The Mathematics of Change, page 2 Sitting in our folding chairs. Facing us was the big main administrative building: Nassau Hall (because Princeton used to be just Nassau College). Also facing us, in front of Nassau Hall, was the president of Princeton, President Bowen. Not, uh, a real . . . ethnic-looking guy -- but focused. He, in turn, was flanked by two stone tigers. Tigers being the mascot of Princeton -- representing, as they do, the ancient ferocity of the WASP peoples. Up behind President Bowen were the elder statesmen among the Board of Alumni -- old gents with their canes, pointing them up at the sky: "We own those clouds! We own those trees! Ha ha ha ha ha!" President Bowen said, "Welcome. Welcome, class of 1980. You know, each incoming class is unique. In fact, this uniqueness is what each incoming class . . . shares. Can any among you tell me what is unique about 1980?" Well, to me it was obvious right away: I mean, 1980 was divisible by 4! It's a trick, you see. If you take any number, and the last two digits are divisible by 4, then the entire number is divisible by 4. 80 can be divided by 4, therefore 1980 is divisible by 4 -- so 1980 can be divided by 4, which of course is 2 times 2. Now, another trick: if you add any number across -- in this case, 1 plus 9 is 10 . . . plus 8 is 18. If you add all the numbers across and that sum is divisible by 3, that means the entire number is divisible by 3. Since 18 is divisible by 3, that means 1980 is divisible by 3. These are the kinds of tricks my dad taught me. He would pick me up from my mom's and take me the six or seven blocks to the 168th Street station of the subway. We'd go down and down into the incredibly fetid air of that subway station. We'd get onto the Broadway Local. And we'd sit down on seats that were woven and cushiony in those days, not plastic the way they are now. We'd sit down in our seats and right away Dad would turn to me and say: "Your mother is a very crazy woman." I'd go: "I know." So we'd get that out of the way. And now we could move on to math. Math being one of my father's twin passions, the other one being Communism. My dad, a lifelong Communist, also a lifelong lover of mathematics. In fact, he was a math teacher -- in grade school, in middle school, and often in Special Ed. And all around us on the subway, people would be bopping each other, being bopped, mugging each other, getting mugged, spraying things on the inside, on the outside of the train, hanging off of things, hanging out of things and into things . . . But we were in our own world -- our world of math. And Dad would take out his little index cards and show me tricks . . . He started by teaching me the numerals. The first numeral I can remember learning was . . . 1. 1: tall, fragile, delicate . . . very much like my mom. And my mom had recently slipped on a patch of ice and bruised her knees, and for some reason this affected me very deeply. So from that time on, I always made sure to give my 1's those little feet, for support. And I'd give them a little cap, you know, just for style. 1: very fragile. Not like 3, with its beautiful, nurturing curves -- I like 3! But then, one day I happened to be approaching my beloved 3 from the wrong direction and . . . ow! It pierced me! That's when I learned about the terrible schizophrenia that numbers can have. They can help you, they can hurt you -- it all depends from which side you approach them. 3: very unpredictable, very scary. Not like 6. 6, shaped very much like my dad, with his nice potbelly. 6, which would then spin round and round and round and would end up as . . . 9. 9, the most beautiful and mystically powerful of all the numerals. 9, with its huge brain staring down beneficently on all the smaller numerals. 9, so mystical, so powerful . . . For example: 9, though it was of course the largest numeral, could, in an instant, turn itself into 0. This was part of a trick called "Casting Out Nines" that my dad showed me on the train one day. He said, "My son, let's pick two numbers at random. How about . . . 1848 and 1917? Just any two numbers at random. "Now, first, let's add them together: 1848 plus 1917 equals 3765. Now, we check our calculations by Casting Out Nines. . . ." What Casting Out Nines involves is that you add across and you keep adding across until you have a single-digit numeral. And then you add down and you check. And if at any point you get a 9, then you can turn it into a 0 -- that's why it's called "Casting Out Nines"! Mystical . . . Powerful . . . It was armed with the power of 9 that I was able to blow away my competition . . . in second-grade math class. Our teacher, Mr. Cavanaugh, was a small, compact, very intense guy. Always seemed to be about to blow some very big gasket, Mr. Cavanaugh. (Also, I think, a very patriotic guy.) And whenever he got the sense that people weren't paying the proper amount of attention, he would slam the blackboard with his fist: "DAMMIT!" And when he slammed the blackboard particularly hard, a lot of the pieces of chalk would fall off the bottom ledge onto the floor and they would shatter. And then he'd pick up the little half-pieces of chalk and hold them in his middle fingers, and he'd go: "PAY ATTENTION! PAY . . . ATTENTION! . . . Okay, let's do some addition. Let's take two numbers at random. How about . . . 1492 and 1776. Just any two numbers at random. Now, who would like to add these two numbers together? I know you would, Mr. Kornbluth. Anyone else? How about you, Mr. Warren?" Oh, I loved it when he called on Ruben Warren! Ruben Warren was my tormentor in second grade. He was the big bully in our class. Ruben Warren was always picking on me because I was a little chubby and had kind of a big head and was really good at math. Ruben Warren was the master of his domain, which was the schoolyard. So I loved to see it when he had to go up to my turf: the blackboard. Ruben got up. Had to release the wrist of the kid sitting next to him. Walked up to the blackboard, with his little red Prince Valiant haircut . . . "Okay, um, 2 plus 6 is, uh . . . 8 -- right. (Thanks, Billy -- you'll live.) Now, uh, 9 plus 7 is, um . . . (Billy?!) Okay, put the 6 down here, carry the 1 . . ." Mr. Cavanaugh leapt up. "NO, Mr. Warren! You don't write out the number that you carry. That's weak! WEAK! Hold it in your mind, Mr. Warren. HOLD IT IN YOUR MIND! Mr. Warren, what do you plan to be when you grow up -- besides incarcerated, that is?" "I dunno, maybe a astronaut?" "Well, Mr. Warren, then let's say, God forbid, NASA hires you. They go, 'Excuse me, Mr. Warren, but how far away is Sputnik?' Instead of saying, '1492,' you go, 'It's eleven thousand four hundred ninety-two.' You see? You made a mistake because you wrote out the number that you carried. That's weak -- weak! HOLD IT IN YOUR MIND! . . . "And what are you chortling about, Mr. Kornbluth?" I said, "Well sir, I long ago did that simple addition -- and since then I've also checked my addition using a simple trick my dad taught me called Casting Out Nines." "Would you care to share this trick with the rest of us, Mr. Kornbluth?" "Certainly, sir." So I went up to the blackboard (covertly giving Ruben the finger). "Okay, so here's how I did it. On the first line I cast out this 9 right here. And add across: 1 plus 4 is 5, 5 plus 2 is 7. Now, on the second line, 1 plus 7 is 8, 8 plus 7 is 15, 15 plus 6 is 21, which I then add up to get . . . 3. Now I add the 7 and the 3 together to get 10, which adds up to 1. Okay, now on the bottom we cast out this 6 and 3, which add up to 9, so we have 8 and 2, which add up to 10 -- which gets us back to 1!" "Mr. Kornbluth, you seem pretty smug. Are you aware, though, that this little trick your father taught you -- this Casting Out Nines -- are you aware that this does not in fact prove that your addition was correct?" "Uh . . . no?" "No, Mr. Kornbluth. All it proves is that your addition could have been correct. . . . Ahh, so now we see that smile start to fade. Oh, Mr. Kornbluth, what I'd give to be there that one fine day when you finally . . . hit the wall." Hit the wall? "Dad, what's 'hitting the wall'? Dad, why doesn't Casting Out Nines work? Dad, Mr. Cavanaugh said . . ." But my father didn't hear me -- because we were moving, moving again. Dad didn't hear me, because while he was carrying the big bookcase down the stairs -- from the sixth floor down to the first floor -- he was repeating his moving mantra over and over: "It's not that it's heavy, it's that it's bulky. It's not that it's heavy, it's that it's bulky. . . ." It's like going up to Atlas and saying, "Excuse me, Mr. Atlas, how's the world doing?" "It's not that it's heavy, it's that it's bulky." The idea being that no real man should have any trouble with anything heavy, but bulk -- no one could deal with bulk. "It's not that it's heavy, it's that it's bulky." My father, as I've said, a Communist, a man not given to those little intermediary steps. While a lot of his friends from City College when he was an undergraduate had taken these little tiny steps towards some sort of success, Dad was waiting to take that one big leap -- that leap that he would take, that the people of America and the world would take, towards happiness and Communism and freedom. My dad, not given to those little intermediary steps. For example: he's carrying down the bookcase -- he hasn't taken out the books. "It's not that it's heavy, it's that it's bulky. It's not that it's heavy, it's that it's bulky. . . ." "Dad, what's 'hitting the wall'?" "It's not that it's heavy, it's that it's bulky." And Dad hit the lobby and tripped on a loose tile and almost slammed his head into the bookcase. "Oy! . . . What's that, my son? You're asking about 'hitting the wall'? It's a term for something that happens to some people in math. It'll never happen to you. . . . Why? I'll tell you -- No, I'll show you." And he started running back up the stairs. I followed him -- up and up, past the sixth floor, up through the big metal door I'd never been through, up onto the roof. And Dad took me all the way out . . . to the edge. "My son, do you recognize what you see before you?" "Of course, Dad. That's the George Washington Bridge. It's only a few blocks away. It's the one thing that connects me with Palisades Amusement Park." "My son, what would you say if I told you that one day you will possess that bridge?" "You mean, like, the way all people will one day possess all things? Or, uh, . . . me personally?" "I mean you, personally, my son. You'll possess that bridge by holding it . . . in your mind. And your tool to do that will be . . . math. "My son, look at those mighty cables holding up the bridge -- straight lines! You'll possess those through . . . geometry. Look at those pebbles strewn along the bottom of the bridge -- you'll possess those, as abstractions, through . . . algebra. And my son, those curves, those mighty curves that hold up the bridge -- to possess those you'll need . . . calculus -- the mathematics of change. "Oh, my son, I loved calculus so much back when I took it at City College -- until one day when I . . . I hit the wall. "But you won't hit the wall. No, my son. You will burst through every wall that has hitherto existed. You will go on to become the GREATEST MATHEMATICIAN WHO HAS EVER LIVED!" Wow! At the age of nine, all of a sudden I have a purpose -- nay, a destiny. I'm going to become . . . the greatest mathematician who ever lived. And it's all going to start with calculus -- and I can't wait to get there. But in order to get there, I have to go through all this baby math stuff! Which is what I did. I went through, you know, addition, subtraction, multiplication, division . . . Then I went on to the Bronx High School of Science, where the first year we learned all about . . . geometry. In geometry we learned from the ancient Egyptians -- indirectly -- that you need not traverse your land to know the area of it. No. All you need to traverse is just two of the sides, and you multiply those sides together and you get the area. I thought this was very beautiful, very elegant. Others in my class did not find it so. Others hit the wall in geometry. Fortunately, it wasn't too late to transfer back to Performing Arts. The next year we learned all about . . . algebra. In algebra we learned from the ancient Greeks -- again, indirectly -- that you need not traverse any of your land to get the area. You can just sit in your sandbox all day and work things out with a stick, using abstractions like x, y, and z . . . I mean, sure, every once in a while you might get attacked by a Roman soldier with a knife, but basically I loved it. Others, however, they hit the wall at algebra. But this was nothing compared to the carnage at trig. Trigonometry, the mathematics of triangles. Oh my God, everything turns into a triangle, everything. Take this rectangle -- that's two triangles, no, it's four, no, it's eight! . . . It's lots and lots of triangles! All around me, kids were impaling themselves on triangles. Some were going inside triangles, never to emerge. I, however, loved trig -- especially seeing as now I was just one year away from calculus, the mathematics of change. But I didn't want to take little kiddie calculus -- you know, the A.P. class you can take senior year in high school. I wanted to take real, grown-up, adult calculus -- the kind you can take in college. So I applied for early admission to Princeton, with its famous math department. And I got my application, and it had this part to fill in where you were supposed to say, essentially, what you wanted to be when you grew up. Well, all I wanted to do was do math. That's all I wanted to do. But there was this great big space to fill in -- so I wrote, I want to do math! And evidently they saw the fervor of my conviction, and I was accepted. So there I was, trembling in my folding chair among all the other tremulous variables. Madly in my head, I'm factoring out 1980, when I realize that President Bowen has moved on to another topic! Evidently that question had been . . . rhetorical. Continued in Josh Kornbluth's "Red Diaper Baby" (available from Mercury House -- email mercury@hooked.net). Josh Kornbluth is a writer and performer who has presented his monologues "Red Diaper Baby," "Haiku Tunnel" and "The Mathematics of Change" in New York, San Francisco, Los Angeles and across the U.S. He is currently at work on a play about Benjamin Franklin, which will premiere next year.