[math-fun] a dismal arithmetic
Decimal hacks aren't usually my taste, but perhaps you'll find this trivia fun (inspired by the newest volume of "Winning Ways"...) To address the dismal state of math education, let's engineer a "dismal arithmetic" to be easier than decimal, so no student will be left behind. You'll still operate on pairs of digits, but you don't have to worry about carrying or in fact doing anything harder than comparing. Instead, for each pair of dismal digits, to Add, take the lArger, but to Multiply, take the sMaller That's it! For example: 169 + 248 ------ 269 and 169 x 248 ------ 168 144 + 122 -------- 12468 You can check that these associate, commute and that multiplication distributes over addition: 357 x (169 + 248) = 357 x 269 = 23567 = 13567 + 23457 = (357 x 169) + (357 x 248) Curiously, while 0 is the additive identity, 1 is *not* the multiplicative unity (Quiz: what *is* the unity?). In fact, 1 even has non-unit divisors, eg 1 x 2 = 1! (Quiz: what's the smallest dismal prime?) If we count the number of dismal partitions of n into distinct positive parts not greater than n we get the bizarre sequence 1 1 2 4 8 16 32 64 128 256 1 5 22 92 376 1520 6112 24512 98176 392960 2 22 200 1696... [not yet sloaned--superseeker thinks it might see a generating function, is it hallucinating?] (Quiz: notice that p(12) = 22 = p(21); does p(13) = 92 = p(n) for any other n?) Perhaps some funsters will enjoy developing this dismal enterprise further...
On a related note, The word "dismal" apparently derives from "decimal" in relation to the tithing day, or "decimal day", of the Roman Catholic Church, when one tenth of one's yearly income was given to the Church, a dismal day indeed.
=David Wilson The word "dismal" apparently derives from "decimal" in relation to the tithing day, or "decimal day", of the Roman Catholic Church, when one tenth of one's yearly income was given to the Church, a dismal day indeed.
Fascinating! So I guess the economists aren't the only practitioners of a "dismal science"! According to the web sources I looked at, "dismal" is actually derived from Latin dies + mali, "evil days", and yet: "These days of evil omen were known as Dies Aegyptiaci (Du Cange, Glossarium, s.v.) or Egyptian days, either as having been instituted by Egyptian astrologers or with reference to the "ten plagues"; so Chaucer, "I trowe hit was in the dismal, That were the ten woundes of Egipte" (Book of the Duchesse, I 206). There were two such days in each month." --www.1911encyclopedia.org So there's also an association with 10 plagues. (Us enlightened moderns of course call our biweekly tithe "payroll tax withholding" generally >>10%!) Crossing these words seems irresistible: "However, a number of learners continue to have difficulties with English words, and continue to make confusions like "dismal numbers" for "decimal numbers". Indeed, some may never get past this problem." --Ryan, A. and P. Meara, "A diagnostic test for 'vowel blindness' in Arabic speaking learners of English.", PRYFISGOL CYMRU ABERTAWE [!] Also, on the Eastern US seaboard "dismal" is nouned as a synonym for "pocosin", which is not an exotic trigonometric, but rather a swamp or marsh. Apropos bogs, here's answers to those "Quiz" questions (would dismals make for fun elementary-grade "enrichment" puzzles?): Q: What is the unity? A: 9. (Multiplying by 1 changes each non-zero digit to 1). Q: What's the smallest dismal prime? A: 19. The dismal primes are the numbers >9 ending in 9. Q: Notice that p(12) = 22 = p(21); does p(13) = 92 = p(n) for any other n? A: Yes, p(31)=92; p(n) depends only on the multiset of the digits in n, independent of order. (Extra credit: what is p(111)? Hint: A003465)
I was blabbing from memory, and inaccurately substitued the Roman Catholic Church for feudal lords. Anyway here is my source, which seems less confident than the "dies mali" etymology: "Webster's Universal Dictionary of the English Language", 1937: dis'mal, a.; comp. dismaller; superl. dismallest. [M.E. dismal, diesmal, dysmall, dismal; origin uncertain, perhaps originally signifying a decimal or tithing day, when tithes were to be collected by the feudal lords, from OFr. disme, a tithe; L. decimus, a tenth; from decem ten] Having or exhibiting depression, melancholy, or gloom; lacking cheer; presaging ill; characterized by darkness or dolefulness; as, a dismal appearance; dismal prospects. ----- Original Message ----- From: Marc LeBrun To: math-fun Sent: Wednesday, October 15, 2003 2:08 PM Subject: Re: [math-fun] a dismal arithmetic
=David Wilson The word "dismal" apparently derives from "decimal" in relation to the tithing day, or "decimal day", of the Roman Catholic Church, when one tenth of one's yearly income was given to the Church, a dismal day indeed.
Fascinating! So I guess the economists aren't the only practitioners of a "dismal science"! According to the web sources I looked at, "dismal" is actually derived from Latin dies + mali, "evil days", and yet: "These days of evil omen were known as Dies Aegyptiaci (Du Cange, Glossarium, s.v.) or Egyptian days, either as having been instituted by Egyptian astrologers or with reference to the "ten plagues"; so Chaucer, "I trowe hit was in the dismal, That were the ten woundes of Egipte" (Book of the Duchesse, I 206). There were two such days in each month." --www.1911encyclopedia.org So there's also an association with 10 plagues. (Us enlightened moderns of course call our biweekly tithe "payroll tax withholding" generally >>10%!) Crossing these words seems irresistible: "However, a number of learners continue to have difficulties with English words, and continue to make confusions like "dismal numbers" for "decimal numbers". Indeed, some may never get past this problem." --Ryan, A. and P. Meara, "A diagnostic test for 'vowel blindness' in Arabic speaking learners of English.", PRYFISGOL CYMRU ABERTAWE [!] Also, on the Eastern US seaboard "dismal" is nouned as a synonym for "pocosin", which is not an exotic trigonometric, but rather a swamp or marsh. Apropos bogs, here's answers to those "Quiz" questions (would dismals make for fun elementary-grade "enrichment" puzzles?): Q: What is the unity? A: 9. (Multiplying by 1 changes each non-zero digit to 1). Q: What's the smallest dismal prime? A: 19. The dismal primes are the numbers >9 ending in 9. Q: Notice that p(12) = 22 = p(21); does p(13) = 92 = p(n) for any other n? A: Yes, p(31)=92; p(n) depends only on the multiset of the digits in n, independent of order. (Extra credit: what is p(111)? Hint: A003465) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Marc LeBrun