My mother taught me to subtract with the algorithm she learned, long before the inferior algorithm I was taught in school. Borrowing from the minuend requires lots of memory; carrying into the subtrahend does not. On 26-Jan-21 21:09, Lucas, Stephen K - lucassk wrote:
Dan, your method of subtraction is known in the literature as the “Austrian” method, one of the 3 main ways of subtracting in America in the 1930s. The history of why we ended up subtracting using reallocation is that forever, subtraction “using crutches” or putting markings on the paper was considered inappropriate. Then a single county administrator decided to experiment, with some of his school using crutches, some not. Those using crutches understood as well as those who didn’t what was going on, and got the answers right substantially more often. Almost overnight, a\American schools used crutches, and still to today. By sheer chance, that county used reallocation. So we all do now.
I have a history paper on 10 different ways to subtract through the ages, must get that submitted at some point...
Steve
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Stephen Lucas, Professor Department of Mathematics and Statistics MSC 1911, James Madison University, Harrisonburg, VA 22807 USA<x-apple-data-detectors://1/0> Phone 540 568 5104<tel:540%20568%205104>, Fax 540 568 6857<tel:540%20568%C2%A06857>, Web http://educ.jmu.edu/~lucassk/ Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other)
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato)
On Jan 26, 2021, at 8:46 PM, Dan Asimov <asimov@msri.org> wrote:
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Thanks!
I wonder if there's a more concrete way to describe the power of 2.
I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day.
(I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.)
—Dan
On Tuesday/26January/2021, at 5:32 PM, Victor Miller <victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://urldefense.proofpoint.com/v2/url?u=https-3A__oeis.org_A001700&d=DwIG... >) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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