﻿ Illustration of the Quotient Theorem of Digital Arithmetic
San José State University

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Illustration of the Quotient
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.

• The sum of the digits of first multiples of nine is equal to nine; e.g. 1+8=9, 2+7=9, 3+6=9, …. When that sum is greater than nine it a multiple nine; e.g. 9+9=18. When the process is repeated until a single digit is obtained that single digit is the digit sum of the number. Thus the digit sum of almost all multiples of nine is nine. The only exception is zero. But in this digital arithmetic zero and nine are equivalent.
• The digit sum of any number is equal to its remainder upon division by nine, with the special provision that 9 is the same as 0

. 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.

• The digit sum of a product is equal to the digit sum of the product of the digit sums,

#### D(M*N) = D(D(M)*D(N))

For example, 25*15=375, D(25)=7, D(15)=6, D(7*6)=D(42)=6 and D(375)=D(15)=6.

• The cumulative sums from the left of a number reveal its quotient and remainder upon division by nine. Consider 125. The cumulative sums are 1, 3 and 8. The note that 125=13*9+8. Thus the first two cumulative sums of 1 and 3 are combined to get the quotient of 13 and the last cumulative sum (the sum of the digits) is the remainder upon division by nine.

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