Sunday, March 7, 2010

Insight on a Sleepless Night

Just before I fell asleep a couple of nights ago, I made an interesting discovery. (A great first line of any post.)

Let me preface this by stating that I have no great interest in mathematics. With a bit of paper, I can add and subtract and multiply and divide. The longer the numbers, the less confidence I have in my results. By the time, I reached eighth grade, sitting in algebra class, I could never understand why all the fuss with the identity of x and y.

One thing that has always perplexed me- something my math teachers never gave a satisfactory answer to- was why are numbers called "squared" and "cubed." It is such a practical question that I am sure I must have been THAT one student who distracted everybody by asking. Then, lying on my bed, it suddenly the answer appeared to me in a visionary insight. And now, I shall share it with you.

First of all, you have to imagine yourself a teacher in ancient times without the use of a white board or a black board. You must explain some very high-minded concepts- a challenge. Now, imagine a sugar cube as a unit, representing the number one- a single item. Got that? Pretty easy. Take three of them SQUAREDand lay them side by side horizontally, and you now have- you guessed it- the number three. Lay another two on top of the first so that now you have three up and three across. Fill in the  empty spaces of that "L" shape. Now, you have 3 x 3 which adds up to 9, forming a square, or 3 squared. See illustration.

So far this was only a theory until I came to the term, "cubed."

Untitled-1Now you must visualize in three dimensions. Take your 3-squared and make three of the same layers upwards. This shape represents 3 x 3 x 3 = 27 which is three-cubed. I guess there wasn't any other expression after that so they referred to 3 x 3 x 3 x 3 as three to the fourth power. I never had any problem with that, though.

Probably this was explained- exactly as I have here- in every other school in the world. Alas, my teachers parachuted in with both feet with x - 4 = y 4 . I'd stare at the blackboard until my eyes hurt and nothing happened. Apparently I need to begin at the sugar cube level.

And I wouldn't say I went to the best school in America and, in any case, I did everything in my power to avoid learning mathematics.  So, blaming the school for gaps in my education is probably a bit unfair.

The school seemed modern and the teaching above standard. However, I distinctly recall in elementary school several of the students pointing out to our aging school teacher how the continents seemed to fit together if you could smooch them, like green and brown play-doh. The instructor stared at the map for a second and. with a dismissive nod.  told us it was just a coincidence. Years later, individually all of the scattered members of Miss Humphrey's class of 1968 felt dismayed and not a little betrayed to learn all about plate tectonics.

Someday, if you are good, I will share with you my discoveries about the hidden mysteries of numbers.


  1. That's a pretty cool explanation for the terms, isn't it? I too hated mathematics, primarily as rebellion against my mother who loved maths. I did my best to slide through it as easily as I could, so I came up with a way to calculate the 9th times table with my fingers. My mom was astonished with the fact that I could remember my 9th tables without fail, but would not get through the 7th and 8th. I told her, eventually, that I cheated. She was so impressed that she told my teacher, and to my knowledge the method is still taught at my grammar school. :D

  2. That nine trick.. would that be the up and down technique? If it is, you will automatically know what I mean.. if not, I suppose you will think I'm a perv. That nine thing was what I hinted at in the last sentence of the post. My father drilled me with the multiplication tables and I never had much problem with that. The nine trick I later learned in one of my brief forays into numerology.

  3. I too am a complete math moron. I used to drive my teachers crazy because I'm so dumb at math they thought I was doing it on purpose. On the GREs I was in the 99th percentile on verbal and the 10th percentile in math. I'm sure your explanation of cubed is very good but I kind of went into hibernate when I saw the numbers.

    One reason I couldn't pay attention in math was my inability to accept absolutes. When I learned that numbers go infinitely up and down, or that anything times zero is zero, I thought "Well, how can they be so sure of that?" and then I'd start imagining being the person who proved otherwise. By the time I pulled out of these musings I was halfway through failing geometry for the second time.

    My last high school math teacher was kind. I'd failed trig, but she said, "I'll give you a D+ and pass you so long as you promise never to take a math class at this school again." So that was a win-win situation.

  4. Nomand, no I don't think it was the up and down thing. :D Never heard of it... It's just happens so: Let's take 7x9 for example. Look at your 10 fingers, namely the 7 of them. Use the 7th finger to "divide" your fingers, and put the numbers of what it divides down: 6 and 3. Therefore, 7x9=63. Same way: 5x9 5th finger divides, 4&5, 45. Always put the left side first and right side second.

    God knows how I came up with that...

    Stranger - that was a funny story. XD

  5. I guess I too should explain the "up and down" thing. If you write a column of numbers 0-9 down and then, next to that, starting at the bottom, working up again 0-9 you will have all the multiples of 9.
    Anyway, nine is one of my favorite numbers. All the multiples of nine when added together equal nine. 18, 1 + 8 = 9 or 63, 54, 45, all of them. That is true no matter how high the multiple goes. 122 X 9 = a number that when you add up all the numbers will equal nine. (1098)

    Nine is considered a magical number in numerology as well. If you are into that sort of malarkey.


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