The gradual development of Hindu Arabic numerals independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing Syriac bishop Severus Sebokht described the excellence of this system as " . . . valuable methods of calculation which surpass description ".
32.
1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0 . ( Sometimes tallying is referred to as " base 1 ", since only one mark the tally itself is needed, but this is not a positional notation .)
33.
That means that instead of " ab " someone typed " ba " . ( That's positional notation, of course, not multiplication . ) Your question is why does ab minus ba always equal a multiple of 9 . Now, can we agree that ab = 10a + b, and ba = 10b + a.
34.
Again, I urge you to consider roman numerals where you have I, II, III, IV, V, VI, VII, VIII, IX, X for the numbers from 1 to 10, no positional notation ( you need completely different sybols for 5, 50 and 500 or 1, 10, 100 and 1000 ) and no concept of zero!
35.
Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu Arabic numerals on a trip to what is now B�ja�a, Algeria with his merchant father . ( Europe was still using Roman numerals . ) There, he observed a system of arithmetic ( specifically algorism ) which due to the positional notation of Hindu Arabic numerals was much more efficient and greatly facilitated commerce.
36.
Next, the answer is'no': the convention of decimal representation ( and all other positional notations ) does not assign any standard value to an infinite sequence of digits ( exception : if a sequence contains only zeros from some place to infinity, you can drop those zeros and get a finite sequence of digits ) .-- talk ) 08 : 32, 27 December 2013 ( UTC)
37.
The wire frame may be used either with positional notation like other abaci ( thus the 10-wire version may represent numbers up to 9, 999, 999, 999 ), or each bead may represent one unit ( so that e . g . 74 can be represented by shifting all beads on 7 wires and 4 beads on the 8th wire, so numbers up to 100 may be represented ).
38.
Positional notation ( also known as " place-value notation " ) refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude ( e . g ., the " ones place ", " tens place ", " hundreds place " ) and, with a radix point, using those same symbols to represent fractions ( e . g ., the " tenths place ", " hundredths place " ).
39.
In a typical large integer implementation, each integer is represented as a sequence of digits in positional notation, with the base or radix set to some ( typically large ) value " b "; for this example we use " b " = 10000, so that each digit corresponds to a group of four decimal digits ( in a computer implementation, " b " would typically be a power of 2 instead ).
