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Theorem of Digital Arithmetic
This illustrates a property of the digital representation of numbers to base ten. It is a special case of a far more general theorem that applies not only to number representation to other bases but to polynomials in general. Ordinary numbers are polynomials in powers of ten. Since the base is ten the property concerns nine, one less than the base,
Consider the known properties of digital arithmetic with respect to nine.
. Let the digit sum of N be denoted as D(N). For example, D(123)=6 and 123=13*9+6. Similarly, D(989)=D(26)=8 and 989=109*9+8.
For example, 25*15=375, D(25)=7, D(15)=6, D(7*6)=D(42)=6 and D(375)=D(15)=6.
It is not always so simple Cumulative sums may be two digit numbers. The excess above nine must be added to the coefficient of the next higher power of ten. Consider 989. The cumulative sums are 9, 17, 26. First the 1 of 17 is added to 9 to give a quasi-quotient of 107. It is true that 989=107*9+26. The 18 of 26 is added as 2*9 to give the quotient of 109. Thus 989=109*9+8.
The general proof of this property is given in Sum of digits
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