[HN Gopher] Early Times (Multiplication)
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Early Times (Multiplication)
Author : beardyw
Score : 26 points
Date : 2022-07-01 06:47 UTC (16 hours ago)
(HTM) web link (www.futilitycloset.com)
(TXT) w3m dump (www.futilitycloset.com)
| texaslonghorn5 wrote:
| Arthur Benjamin's book _Secrets of Mental Math_ has similar
| tricks to this one.
|
| One I like is multiplying 10x+y by 10x+z where y+z = 10, then the
| answer is 100x(x+1) + yz (such that the two multiplications can
| be concatenated). So 53*57 = [5*(5+1)][3*7] = 3021.
|
| The article mentions its trick
|
| > saves the student from having to learn the scary outer reaches
| of the multiplication table -- they only have to know how to
| multiply digits up to 5.
|
| Similarly, another cool trick I like allows quick calculation of
| all squares up to 125^2 (you could go higher too if you extend
| it, but through 125^2 is simpler), and you only need to memorize
| the squares up to 25^2. The trick is to expand (x)*(x) into
| (x+y)(x-y) + y^2, picking y such that one of the sum/difference
| factors becomes 50 or 100. Then, you can easily halve the other
| factor (which will also be even) and shift it two places (if 50),
| or simply shift it two places (if 100), and then add the square
| of the small y which has been memorized.
|
| For example, 78^2 = (78-22)(78+22) + 22^2 = 56*100 + 484 = 5600 +
| 484 = 6084.
|
| Or 74^2 = (74+24)(74-24) + 24^2 = 98*50 + 576 = 4900 + 576 =
| 5476.
|
| Or 123^2 = (123+23)(123-23) + 23^2 = 146*100 + 529 = 14600 + 529
| = 15129.
|
| Since the distance to the nearest multiple of 50 is at most 25,
| you only need to remember your squares through 25^2 = 625, and
| the rest of the work is division by two, place shifting, and
| addition.
| selimthegrim wrote:
| Certainly a top shelf algorithm, unlike its namesake.
| vivegi wrote:
| My favorite ancient multiplication method is the Egyptian
| multiplication (also called Russian peasant method and other
| names).
|
| Column a is halved in each step. Column b is doubled in each
| step. Column r is the remainder of a after dividing by 2. b.r is
| product of b and r. Product is the sum of column b.r
|
| a b r (=a%2) b.r
|
| 19 73 1 73
|
| 9 146 1 146
|
| 4 292 0 0
|
| 2 584 0 0
|
| 1 1168 1 1168
|
| ---------------------------------------
|
| Sum 1168 + 146 + 73 = 1387
|
| ---------------------------------------
|
| Interesting observation: Reading the digits in column r from
| bottom to top gives (10011) in base 2 which is 19 (the
| multiplier)! That is why the method works.
| svat wrote:
| This trick, like several others like it, is easier to remember
| and work with if one adopts the notation x for ten's complement
| (10-x), e.g. 4 means 10-4=6.
|
| So the trick this post gives is (essentially) that:
| [x] x [y] = [x+y][xy]
|
| where [...] denotes a single digit, as it simply says that
| (10-x) x (10-y) = 10(10-(x+y)) + xy
|
| In this notation, the example given becomes: 6
| x 8 = 4 x 2 = 68 = 48.
|
| Note that this also works when there's a carry (write the "carry"
| digit as a superscript on the left): 6 x 7 = 4
| x 3 = 712 = 312 = 42.
|
| (I think I first saw this convention with the "vinculum" in Swami
| Bharatikrishna Tirtha's book _Vedic Mathematics_ which is
| simultaneously a work of fraud and genius: the book contains
| several such dazzling tricks that are convenient in special
| cases, which the Swami must have collected /invented and mapped
| to a small set of mnemonic rules ("sutra"s), then (dubiously)
| described as "Vedic" knowledge as part of an anti-colonialist
| project. Actually looking at it now, it uses the vinculum
| slightly differently, e.g. uses 31 to mean 29 rather than 39.
| Another book in that genre is _The Trachtenberg Speed System Of
| Basic Mathematics_ though it doesn 't go as far.)
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