Had I had any explanations of math beyond basic arithmetic I might have learned something about it. But I forgot that bag of tricks soon after learning them because they had no application I could imagine at the time.
Am I the only one that still occasionally(*) “casts out nines” to check the results of a column addition or long multiplication?
(* - Well, I used to do this routinely. But I’m not sure I’ve done it in the last decade or two unless it was to teach the trick to my kids.)
ETA: Yes! I’ve done it to check Yahtzee score totals!
Given the number of people I see who fail to understand major, important issues, or who struggle with reading comprehension, because they cannot get really fundamental set theory, I’d say the schools aren’t teaching enough of it.
Math isn’t just math. Math serves as practice for the mind to understand logic.
When I was a kid, they never taught us about this in school. I ran across the technique in a math book I checked out from the library (yes, I was that kind of kid) and thought, “Wow, that’s cool! And useful! How come they didn’t teach us this?”
I’m looking at it now, and I am really glad we didn’t learn this; it’s as complicated a process as doing the problem in the first place, and would add a new way to make an error.
A lot of logic is mathematical. They even call it “mathematical logic” (a term which can include set theory). But I do not see how one can make it through elementary school without even a superficial introduction to arguments in logical form, even for something like simple Euclidean geometry or writing certain types of essays. It is also a core liberal art, to the extent that influences primary and secondary curricula.
Often the best way to check whether you got the right answer is to do it in two different ways.
I don’t know where you looked at “casting out nines,” but it’s not complicated, and catches errors with very high probability. One might need to practice with it a bit before drawing conclusions.
For example, let’s suppose I wonder how many centimeters there are in the circumference of a circle with one-inch diameter. It might be easier to work this by hand instead of looking for a calculator.
3.14159265 × 2.54000000 = 7.979645331
Whoa! That involved so much work that an error might have crept in! Let’s cast out nines:
[del][COLOR=“Red”]3.141[/del][/COLOR]5[del][COLOR=“red”]9[/del][/COLOR]265 × 2.[del][COLOR=“red”]54[/del][/COLOR] = 7.[del][COLOR=“red”]9[/del][/COLOR]7[del][COLOR=“red”]9[/del][/COLOR]6[del][COLOR=“red”]45[/del][/COLOR]331
No arithmetic. Just wherever you see a ‘9’ (or a ‘45’ or ‘18’ etc.) you cross it out. (In practice there are even easier ways to cast out nines.) Let’s rewrite the work with the crossed out numbers not shown:
5 2 65 × 2 = 7 7 6 33 1
Now continue the process. While looking for nines on the left you notice the whole group of digits sums to 18. 1+8 … a multiple of nine; cast out the whole thing. Similarly the right side group sums to 27; 2+7 is another multiple of nine. We’re left with
0 × 2 = 0
Yep, the arithmetic checks. Checking it was much easier than the original 3.14159265 × 2.54000000 = 7.979645331.
That is because
For y = the sum of the digits in x,
y mod 9 = x mod 9
That trait seems to hold true for every integer base (b).
y mod (b-1) = x mod (b-1)
I have no Greek at all, which is perhaps why the Euclid I saw did not use negative numbers.
Knowing Greek wouldn’t help. The oldest surviving copies we have of The Elements are in Arabic.
Of course, the language isn’t the important part, anyway.