tag:blogger.com,1999:blog-264226589944705290.post7907640111371437621..comments2021-12-14T05:53:12.175-08:00Comments on God Plays Dice: Arthur Benjamin's mental arithmeticMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-264226589944705290.post-42687474162664277692008-01-09T07:50:00.000-08:002008-01-09T07:50:00.000-08:00Yep I was really dumb. I became curious about what...Yep I was really dumb. <BR/><BR/>I became curious about what happened June 13 '45 did a search; saw an article about a concert on June 13 written on MONDAY the 16th and BINGO Friday the 13!! Well I did not read the article, it was written <B>APRIL</B> 16th '45.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-10773879853382520552008-01-08T16:02:00.000-08:002008-01-08T16:02:00.000-08:00Isabel,In high school I had a close friend who cou...Isabel,<BR/><BR/>In high school I had a close friend who could out calculate me quickly and accurately. I told him a few years ago that I was trying to learn how to compute products by the difference of squares. Since he had taught himself to multiply, he agreed that this was the method he used. <BR/><BR/>I am fairly bad at mental arithmetic (like many mathematicians), but the exercise helps sharpen my ability. The largest obstruction remains subtraction. I can't seem to do borrowing and keep focused on the problem. Sometimes I simply forget a given difference. I found that by subtracting squares from squares and n(n+1) from m(m+1) to be helpful. I mentioned the method in detail for your younger readers.<BR/><BR/>Maybe some of Vlorbik's fans can pick this thread up.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-22863672533358060882008-01-08T15:21:00.000-08:002008-01-08T15:21:00.000-08:00scott,although I haven't really tried to learn the...scott,<BR/><BR/>although I haven't really tried to learn these tricks, the way I do mental arithmetic probably has more in common with such "tricks" than with the various grade-school algorithms. I haven't consciously memorized things, but the fact that I do a lot of arithmetic means that results which come up often in the sorts of calculations I do come to mind quickly.Michael Lugohttps://www.blogger.com/profile/15671307315028242949noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-30420758298960662192008-01-08T15:17:00.000-08:002008-01-08T15:17:00.000-08:00m,that can't be right. If you were born on Friday...m,<BR/><BR/>that can't be right. If you were born on Friday the 13th, then your 13th birthday would have been on either a Sunday or a Monday. A date falls one day later each year than in the year before, except two days later in leap years; thus the day of the week of your 13th birthday would be thirteen days after the day of the week of your birth, plus three or four extra days for the leap years.Michael Lugohttps://www.blogger.com/profile/15671307315028242949noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-15181280098175812742008-01-08T15:13:00.000-08:002008-01-08T15:13:00.000-08:00Listening to him got me curious to what day of the...Listening to him got me curious to what day of the week I was born; Friday the 13th; which is funny because I was 13 on Friday the 13thAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-28716947283860238222008-01-08T08:03:00.000-08:002008-01-08T08:03:00.000-08:00I have had limited success in learning some of AB'...I have had limited success in learning some of AB's tricks.<BR/>BTW, knowing that 38x38=1444 is helpful. Numbers a distance 12 away from <BR/>0,50, 100, etc, have their last two digits as 44. Since 60x60 is 3600, you can extrapolate via the mnemonic that 62x62=3844. That 88x88=7744 follows from considerations about 11. <BR/><BR/>The way I see the multiplication algorithm for proximate 2 digit numbers, one memorizes squares from say 1 to 50, and memorizes products of the form <BR/>n(n+1). Every product is the difference of squares. When two proximate numbers are of differing parity, then instead of squaring their mean you compute n(n+1) where the mean is n +1/2. The reason for this trick is that <BR/>(10n +5)(10n+5)=<BR/>100n(n+1)+25. When squaring anything plus a half there is a quarter tagged on.<BR/><BR/>It is possible to train really young children in these algorithms, and especially get them to think algebraically while they are computing arithmetically. <BR/><BR/>Here is an example computation. It doesn't look so flashy in writing, but I am working it out in real time as I go. OK, I want to compute the product 52x57. The mean is 54.5. The square of 54 is about 100 less than 3025, so it must be <BR/>2916. Now I want to<BR/>add 54 to this (to get 54x55) and also subtract 6 from that. So I need to add 48 to 2916. I think that 52x57 is <BR/>2964. The result I have is clearly divisible by 3, so I am pretty confident.<BR/><BR/>Now I went to pull down my (yuck) windows calculator and got the same result. <BR/><BR/>With practice, one remembers more products and implements the most convenient algorithm for the product at hand. <BR/><BR/>For example, for the problem above the numbers in question factor as 4x13 and 3x19. One could compute 12x13x19 instead. For example 13x19=16x16-9=256-9=247.<BR/><BR/>Again these calculations are being done as I type. <BR/><BR/>I started working on these in my spare time, while riding in the car to and from work. I turned off the misearbly terrible local radio, and started teaching one of my sons these techniques. <BR/><BR/>I can't do the calculations as quickly as I might with a pencil and paper, but I can get accurate results without pencil and paper and without calculator. <BR/><BR/>My belief is that such methods should be taught to grade school kids so that they have internalized algebra before working with it.Anonymousnoreply@blogger.com