Today is my birthday, and since this post is about mathematical observations I’ll express my age in this way. My age = the 10th prime number.
In any case, where I work they give out $10 gift certificates to local restaurants for people’s birthdays. Our department celebrates these birthdays by going on the 3rd Thursday of the month (regardless of when the birthday actually is) as a group. Thus, yesterday we went to Chipotle. I ordered my food and the cost came out to $7.95. I gave the card to the woman at the register and she swiped it and asked, “Do you want a receipt?” I said no, because who needs more paper in their pockets? Then she asked: “Do you want to know what your balance is?”
I looked over at the display and saw it said $7.95. So I said, “Well, I assume it’s going to be $2.05.”
She burst out laughing at this point, since the ridiculousness of the situation was fairly apparent. I mean, how hard is it to subtract $7.95 from $10.00? But as I contemplated this, I realized that for a lot of people (read: publik skewl students), this would be hard. After all, if you want a great, although quite mean, prank to pull on a poor high school student, go to McDonalds. When they say, “That’ll be 6.87″ give them $10. Then, after they press the button on the keyboard to make change, say, “Oh, here’s $0.02 too.” Watch their eyes glaze over and the beads of sweat break out on their foreheads…
In reality, though, math has tons of shortcuts that make it really easy. But people don’t consider these shortcuts to be “math.” A quick example: when multiplying numbers by 9, the result will have digits that add up to the number 9. (Actually, if I remember correctly there is a limit to this–it might not do it for large numbers; but for small numbers it certainly does.) Thus, if you want a quick multiplication table for 9s, it’s way easy. Just write the following:
0
1
2
3
4
5
6
7
8
9
Then, next to these numbers, do the opposite, so you have:
09
18
27
36
45
54
63
72
81
90
And there you go. The multiplication table from 1 to 10 (of course, normally you’d drop the opening 0 from 09; but that’s just a format issue–it’s the same number).
This, however, is considered “cheating.” I don’t think it’s any less math, however, than the multiplication by 10 trick. You know:
N x 10 = ‘N’0
(That is, any number multiplied by 10 is the original number with a 0 added to the end of it.)
How is this math, but the 9s multiplication table not?
In any case, there are a lot of math shortcuts out there that make computing stuff in your head so much simpler. But no one teaches these things any more. It’s taught even less so because of an over-reliance of calculators. Ironically, I was talking with a co-worker today who has a birthday tomorrow. He’s two years younger than I am, so I said, “Wow, I’m one day plus 365 times 2 older than you are.” At which point, he said, “I wonder how many days that is” and grabbed his calculator to computer it. I said, “It’s 731″ and got that result in my head the same time he got it on his calculator.
Incidentally, when I do multiplication of big numbers in my head, I don’t do it “correctly.” Most people would multiply 365 x 2 by going right to left (starting with 2 x 5, then adding that to 2 x 60–although they probably would actually think of it as 2 x 6, not realizing that the placement of the number hides the “tens” column–and then ending with 2 x 300). I reverse that. I start with 2 x 300 as 600 (although I do the same reduction trick, where I’m simply thinking of it as 2 x 3); but I also know that the 6 next to the 3 in 365 is greater than 4, so I automatically view the 2 x 3 section as resulting in 7, because carrying the 1 is going to happen anyway. So I start with 700.
Then I do 65 x 2 in the same manner. First, I know the 6 x 2 = 12 (but since I’ve already carried the 1, I only care about the 2; and the 5 portion of 65 is greater than 4, so I change the 2 to 3: thus 730.
Finally, the 5 is in the same manner. 5 x 2 = 10; I’ve already carried the 1 so I ignore it and I have 0. Thus, I have the number already obtained in the order it makes sense: 730. (And, of course, I then add in the 1 extra day at this point to get the final result of 731.)
Doing it in this order, it remains left -> right. I don’t have to do the reversal of right to left, with the subsequent remembering what I’m carrying over, etc. I handle both at the same time and the process occurs quickly. Of course, it takes some getting used to doing math in this order instead of the way we’re all taught, going right to left instead of left to right. But once you’ve trained your brain to ignore the nonsense you were taught in elementary school, math is a lot easier.
