San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Digit Sum Arithmetic for Numbers
with Nonintegral Bases

Some interesting aspects of digit sum arithmetic are shown in Quotient 9. Digit sum arithmetic is based upon a property of polynomials so it would apply to numbers with any base not just other integers beside 10, such as a fractional, irrational or transcendental number. The general proof is given Sum of Digits.

Let NB be a number N represented to base B. That is to say, and NB=cnBn+cn-1Bn-1+…c1B+c0, where ci for i=0 to n belong to a set called digits. For simplicity the set of digits here will be taken to the digits for the base 10. Let S(NB) be the sum of the coefficients of NB.


NB−S(NB) is an exact multiple of (B−1).

This means that the quotient of [NB−S(NB)] divided by (B−1) is a number QB where the powers of B run from (n-1) down to 0.

Illustration using B=3/2.

Let N3/2 = 2313/2=2(3/2)²+3(3/2)+1=1010. S(N3/2=2+3+1=6 and .N3/2−S(N3/2) = 10−6 = 4. B−1=1/2. So N3/2−S(N3/2) divided by B−1 is equal to 8 so N3/2−S(N3/2) is an exact multiple of B−1.

Another illustration using B=3/2. Let N3/2 = 3123/2=3(3/2)²+1(3/2)+2=41/4=10+1/4. S(3123/2)=6. Then [3123/2−S(3123/2)]=4+1/4=2+(9/4)= 1(3/2)²+0(3/2)+2=1023/2. Division by (B−1)=1/2 produces 2043/2. Thus Q3/2=204 3/2 and hence [N3/2−S(N3/2)] is an exact multiple of. (B−1)

Illustration using B=√2. Let N√2=221√2=2*2+2√2+1. Then N√2−S(N√2) is equal to 2√2. Consider B−1=√2−1. Then 2√2/(√2−1)=C(√2+1)/(2−1)=4+2√2 = 2√2 + 4. Thus Q=24√2.

Illustration using B=π. Let Nπ=3π²+2π+4. Then S(Nπ)=9. But

Nπ−S(Nπ) = 3(π²−1)+2(π−1)+4−4
= (3(π+1)+2)(π−1)=(3π + 5) = 35π

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