Math is all about rules, right? Immutable axioms come together to prove still more complex phenomena. Truth does not change with the times or with popular opinion, or even with new scientific insights. Some might find that comforting, others boring, but it wasn’t how I learned math.
Let me tell you one of the first things I learned in math. One is the smallest number. When you lift a finger, you start counting at one. Before that is not a number, it’s just nothing. My math universe consisted of the “natural” numbers, and the only way it grew was when I learned bigger natural numbers. To tell my friends that they were poopy heads times one thousand was much more impressive than times one hundred (roughly ten times, I’d say) and to tell them they were wrong times infinity was deeply gratifying. If I saw an empty desk, there weren’t “zero apples” on the desk, there just weren’t anything.
Soon after, I learned that before I started counting I was at zero, and my understanding of math expanded to the whole numbers. The whole world was packed with zero of anything I could imagine. It was an exciting but somewhat unsettling time, knowing that at any given moment I was both in possession of zero raspberry cheesecakes and moments away from becoming the victim of zero rampaging Tyrannosaurus Rexes. The only time I was not in possession of zero raspberry cheesecakes was when I was in possession of one or more raspberry cheesecakes. If I ate my cheesecake I would be back to zero, and I would have to stop because, of course, there is nothing less than zero.
Enter the integers. If somehow I ate a raspberry cheesecake that was not there, I could find myself in debt one raspberry cheesecake. I tried to imagine what it meant to have a negative number of raspberry cheesecakes, and the image that always came to my mind looked like an undeveloped photograph (appropriately called a “negative”), or a digital picture with its colors inverted in Photoshop. This black cheesecake with cyan raspberry sauce would sit on my desk, I imagined, generating stress and consternation from all around until I hoarded my allowance and bought another raspberry cheesecake to carefully line up with the anti-cheesecake until it popped into place, and both vanished, leaving me again with the ever-present zero cheesecakes.
I still worried that I would be in trouble if I ever had more friends come to my birthday party than I had cheesecakes to give. If one friend came and I took no cheesecakes for myself, I could make do with just one cheesecake. Advancing through grade school, though, I learned that with even just one cheesecake I could serve two, even three friends. I could serve eight friends or a thousand friends, provided they did not mind getting one one thousandth of a cheesecake. This raised a terrible prospect – what if zero friends came to my birthday? Solemn-faced, my math teacher Mrs. Fraction told me that I could not divide my cake among zero people. The results would be undefined, and I shuddered to try and picture an undefined amount of cheesecake. It felt to me like zero cheesecakes, a whole universe of cheesecakes, and a whole universe of black and cyan anti-cheesecakes. I took Mrs. Fraction’s warning to heart and promised I would never allow a birthday to happen with zero friends.
By this time I understood square roots, and the first matter of square roots was that, since a square is always positive, the root of a square may be positive or negative. A negative number does not have a square root, so my teacher Mr. Algebra taught me. From Mrs. Precalculus, I learned that square roots of negative numbers still did not exist, but that didn’t mean mathematicians didn’t sometimes dress up in long white lab coats and thick goggles, shout “eureka!” and pretend that they had discovered them. That is, the mysterious quantity that when multiplied by itself would become an anti-cheesecake was something familiar to all children – imaginary. So, if I could imagine the square root of my raspberry cheesecake imaginary square root of raspberry cheesecake times I would end up with a real life anti-cheesecake. I knew that anti-cheesecake only provoked consternation and judgement and cost a whole positive cheesecake to get rid of, so I never tried this experiment.
So I could take the square root of a negative number, or at least pretend to, but I remembered my vow to Mrs. Fraction. One thing I could never do, never ever, was take my cheesecake and give it to zero people to share. One of my less responsible or possibly less honest friends claimed to have tried this. He was fond of chili cheese dogs with relish. He donned his older brother’s football helmet, then for safety he hid in a pillow fort behind the couch and sent his little sister to place the chili cheese dog with relish on an empty table where there were precisely zero people to share it. My friend reported that the chili cheese dog with relish did not fill the entire universe, it did not turn cyan with red relish and a purple bun. It did not even do so much as disappear until his father came along and subtracted one chili cheese dog with relish, licking his lips and telling my friend to wash the table. My friend concluded that one divided by zero was in fact one, but I suspected there might be something wrong with his experiment, and thank goodness. I would not want to live in a universe packed entirely with chili cheese dogs with relish or even raspberry cheesecake. Eating food is fun, but wading through it to get to the bus each morning and having to eat it just to make room to breathe seemed less so.
All these terrifying prospects returned with a vengeance when Mr. Calculus told the class that we were going to divide by zero. I raised my hand and asked Mr. Calculus if this was wise, but he assured me that the secret is knowing whether you are dividing by negative zero or positive zero. That flew in the face of what Mr. Algebra had taught me that zero is neither positive or negative, but here Mr. Calculus admitted that we were not exactly dividing by zero. We were also not simply imagining that we could divide by zero, Heaven forbid. To divide by zero, you divide by x as x approaches zero. Think of the smallest possible number. Think of it as literally infinitely small. If you divide by that, you know what you get. If it’s an infinitely small positive number you get infinity, if negative, negative infinity. As long as there’s the slightest fraction of a friend coming to my party, I at least know what is going to happen to my raspberry cheesecake. Just to be safe, I still make sure that I’m among the people sharing. One raspberry cheesecake divided by one is one raspberry cheesecake.