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#1 2012-05-14 20:00:41

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,565

Shapes With Ten Things Marked

Here we can share shapes we have come up with that have ten marked locations.
The point is that one may integrate the shape into their mind's eye and utilize the shape to remember digits in base-10.
Any pictures are welcome with the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 noted on them.
You can take the same picture and number it a different order if you want to create another one similar.
This thread could be useful for basically anyone at any level of mathematical understanding.
Each picture may be accompanied by a name if you wish to provide one.

Here is what I call "Ten Piano Circle 2012".

View Image: TenPianoCircle2012.PNG

igloo myrtilles fourmis

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#2 2012-05-14 20:49:40

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,565

Re: Shapes With Ten Things Marked

I call this, "Three Squares".

View Image: TenSidesThreeSquare2012.PNG

igloo myrtilles fourmis

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#3 2012-11-27 11:48:16

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Shapes With Ten Things Marked

Hi! smile
                                8  1
How 'bout base 16:      X      A very manipulative notation for the numbers.
                                4  2
           .           .           .           .  .     .  .  .     .  .  .     .  .  .     .  .
   X    X    X    X    X    X    X    X    X    X    X    X    X    X    X    X   
                 '     '   '     '     '  '   '  '                 '     '   '     '     '  '   '  '
   0     1    2    3    4     5    6    7    8    9    A    B    C    D    E     F
                                                              10   11  12   13   14   15

Notice the multiples of 5 and 10 have only two marks at the ends of one one of the lines
and 15 has both superimposed.

Mulitples of three have only two marks and those next to each other.

Powers of 2 only have one mark.

7,11,13 and 14 have three consecutive marks.

Multiplication of 1 thru 7 by 2 just rotates the figure 90 degrees clockwise.
(Multiplication by 4 rotates 180 degrees and by 8, 270 degrees, but these
will cause "carries" for the most part.)

A "carry" is produced when  8 thru 15 are multiplied by 2.  The 8 position causes a 1 in
the next "X" to the left.
                                       .                 .   
                       Example:   X  x  X  =  X  X
                                                 ' 
                                        8  x  2  =  1  0   (which is 16 in base 16)

                                       .  .               .
                                        X  x  X  =  X  X
                                          '       '         '  '
                                        11 x  2  =  1  6  (which is 22 in base 16)

Addition is done by superimposing the two on top of each other.  When two dots occur on
the same end of one of the crosses they cause a carry of one dot to the next clockwise.
                     .   .       .   .             .              .        .
Examples;    X + X  =   X             X + X  =  X   =  X
                  '               '                 '     '        ''     '
                   5 + 8   =  13            3  +  2          =  5

When writing these by hand it is a little clearer to just put a little perpendicular dash at the end of
the crossed lines instead of trying to put dots.  Then the numbers requiring two columns don't
need as much separation to distinguish them.

Our Arabic numeral system isn't manipulative at all in terms of the digits 0,1,2...,9.
But if we use an abacus (or soroban) then we can manipulate quite well using the beads to
represent digits instead of the Arabic symbols.

Interesting note:  The soroban (Japanese version of the Chinese abacus) has one vs four beads
                          on each column.  Hence digits 0 through 9 can be entered.
           
                          The Chinese abacus has two vs five beads on each column.  Hence, the
                          digits 0,1,2,...9,A,B,C,D,E,F  can be entered.  So the Chinese (which
                          predated the Japanese version) had a HEXADECIMAL abacus.  I wonder
                          if they realized that.  roll


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#4 2012-12-06 19:25:51

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,565

Re: Shapes With Ten Things Marked

thanks for the binary dots in a square, very interesting...


igloo myrtilles fourmis

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#5 2012-12-07 12:49:20

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Shapes With Ten Things Marked

You're sure welcome.  Take a "spoke" away forming a Y and you have octal.  Add a spoke for base 32 etc.  They you can pop some wheelies!  It's interesting that the Arabic system we have is very
UNmanipulative.  At least the primitive  Tally  system was somewhat manipulative ||| + |||| = |||||||.

(1/2)(Good Day) smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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