San José State University |
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applet-magic.comThayer WatkinsSilicon Valley & Tornado Alley USA |
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Digit Sum Arithmetic for Numberswith Nonintegral Bases |
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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 N_{B} be a number N represented to base B.
That is to say, and N_{B}=c_{n}B^{n}+c_{n-1}B^{n-1}+…c_{1}B+c_{0},
where c_{i} 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(N_{B}) be the sum of the coefficients of N_{B}.

Theorem:

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

Illustration using B=3/2.

Let N_{3/2} = 231_{3/2}=2(3/2)²+3(3/2)+1=10_{10}. S(N_{3/2}=2+3+1=6 and
.N_{3/2}−S(N_{3/2}) = 10−6 = 4. B−1=1/2. So N_{3/2}−S(N_{3/2})
divided by B−1 is equal to 8 so N_{3/2}−S(N_{3/2}) is an exact multiple of B−1.

Another illustration using B=3/2. Let N_{3/2} = 312_{3/2}=3(3/2)²+1(3/2)+2=41/4=10+1/4.
S(312_{3/2})=6. Then [312_{3/2}−S(312_{3/2})]=4+1/4=2+(9/4)= 1(3/2)²+0(3/2)+2=102_{3/2}.
Division by (B−1)=1/2 produces 204_{3/2}.
Thus Q_{3/2}=204 _{3/2} and hence [N_{3/2}−S(N_{3/2})] 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 N_{π}=3π²+2π+4. Then S(N_{π})=9. But

= (3(π+1)+2)(π−1)=(3π + 5) = 35

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