Mum
Mathematical Ulterior Motives
- the mother of all ulterior sites -
|
Mum |
- the mother of all ulterior sites - |
|
My Dad didn’t care for speculations. "Give me facts, son!" was his constant demand. "If we lay the tiles out like this, how many tiles will we need?" That was the innocent question that started it all. If you look at this drawing, you may get an idea what my Dad wanted:
He wanted four tiles in the bottom row, then one less tile for each row above. He wanted three rows in all. "That will be 4 + 3 + 2 or 9 tiles," I volunteered. "Do you have to add to do that," he complained. "Don’t you learn multiplication nowadays?" "Yes, we do," I started, "but this is not a rectangle, Dad. Multiplication is for rectangles only." "Well, you better redefine multiplication, son, because this summer we are going to lay a lot of these patterns and I need to know in an instant how many tiles we need to bring for each of them. I want 4 * 3 to be 9 from now on!" He smiled and tried to hide his own surprise of what he had just said. From a matter-of-fact man, this bordered on creativity and, I thought, insanity. I didn’t know how often multiplication had been redefined in the past, but that I should be given such an ordeal made my head feel dizzy. I started to think To not redefine multiplication was out of the question. My Dad didn’t give me tasks he was not man to see I would carry out to his satisfaction. Therefore, I started to wonder. If 4 * 3 was 9, what about 1 * 1 and 5 * 2? I made drawings in the mud. The results I copied to my brown lunch paper. It was not perfect, but it would have to do. Here is what I found the first day:
I didn’t fill out the top cells because 3 * 5 did not
make sense. To put 3 tiles in the first row, 2 in the second, 1 in the third and 0 in the
fourth and –1 in the fifth, was nonsense. The first critique My Dad surprised me by how pleased he was with my table. It lasted only two minutes though. "Can’t you find a formula, son? You don’t expect me to memorise this table?" "Well, why not?" I said. "Do you have a formula for normal multiplication?" "Don’t be smart with me lad! Give me a formula. End of discussion!" Why is there no formula for normal multiplication, I wondered. Or could it be that there was a formula after all, and my teacher had just hid it from us? May be there was an agreement that the formula should not be revealed as it would make school too easy. These were my thoughts that summer day more than fifty years ago. (Later I found some of these secret formulas,
but that is another story. See "Multiplying on Mars" at
http://members.tripod.com/mumnet/action/action005.htm.) Looking for a pattern 10 * 6 = 10 + 9 + 8 + 7 + 6 + 5. Spirit in the sky, spirit in the water, what is the secret behind these numbers? It struck me almost instantly! These were not numbers, they were my brothers and sisters. I was 10, Carlos was 9, Patricia 8, Eugene 7, Samuel 6 and Gabriela 5. When we played games I teamed up with Gabriela, the youngest, Carlos and Samuel made another team and the middle kids, Patricia and Eugene was the third team. The sum of the ages for each team was the same, namely 15. And, since there were 3 teams the total was 15 x 3 = 45. (I use * for Dad’s new multiplication and x for the one I learnt in school). 10 * 6 = 3 x 15, where does that lead me? Well, 3 is half of 6 and 15 is the sum of 10 and 6 if you subtract one! I felt I was stretching it, so I checked my idea immediately. 4 * 3 I knew was 9. Now, let’s try the pattern just described. 1.5 is half of 3 and the sum of 4 and 3 minus one is 6. And, I was jumping up and down laughing in joy and bewilderment, 1.5 x 6 = 9! Before I could tell my Dad I needed a proper formula, which meant letters. Letters was for me for stories, not for mathematics, but my Dad had scorned me before for thoughts like that, so I made an attempt. m * n = (n / 2) x (m + n – 1). The formula may look
simple when it shines in all its brackets, but it took me exactly two weeks to find it.
Two weeks fighting with the evils of arithmetic in all its armour. I will spare you for
the details. Beautiful it was not. Second critique "It is ugly, but if it works I’ll guess it will do." My Dad’s reaction was not what I had hoped for. I felt like getting a blow in round one of a boxing match. The knock out came ten seconds later. "7 * 2 = 7 + 6 = 13. But your formula gives (2 / 2) x ( 7 + 2 – 1) = 1 x 8 = 8." I was dead. How could my wonderful formula behave like this in front of Dad? When I had tested the formula it had behaved beautifully. I checked my Dad’s calculations, but could not find anything wrong. "Give me a hand with these tiles." Dad’s
voice brought me back to reality. A more cautious look 10 * 6 = 10 + 9 + 8 + 7 + 6 + 5. I lined up my sisters and brothers again, to see where I had gone wrong. Again, when we played we made three teams whose total age was all 15. And 3 x 15 = 45, so where did I go wrong? My formula m * n = (n / 2) x (m + n – 1) had two parts. The number of teams (n / 2) and the total age of a team (m + n – 1). Which of them was the impostor? (That both could be wrong, was too much for my ten year old stomach to consider!) Dad’s knock out came from 7 * 2 = 7 + 6 = 13. Here there is one team and 2 / 2 = 1, so the n / 2 part seems to be not guilty. The total age of the team, however, is not 7 + 2 – 1! The (m + n – 1) was the rotten apple in my basket. I felt relieved! Now, all I had to do was to fix the formula. My new formula would look like this m * n = (n / 2) x (age of oldest + age of youngest). The oldest kid would me m years old, but what about the youngest? After a few days pondering I came to the expression m - n + 1. I can’t justify it in just a few words, so you have to take my word for it. Or, think about it for a while and I am sure you see I am right. My new formula was m * n = (n / 2) x (m + m - n + 1) which I rewrote to m * n = m x n – n x (n – 1) / 2 as that looked much nicer. Before I saw my Dad I tested the formula on all the numbers
in my table! I didn’t want to have another surprise in front of him. Especially I
tested the case 7 * 2 = 7 x 2 – 2 x (2 – 1) / 2 = 14 – 1 = 13. It worked! Third critique To put it short: I came, saw and won! Dad was even impressed. "This is not as ugly as the last formula you cooked up," he smiled. "And it even works! You are one smart kid. I am proud of you!" I could have died there and then. How could there be a
sweeter moment in my life? I felt good inside and sensed the smell of the earth as my
father put his arms around me. I had arrived! Summer holiday Dad had to go away for a few weeks. His tile laying skills were needed elsewhere. That meant fishing and playing for me. Life was good after all. In school I had told anyone who cared to listen about my tile multiplication. What impressed my friends was not the usefulness of the formula, but that I had a formula that was just mine. Joe’s dad had a tractor and I had always envied him that. Leteisha had a brand new bicycle, and I knew I would never have one. Having a formula, was something new. I carved it into several trees along the road. "m * n = m x n – n x (n – 1) / 2 is true. Francisco." My teacher had asked me something I did not understand at first. "Does your multiplication have prime numbers?" she had said. I don’t think she said it to be mean. Anyway, I didn’t understand it before I was sitting at the river side not catching anything. "Does my multiplication have prime numbers?" I said out loud. Who could I ask? I could ask no one. I was the expert on the redefined
multiplication. If this question could be answered I had to find the answer myself. The
thought made me feel good, but I had no clue where to start. Prime numbers I knew that 2, 3, 5, 7 and 11 were the normal prime numbers. They had exactly two divisors and therefore they had been singled out as being special. The only number with one divisor was 1 and it had an even more elevated status I felt. I looked at my table. Was 7 a prime number in my system? No, it was not! 7 was 4 * 2, so it was composite. My system was different! What a joy! I soon saw that 3, 5 and 11 were not prime either! The first prime I found was 2. Then followed 4, 8 and 16. After my Dad had left I had time to extend my table. As you can see I had decided that 3 * 5 = 5 * 3. I wanted my multiplication to be symmetrical, to look nice. Just like the normal multiplication table. (What I mean is this. If m > n then m * n = m x n - n x (n - 1) / 2, otherwise m * n = m x n - m x (m - 1) / 2.)
A number would be prime if it could not be found in the
grey area. The first prime numbers seemed to be 2, 4, 8, 16, 32 and 64. In the nearest
tree I carved "2n is a prime!" Odd and even numbers During the holiday I studied many things and made many discoveries. They were all surprising and made the days fly by. All my friends thought 2, 4, 6, 8, ... were the even numbers since they have 2 as a factor. My even numbers were odd looking: 3, 5, 7, 9, ... Even and odd seemed to have changed camps. I didn’t
wonder why they had. Just enjoyed the fact. Square numbers I had always liked the square numbers. Don’t know why exactly, but they had a certain cleanness about them. 1, 4, 9, 16, 25, ... were so easy to make, and so nice to look at. My square numbers, seemed at first less impressive: 1, 3, 6, 10, 15, 21, ... While the squares increased by 3, 5, 7, 9, ... the normal odd numbers, my numbers increased by 2, 3, 4, 5, 6, May be you could say that my square numbers were actually simpler than the normal square numbers? I couldn’t be impartial I know, but I liked them. (Later I found these numbers in a normal book where they
were called ‘triangular numbers’. I think I can see why.) Factorisation I set myself the task to prime factorise all numbers below 100. I thought I could do it in a day or two, but I did not know what I was up against! I started like this: 1 the unit What on earth is going on here?
Order re-won It took me many days to figure out what was wrong with the factorisation. These days were not happy days. I thought that may be my invention was just a whole lot of garbage. I was considering removing the bark from the trees I had carved so proudly a few weeks earlier. How I came to see the light I don’t know. I dreamt
about 2 * 2 * 2 * 2 and how it could give two different results. Then I saw that 6 = (2 *
2 ) * ( 2 * 2) while 9 = (2 * 2 * 2) * 2. The order multiplication was done was important.
Multiplication, of course, is a two number affair. Strictly spoken 2 * 2 * 2 does not make
sense. Do we mean I started afresh: 1 the unit So, 6 and 9 have different factorisations. 6 is (2 * 2) * (2 * 2) while 5 * 2 is ((2 * 2) * 2) * 2. Anything else would have been a disaster! And that 9 can be prime factorised in two ways may be sad, but it also adds richness to the system. Which numbers have more than one prime factorisation? Are there numbers with three different factorisations? How are these numbers different from the single factorised numbers? (I still don’t know the answers to these questions. In
my life other questions came along. Questions seemingly more pressing. But may be, just
may be, someone finds some time to find the answers. And then, one day when I walk along
the road I will see your carving on a tree. That would be nice.) Division My sister Patricia and I shared an apple at one of our many fishing expeditions. "How is a half in your tile system?" she asked. By this time, I have to admit, I was getting more and more confident. I hadn’t thought about division before, but I knew I could sort it out there on the spot. If I just didn’t rush it. In the ground I wrote 1 / 2. "2 can’t go into 1 so let’s write 0, and try 2 into 10," I started. "2 goes into 10 five times with remainder one," smiled Patricia. You have to agree. To be only 8 she was a smart girl! So, that afternoon Patricia and I discovered that half of an apple is 0.5555... To check it we tried, just for fun, 0.5555.... * 2. Since 5 * 2 = 9 we got 0.9999.... and agreed that was close enough to 1 for our purposes. Here are a few other simple fractions we calculated:
More division I found division very hard so I played with some simpler examples. 20 / 5 = 6 since 5 * 6 = 20, and 40 / 10 = 5 since 10 * 5 = 40. Isn’t it strange that 20 / 5 is different from 40 / 10? Actually it is not strange at all, since 20 * 2 is not 40 and 5 * 2 is not 10. The old ways of seeing had played me a trick! 20 / 5 = 6 and (20 * 2) / (5 * 2) = 39 / 9 = 6, so I could
breathe a sigh of relief! Multiplying negative numbers My Dad had received the news about my won battles in the redefined multiplication system and gave me a task on the phone one evening. "Son, the future is negative. Have you looked into negative numbers yet? That’s where the money is!" That my Dad showed such interest was gratifying. Immediately I entered into this new domain. If a tile layer can think of negative tiles, well, then why can’t I? I found this very exhilarating. We were leaving mother earth with course for I don’t know what. By using my formula for m * n it was easy to produce this extended multiplication table:
I was familiar with the lower right corner of the table, but the rest was new terrain. I was excited to find that -5 * -9 = 0, and that -2 * -2 =
1. Final comment I wanted to study square roots, equations and all kind of stuff, but the summer came to an end all too quickly. To be honest, it was only by visiting my old home last week that I remembered what had happened that summer in 1949. When I walked up to the house I noticed a carving on a tree. I went closer. "2n is a prime!" was the message. It made me remember, and since the memories were so good I wanted to share them with you. May be you, or your students, would like to carve a tree as well. For me it was the most beautiful moment. References:
Reactions: This space is for your reactions to the above. Send an e-mail to Mum.
|
 |
||
 |
|
|
| mumnet.tripod.com | © 1999-2004 Jan Einar Nordgreen |
|
