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From: Ilmari Karonen on 12 Aug 2010 16:25
On 2010-08-09, John Moser <john.r.moser(a)gmail.com> wrote:
>
> I want to collect a number of practical fast math generalizations.
> Things like reducing brute force multiplication to basic algebra (i.e.
> 39183 * 293 == (3 * 10000 + 9 * 1000 + 100 + 8 * 10 + 3) * 293) or
> multiples of smaller numbers (like squares -- computing squares is a
> quick and dirty friendly numbers operation).
>
[snip]
>
> This is not what we were taught to do in school as children... and I
> think that's why most people can't add 2 3 digit numbers without a
> calculator these days.

Since there seem to be no other replies so far, I thought I'd try to
hijack your thread a bit. The reason being that a recent post
elsewhere made me think about something tangentially related, and I
figured I might as well try to put it in words here.

I think that the most important "tricks" for mental math really are
the simplest, most basic ones -- the ones that are so simple that we
(by which I mean the readers of this group, most of whom presumably
have at least some basic level of numeracy) do them without even
thinking about them, and wouldn't really consider them worth calling
"tricks", at least until we meet someone who _doesn't_ have them
properly internalized.

For example, when I was around five or six years old, a somewhat older
friend taught me something she'd presumably learned at school called
the "rule of 10" (or something like that; I'm loosely translating from
Finnish). What was it? Just the table for subtracting single digit
numbers from 10:

10 - 1 = 9
10 - 2 = 8
10 - 3 = 7
...
10 - 9 = 1

Many of us would probably consider that "too trivial to mention", but
in hindsight, I would call it one of the most important "mental math
tricks" I've ever learned, and certainly the first one (or second, if
you'd consider the first to have been learning to count up past 10 in
the first place). Why is it so useful? Because it lets you
complement numbers to turn subtraction into addition or vice versa:

15 - 7 = 15 - 10 + 3 = 5 + 3 = 8.

(Of course, you also want to memorize "5 + 3 = 8", and really all such
rules for single-digit addition and subtraction, but that part isn't
so critical; even if you didn't remember it, and had to count up "6,
7, 8", that would still be faster than counting down from 15 to 8.)

With more digits, the corresponding "rule of 9" ("9 - 1 = 8, 9 - 2 =
7, etc.") becomes even more useful, since, coupled with the rule of
10 and another basic trick, it allows borrowless subtraction, as in:

100000000 - 22222222 = 77777778,
100000000 - 12345678 = 87654322,
100000000 - 84743582 = 15256418,
100000000 - 84743000 = 15257000.

If the "other trick" isn't obvious to you already, you ought to be
able to figure it out from the examples and the subtraction tables
given above. I should point out, though, that it's not limited to
subtraction from round powers of 10 (although it's most convenient
when you're close to them):

145032798 - 78436789 =
100000000 - 78436789 + 45032798 =
21563211 + 45032798 = 66596009.

Of course, all these single-digit subtraction rules and such really
are literally elementary stuff -- you're *supposed* to have them fully
memorized by the time you get to third grade and start learning the
single digit multiplication tables (which also, however tedious and
boring to learn, really are immensely useful for mental math).

Their real value is not as "calculation tricks" on their own, but as a
sort of computational substrate upon which, if learned properly early
enough, more advanced calculation tricks can be built. The problem is
that many people do not, in fact, ever properly internalize them; and
if that foundation is shaky, it's very hard to build anything solid on
top of it.

(Ps. Another basic "math trick" that occurred to me while working out
the examples above is that I, at least, find it *much* more convenient
to do addition starting from the left than from the right -- as I
believe it is usually taught in school -- especially when doing it in
my head. The occasional need to back-propagate carries is still IMO
much less hassle than, well, carrying them around all the time, and
meanwhile, starting from the left lets you write out the answer in the
usual left to right order, not to mention letting you stop halfway
through if you only need an approximate answer.)

--
Ilmari Karonen
To reply by e-mail, please replace ".invalid" with ".net" in address.
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