base-10 using -4 to +5

John E. Franklin · 2005-09-02 14:30:04

I just thought this up. Use base-10 but limit yourself to the digits negative
four to positive five. In the left column below I am counting from 1 to 20.
In the right column I show the square of the number.

  1       1
  2       4
  3     1-1
  4     2-4
  5     2 5
1-4     4-4
1-3     5-1
1-2   1-4 4
1-1   1-2 1
1 0   1 0 0
1 1   1 2 1
1 2   1 4 4
1 3   2-3-1
1 4   2 0-4
1 5   2 2 5
2-4   3-4-4
2-3   3-1-1
2-2   3 2 4
2-1   4-4 1
2 0   4 0 0

John E. Franklin · 2005-09-02 17:04:17

John E. Franklin · 2005-09-02 20:07:07

I went to bed and then I had a great idea so I got back up and worked on this through the night.
I hope you like it as much as I do. Imagine the possibilities!

Here is an example comparing our number system to the new Winter number system.

  285
  x64
-------
18240


or in this new Winter number system we have

    3-2 5
  x 1-4 4
----------
   1 1 4 0
-1-1-4 0
 3-2 5
-------------
 2-2 2 4 0

Last edited by John E. Franklin (2005-09-03 13:01:04)

MathsIsFun · 2005-09-02 21:32:05

It is interesting ... we just use decimal (apparently) because we have 10 fingers, and so we look for patterns in our decimal notation.

What if there were another notation that revealed more order, and was actually more "natural"?

John E. Franklin · 2005-09-03 01:29:21

1/7 is normally 0.142857 repeating. Now it is 0.143-1-4-3 repeating! I wonder what I'll find next?

John E. Franklin · 2005-09-04 03:27:20

Here is pi (to 30 places after the decimal) shown 
normally and pi in new Winter sytem.
I called it the Winter system because I like winter and
also you can remember that you get below zero temperatures
in the winter like the numbers used here.

3. 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9
3. 1 4 2-4-1 3-4 5 4-4-1 0-2-1 3 2 4-2 5-4 3-4 4 3 4-2 3 3-2-1

You can obtain the above conversion just by working 
from left to right or from right to left on the normal 
number pi.  If you go right to left, it is very easy.  
Simply carry the 1 if a conversion to negative is made.
So starting at the end of the above pi, look at the 9.
Since 9 isn't allowed in the Winter system, you use need
to think ten minus one.  Nine in the Winter system is a
two digit number and is written 1-1.  A 1 in the ten's place
and a -1 in the one's place.  So you take the -1 and write it
down below the 9.  Then you carry the 1 from the ten's place.
Now we convert the next digit to the left seen above
and it is a seven.  Since we had carried a one, we add it to
the seven and get eight.  Then we convert eight to the Winter
system and think "eight is not allowed because it is above 5,
so eight is ten minus two".  So "8" is "1-2".  So write down
the -2 and carry the 1.  If you run across a case where you
carry a one onto a 9, then you think okay, ten, and ten is 
still 10 in Winter system, so write down zero and carry the 
one.  So that's my explanation of going from right to left
and converting from normal numbers to the Winter system.

If you go left to right you have to look ahead a few digits
to see what's coming next.  The explanation of that is
more complicated than going right to left, but all I think 
I should say is the following and let you work it out for
yourself if you want to do it that way.

A "0" digit may remain "0" or may change to "1". 
A "1" digit may remain "1" or may change to "2". 
A "2" digit may remain "2" or may change to "3". 
A "3" digit may remain "3" or may change to "4". 
A "4" digit may remain "4" or may change to "5". 
A "5" digit may remain "5" or may change to "-4". 
A "6" digit will change to "-3" or "-4".
A "7" digit will change to "-2" or "-3".
A "8" digit will change to "-1" or "-2".
A "9" digit will change to " 0" or "-1".

Good luck playing around with the new Winter number
system.  If you have any questions, just ask!

Last edited by John E. Franklin (2005-09-04 03:32:55)

MathsIsFun · 2005-09-04 09:24:01

1/3 = 0. 3 3 3...
2/3 = 1.-3-3-3...

Also 0.999= 1. 0 0-1

John E. Franklin · 2005-09-04 13:48:16

And the addition table.

fatulant · 2005-12-03 10:41:44

links don't work

Math Is Fun Forum

#1 2005-09-02 14:30:04

base-10 using -4 to +5

#2 2005-09-02 17:04:17

Re: base-10 using -4 to +5

#3 2005-09-02 20:07:07

Re: base-10 using -4 to +5

#4 2005-09-02 21:32:05

Re: base-10 using -4 to +5

#5 2005-09-03 01:29:21

Re: base-10 using -4 to +5

#6 2005-09-04 03:27:20

Re: base-10 using -4 to +5

#7 2005-09-04 09:24:01

Re: base-10 using -4 to +5

#8 2005-09-04 13:48:16

Re: base-10 using -4 to +5

#9 2005-12-03 10:41:44

Re: base-10 using -4 to +5

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