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Decimal System and Double Digits
Name: Ken
Status: other
Grade: other
Country: Canada
Date: April 2011
Question:
If the origin of the decimal system reflects counting on
ten fingers and if zero came into use after the decimal system had
been established why did we not create a single symbol for our tenth
digit rather than use the double digit 10? If T were to represent
the tenth number this would have created a counting system where the
number series 1,2...9,T is followed by the same series having a 1 to
the left then followed by the same series having a 2 to the left,
etc. The T would be the last number in a series of ten single digits
rather than be the first number in a series of double digits. The
symbol zero would be used only between negative one and positive one
because it represents the existence of nothing and, therefore, would
have no other function.
Replies:
Ken,
Some cultures did create a symbol for ten. In a base sixteen system,
"A" is the symbol for ten. "10" represents sixteen. "10" means one of
what the left column represents and zero of what the right column
represents. If "T" means ten, then "1T" means one ten and ten ones. In
a digital system, "1T" would be ten more than "T". "9T" would be
ninety more than ten, or one hundred. What will "TT" represent?
Would you have "T" plus "1" equal "11"? If so, try to find "74237"
minus "73457" on paper. You will have seven minus seven in the first
column and in the last column, but would have to do them differently.
Multiplication also yields a difficulty. What is "7" times "T"? What
is "TT" times "4T4"? Arithmetic is much easier of ten is represented as
one ten and zero ones, i.e. "10".
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
Ken,
So, let us see if I understand. (You have
probably thought this out a lot more than I have.)
Counting: 1,2, …, 9,T, would be the first ten digits, then
11, 12, …, 1T would be the next ten.
The first 2-digit number would be 11 and the
last would be TT, which would represent the
decimal number 110. Right? So the first 3-digit
number would be 111 and the last would be TTT,
which would represent the same number as 1110 in decimal.
In arithmetic, carrying would be done a little
differently and so would borrowing. For example
1T + T = 2T requires a carry and 1T - T = T
requires a borrow. Hmm, maybe not, since would
probably be memorized as part of the addition and subtraction tables.
So if I suppose I have an addition table that
runs from 1+1 =2 to T + T = 1T, how do I use it,
algorithmically, to add and subtract arbitrary
numbers? That is, how do I borrow and carry? Or would you do it that way?
For example, in base 10, 53 - 24 = 40 + 13 - (20
+ 4) = (40 - 20) + (13 - 4) and both 40 - 20 (as
multiples of 10) and 13 - 4 can be read from a
table. How would the same problem read in
T-notation? Maybe 53 = (3T + 13) - (1T + 4) =
(3T - 1T) + (13 - 4)? The 13 - 4 is OK, but how
to get 3T - 1T - 1T from table. And situation is worse for 53 - 23.
(Computer Scientists tend to start their lists
from 0, instead of 1, for similar "edge" problems.)
But I think everything works. Of course, as you
pointed out, you would still need the symbol 0, in algebra, for example.
What would be the advantage of this? What would
be harder and what would be easier? Or is it
just an alternative system? I think maybe it
would be harder to learn. How would you justify
algorithmically T + T = 1T. Ie, the T + T = T
pattern (not arithmetic, just a pattern, T+2T =
3T, 5T- T = 4T, etc) without a zero in the
arithmetic? Maybe an algorithm other than vertical line-em-up arithmetic?
The difficulty for me is to give up on the
apparent multiples of 10 being actual multiples
of 10. Ie, 10 is 1 ten, 20 is two tens, etc. You
don't have that in T-notation. 1T is two, not
one multiple of T, 4T is five, not four multiples of T, etc.
Have you ever looked up a topic called syncopated algebra?
Regards,
Paul Beem
The decimal numbering scheme is positional. The actual value of the digit depends upon its place. Hence a 1 in the units place is valued as 1. A one in the tens place at ten, and the same symbol, 1, is worth 100 in the third place left of the decimal.
I suppose you could use z to represent a value of 10, but what would you do at twenty, thirty or even ten thousand?
Would you count 1, 2, 3, 4, 5, 6, 7, 8, 9, z, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2z, 21...?
I guess you could write ten thousand z,000. and Avogadro’s number as 6.02 x z23? But you might run into trouble with 10 x 10 (z2)
Our present notation works quite well and it did not work so well before that zero was introduced.
The real difficulty with zero is its bothersome properties: “anything times zero is zero” and “division by zero is undefined.” Replacing 10 with z does not affect this since the troublesome zero is not the zero(s) in ten or one hundred, but that zero at the origin where positive numbers turn into negative.
Hope this helps.
R. W. "Bob" Avakian
Ken
Because there are ten symbols in the decimal system, there is no need to use
another symbol T.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
Look at the base 16 symbol Hexadecimal system
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F,
Sincere regards
Mike Stewart
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