0 except the constant polynomial 0, whose degree is undefined. And that’s a perfectly valid choice for defining the degree of the zero polynomial. But an equally good choice (and one that’s more useful in some contexts) is to define the degree of the zero polynomial to be ℧. In fact, this is what mathematicians typically do, except that instead of writing ℧, they write −∞. Now I can fess up: I chose to use the ad hoc, nonstandard symbol ℧ to defer readers’ anxieties about the concept of infinity. In the context of defining the degree of the zero polynomial, there’s nothing supernatural about −∞; it’s just a new player we’ve added to the real numbers, governed by rules that we’ve specified. I’ll switch from writing ℧ to writing −∞ the way everyone else does, but the rewrite is purely notational.

Don’t think of −∞ as the negative of something else called ∞ (at least not yet); think of it as a primitive, elemental symbol. I’ll pronounce −∞ as “minus infinity” (at the risk of offending those who will say I should call it “negative infinity” instead).

In pure mathematics, we’re at liberty to introduce new entities and create new systems, as long as we adhere to certain meta-rules, like the one that says we can’t allow our new entities to undermine the properties of the old ones. (This meta-rule is what told us that our new system shouldn’t have a rule that would let us cancel ℧ from both sides of the equation ℧ + 0 = ℧ + 1.)

Let’s see what declaring deg 0 = −∞ buys us. Consider the rule that says that when you multiply two polynomials *p*(*x*) and *q*(*x*), the degree of the product equals the sum of the degrees; in symbols, deg *p*(*x*) *q*(*x*) = deg *p*(*x*) + deg *q*(*x*). (Example: If you multiply the degree-2 polynomial *x*^{2}+1 and the degree-3 polynomial *x*^{2}+1 you get the polynomial *x*^{5}+*x*^{3}+*x*^{2}+1 whose degree, 5, is the sum of 2 and 3.) What does this rule tell us when *p*(*x*) is the zero polynomial and *q*(*x*) is the polynomial *x*^{2}? Then the equation deg *p*(*x*) *q*(*x*) = deg *p*(*x*) + deg *q*(*x*) becomes deg 0 = deg 0 + deg *x*^{2}, or deg 0 = deg 0 + 2; if deg 0 is 0, then we get 0 = 0 + 2, which is bad news, but if deg 0 is −∞, then we get −∞ = −∞ + 2, which is old news (by now).

If you check out the section “Degree of the zero polynomial” in the Wikipedia page on “Degree of a polynomial”, you’ll find that there’s an alternative convention deg 0 = −1; this convention has its plusses (sorry), but I won’t discuss them here, other than to mention a related convention: topologists often find it convenient to call the empty set a (−1)-dimensional topological space.

**TROPICAL SEMIRINGS**

The set of ordinary real numbers, with the bonus element −∞ thrown in, is called the tropical semiring, when we equip it with the binary operations “plus” (addition) and “max” (maximum). We’ve already talked about how plus acts: *a *+ *b* is just the ordinary sum of *a* and *b* when *a*, *b* are ordinary numbers; −∞ + *a* = *a* + −∞ = −∞ for all ordinary numbers *a*; and −∞ + −∞ = −∞. If we write max(*x*,*y*) as *x* ∨ *y *(see Endnote #3 regarding this notation), then the rules are: *a* ∨ *b* is just the ordinary maximum of *a* and *b* when *a*, *b* are ordinary numbers; −∞ ∨ *a* = *a* ∨ −∞ = *a* for all ordinary numbers *a*; and −∞ ∨ −∞ = −∞.

Part of the game of tropical mathematics is that you can view plus and max as “demoted” versions of times and plus; for instance, just as multiplication is distributive over addition (that is, (*x*+*y*) × *z* = (*x*×*z*) + (*y*×*z*)), addition is distributive over max (that is, (*x* ∨ *y*) + *z* = (*x*+*z*) ∨ (*y*+*z*); see Endnote #4). And, just as 0 is the identity element for + in the ordinary ring of real numbers, −∞ is the identity element for ∨ in the tropical semiring.

There’s a related extension of the real number system that (confusingly) is also called “the” tropical semiring: it’s the set of ordinary real numbers, with the bonus element +∞ thrown in, equipped with the binary operations plus and min, where min(*x*,*y*) is written as *x* ∧ *y*. As you might expect, for all ordinary numbers *a* we have +∞ + *a* = *a* + +∞ = +∞ and +∞ ∧ *a* = *a* ∧ +∞ = *a*, and +∞ + +∞ = +∞ = +∞ ∧ +∞. This is often called min-plus algebra, as opposed to the max-plus algebra I mentioned above. People who work with min-plus algebra typically write +∞ as just ∞.

You might think that the max-plus and min-plus systems could profitably be combined into a single more symmetrical system, where we include both +∞ and −∞. But it’s less useful to combine the two systems than you might think, at least in the context of algebra. The trouble is that there’s no good way to define the sum of +∞ and −∞ without violating some cherished rules of algebra, like the associative property (*x*+*y*)+*z* = *x*+(*y*+*z*), unless we violate the very symmetry we’re seeking to gain by combining the two systems. (See Endnote #5.)

What mathematicians actually do to reconcile the two systems is to view each of them as a flipped version of the other. (See Endnote #6.)

Tropical mathematics may seem like fairly arcane stuff, but if we can believe what the Wikipedia entry tells us, it’s actually been applied to mathematical finance! I don’t know anything about finance, so I can’t comment. But I do know that tropical mathematics is one of the tools that is likely to give us a better understanding of pictures like the one below, taken from Jordan Ellenberg’s excellent article on abelian sandpiles. (For a dynamic preview of this mathematics-in-progress, see Nikita Kalinin’s lovely video.)

**NEGATIVE INFINITY IN CALCULUS**

If you’ve ever taken a calculus class, you’ve probably read the above with a certain amount of bewilderment or impatience, because I haven’t talked about the kind of negative infinity you know and love, or at least feel reasonably comfortable with: the kind that appears in statements like “As *x* goes to 0, −1/*x*^{2} goes to −∞”. Calculus students are often instructed to think of +∞ and −∞ as directions rather than destinations; the meaning of “As *x* goes to 0, −1/*x*^{2} goes to −∞” is declared to be that, no matter what your notion of a gazillion is, −1/*x*^{2} will eventually become less than negative one gazillion, once *x* becomes sufficiently close to zero. Or so we preach. But then we turn around and write things like lim_{x→0} −1/*x*^{2} = −∞, which seems to be treating −∞ as a number, or if not a number, something awfully like a number.

The under-advertised truth is, there’s a rigorous system called the extended real number line that augments the real number line by adding −∞ and +∞ as two extra elements. In this system, the sum of −∞ and +∞ is undefined, but certain other arithmetic operations on these extra elements are permitted; for instance, *+*∞ times −∞ equals −∞. (The intuition here is that if you have a number *M* that’s “near” +∞ in the sense that it’s very positive,and another number *N* that’s “near” −∞ in the sense that it’s very negative, then *M* times *N* will be very negative.)

See if you can figure out what the “right” rules for doing addition and multiplication in the extended real number line should be. The answer is in Endnote #7.

Although addition can be problematical when it comes to combining −∞ with +∞, max and min don’t pose the same problem in the extended number line as addition does. For instance, −∞ ∨ +∞ = +∞, and −∞ ∧ +∞ = −∞. With this in mind, we can define operations called Max and Min, acting on sets of extended real numbers, with the property that, for any finite non-empty set *S*, Max(*S*) is the maximum element of *S* and Min(*S*) is the minimum element of *S*. For instance, Max({−1,+2,+∞}) = +∞ while Min({−1,+2,+∞}) = −1. This leads us to a curious puzzle: what’s the natural way to define Max(*S*) and Min(*S*) when *S* is the set that contains no real numbers at all? See Endnote #8.

Lastly I’ll mention a topic I plan to return to in the future, the projective real line. This is a mathematical system in which −∞ and +∞ are the same thing! You can think of the projective line in many ways; one way is to imagine “gluing” the numbers −∞ and +∞ together, much as last month I wrote about “gluing” 0.999… and 1.000… together. Projective geometry is no whimsical fantasy of renegade mathematicians looking for rules to break; it was developed in Renaissance Italy as a way to formalize the laws of perspective needed by artists and surveyors. Parallel lines on the ground, viewed from a point above the ground, appear to meet far off in the distance; in projective geometry, we add ideal points where the parallel lines actually meet. See chapter 13, “Where the train tracks meet”, in Jordan Ellenberg’s “How Not To Be Wrong”.

**FREE MATHEMATICS!**

Pure mathematics doesn’t force us to choose between the ordinary real number system, the max-plus semiring, the min-plus semiring, the extended real number line, and the projective real line, the way politics makes us choose between candidates, or religion makes us choose between creeds. In the world of pure mathematics, constructs like these coexist and tell us things about each other. We can define conceptual structures of all kinds, as long as the rules governing them are clear and devoid of internal contradiction. As Georg Cantor wrote, the essence of mathematics (by which he meant pure mathematics) is in its freedom.

In fact, now that I think about it, the phrase “pure mathematics” isn’t the best name for the kind of mathematics that isn’t applied mathematics. The word “pure” is, in a subtle way, negative in character — praising the subject by touting its lack of potential adulterants. I think “free mathematics” would be a better term.

In calling pure math free, I’m not saying that “anything goes”; free math, like free verse, can be sublime, mediocre, or execrable. Ultimately, judgments of the value of non-applied mathematics are rooted in a sense of beauty, and since beauty is subjective, the community’s assessment of the value of individual contributions is going to be as much about taste as it is about truth. There are areas of broad consensus, and then there are contested areas. Fifty years ago, many algebraic geometers felt that the theorems of graph theory, although true, were uninteresting. Now tropical geometry has created a link between graph theory and algebraic geometry, so this kind of snobbery is less prevalent. Some people feel disenchanted when they first learn that math, for all its vaunted objectivity, is subject to trends and fashions. But for me that humanizes the subject.

Getting back to “free” versus “pure”: let’s think about projective geometry, with its notion of a “point at infinity”. In its origins, the subject was definitely a branch of applied mathematics: painters needed to know where to place paint on a canvass to achieve a desired effect! So we can’t call the subject “pure math”. But what are we to make of the audacious idea of augmenting classical geometry by adding ideal points not found anywhere in Euclid’s universe? As a piece of free mathematics, projective geometry is (my last pun, I promise!) unparalleled.

*Thanks to Matt Baker, Sandi Gubin Mike Lawler, Henri Picciotto, and Glen Whitney.
*

Next time: Reading, writing, and rigor.

**ENDNOTES**

#1. Let *R* = 10^{r}. Then ℧ + *r* = log 0 + log *R* = log (0 × *R*) = log 0 = ℧.

#2. If we try to define log −1 in a way that respects the formula log (*a* × *b*) = log *a* + log *b*, we run into trouble. Putting *a* = *b* = −1, we get 0 = log 1 = log ((−1) × (−1)) = log (−1) + log (−1). So log −1 appears to be some number that, when doubled, gives 0. Yet log −1 can’t be 0, since 10^{0} is 1, not −1. Since log −1 isn’t 0, we can ask, is it positive or negative? (Remember that we got a clear answer to this question in the case of log 0: ℧, being less than every real number, is in particular less than 0, that is, negative.) If log −1 is positive, then it’s a positive that, when added to itself, gives a sum that isn’t positive; if log −1 is negative, then it’s a negative that, when added to itself, gives a sum that isn’t negative. It appears that if we want to have log −1 in our number system, and we want log (*a* × *b*) = log *a* + log *b* to hold, we’re going to have to give up the trichotomy principle (every element of the real numbers is positive, negative, or zero) or some of the laws of inequalities (a positive plus a positive is a positive).

Euler’s contemporaries were puzzled by this. Eventually they realized that the right way to think about logarithms of negative numbers involves leaving concepts like is-less-than behind and introducing the complex number system. Euler’s successors realized that to resolve all paradoxes, “log” shouldn’t be thought of as a true function but as a multi-valued function.

#3. The formulas 0 ∧ 1 = 0 and 0 ∨ 1 = 1 might remind you of the rules of Boolean algebra (where “0” means False, “1” means True, “∧” means “and”, and “∨” means “or”); this is not a coincidence, but a sign that, in this instance, mathematical notations evolved to fit well together.

∧ and ∨ are kind of a package deal with ∩ and ∪. Mnemonically, it helps to think of ∧ as being the letter A without the crossbar (where “a” is for “and”) and ∪ as being the letter U (where “u” is for “union”); then you can think of ∨ as a pointy ∪ and ∩ as a rounded ∧.

(On some other world beings might use ∧ to mean “max” and “or”, and ∨ to mean “min” and “and”, with the idea being that the two symbols resemble an upward-pointing arrow and downward-pointing arrow respectively, but that’s not how things happened on our planet.)

#4. One way to show that (*x* ∨ *y*) + *z* = *x*+*z* ∨ *y*+*z*; is to split into cases, according to which is bigger, *x* or *y*. First let’s start with the case where *x* = *y*. Then (*x* ∨ *y*) + *z* simplifies to *x* + *z*, whereas *x*+*z* ∨ *y*+*z *simplifies to *x*+*z* ∨ *x*+*z* = *x*+*z*. So in this case the desired equality holds. Next let’s take the case where *x* > *y*. Then (*x* ∨ *y*) + *z* simplifies to x + z, whereas the inequality *x*+*z* > *y*+*z* (a consequence of our assumption *x* > *y*) implies that *x*+*z* ∨ *y*+*z *simplifies to *x*+*z*. So in this case too, the desired inequality holds. And the final case (*x* < *y*) can be proved just as the second case was, with *x* and *y* switching roles.

#5. First, suppose we defined the sum of +∞ and −∞ to be 0. Then two different ways of evaluating the sum (+∞) + (+∞) + (−∞) given to us by the associative law would give us two different answers: ((+∞) + (+∞)) + (−∞) = (+∞) + (−∞) = 0, and (+∞) + ((+∞) + (−∞)) = (+∞) + 0 = +∞. This is a contradiction.

What about other possible values of (+∞) + (−∞)? To rule them out, let’s proceed systematically. Let *x* represent the element (+∞) + (−∞). Then ((+∞) + (+∞)) + (−∞) = (+∞) + (−∞) = *x *whereas (+∞) + ((+∞) + (−∞)) = (+∞) + *x*. So *x* must satisfy the equation *x* = *x* + (+∞). Also, switching the roles of +∞ and −∞, we can use the same argument to show that *x* = *x* + (−∞). Our only choices are to define the sum of +∞ and −∞ to be either +∞ or . But that would break the symmetry of the system (which was our whole motivation for trying to mash together the max-plus semiring and the min-plus semiring). I suppose one could let (+∞) + (−∞) be a new sort of element, but I haven’t explored this possibility and I don’t know if anyone else has either.

#6. The max-plus semiring can be viewed as the min-plus semiring “turned upside down” (though, given that number lines are usually drawn horizontally, the right visual image would be exchanging left and right, as in a mirror). Algebraically, this amounts to the formula that min(*x*,*y*) = max(−*x*,−*y*) for all elements *x*, *y* of the min-plus semiring. More formally, algebraists say that the operation of negation gives an isomorphism between the min-plus semiring and max-plus semiring . This is the setting in which you are invited to see −∞ as the negation of +∞ and vice versa.

#7. In the extended real number line, +∞ plus +∞ is +∞, and −∞ plus −∞ is −∞, but +∞ plus −∞ is undefined, as is −∞ plus +∞. That’s because if you have two number of great magnitude but opposite sign, we know nothing about the sum (could be positive, could be negative, could be zero) if we don’t know which has bigger magnitude.

Multiplication is nicer: +∞ times +∞ is +∞, +∞ times −∞ is −∞, −∞ times +∞ is −∞, and −∞ times −∞ is +∞.

Many functions from the set of real numbers to itself admit a natural extension to the extended real number system. (More formally, if we put a compact topology on the extended real number line, there’s often a unique extension that’s continuous.) If we do this with the function *f*(*x*) = −1/*x*^{2}, we get a function *F* from the extended real numbers to itself satisfying *F*(+∞) = 0, *F*(−∞) = 0, and *F*(0) = −∞. Note however that this won’t work for the function 1/*x*, which goes to +∞ as *x* goes to 0 from the right but goes to −∞ as *x* goes to 0 from the left; there’s no way to break the tie at 0. For the function 1/*x*, the right way to bring infinity into the game is not to use the extended real number system but to use the projective line, for which +∞ and −∞ are the same thing! Then we can legitimately say 1/0 = ∞.

#8. In facing the void and the questions it poses (like “What is the product of all the elements of the set *S* when the set *S* contains no elements at all?”), mathematicians have the habit of thinking contextually. What properties should Max and Min have? One property is that if *S* and *T* are two sets with *S* a subset of *T*, Max(*S*) should be less than or equal to Max(*T*), and Min(*S*) should be greater than or equal to Min(*T*). Call these the “monotonicity property” of Max and Min. The property certainly holds if *S* and *T* are non-empty finite sets, so it would be nice if it applied even when the smaller set *S* happens to be the empty set ∅.

Let’s look at the case where *S* is the empty set and *T* is the singleton set {*r*}, where *r* is an ordinary real number. We have Max(*T*) = Min(*T*) = *r*, so Max(*S*) should be less than or equal to *r* and Min(*S*) should be greater than or equal to *r*. But this has to be true for *every* real number *r*. So we are led to the conclusion that if we want to retain the monotonicity property, we need Max(∅) to be less than every real number and Min(∅) to be greater than every real number. That is, Max(∅) = −∞ and Min(∅) = +∞!

Crazy, right? But historically this kind of craziness has an excellent track record for steering us mathematicians toward the correct definitions (definitions that are not just convenient but also fruitful), even if the conclusions we reach seem bizarre at first.

These conventions about the empty set are related to the convention that *x*^{0} is defined as 1 (at least when *x* ≠ 0), which in turn is related to the seldom-discussed but fundamental fact the product of the elements of the empty set is 1, not 0. The common theme here is the hidden role of the identity element (aka neutral element) of a binary operation. 0 is the identity element for addition; 1 is the identity element for multiplication; −∞ is the identity element for ∨; and +∞ is the identity element for ∧. Whenever you’ve got an associative binary operation * on a set *S*, and that set contains an element *e* with the property that *e***x* = *x***e* for all *x* in *S*, then the most convenient way to define *s*_{1} * *s*_{2} * … * *s _{n}* in the case where

*n*=0 (that is, the case when the *-product stops before it even gets started!) is to define it to equal

*e*.

#9. Lastly, if you’re wondering why I made this an essay focus on minus infinity instead of infinity, it’s for much the same reason as I initially wrote ℧ instead of −∞. The idea of (positive) infinity seemed likely to bring up prior conceptions about infinite sets and whatnot; I wanted to short-circuit those associations. Omega is often used to represent the infinite, so upside-down omega seemed like a natural choice for its negative.