Remainder Arithmetic for Real Numbers

Remainder or modulo arithmetic is useful in understanding polar coordinates, modulo 2p or 360 degrees,  and the periodicity of unit-circle trig functions and the complex-valued cis(q) = cos(q) + i sin(q) = exp(iq) function.

For any pair of real numbers d > 0 and n > 0, there is a natural numbers q > 0 and a real number r such that 

0 < r < d    and    n = qd +r  

Here the quotient q = the number of whole times that the divisor d goes into the dividend n, and r = the remainder. 

Here q and r may be computed via long division exactly or in principle via an infinite sequence of decimal approximations - their decimal expansions.  The case where d is a decimal fraction may be somewhat different (less involved) than the case where d is given by an infinite decimal expansion. We will skip the details.

Two natural numbers n and m are said to be equivalent or equal modulo d, when there remainders on division by d are equal. In this case, we write

 n = m,  modulo d.

Equality modulo a whole number or divisor d is

• reflexive, that is,  each number n = itself, modulo d, or equivalent n = n, modulo d, for each natural number n. 

• symmetric, that is,   n = m  modulo d when and only when  m = n  modulo d, and

• transitive, that is,  if  n = m  modulo d, and  m = t  modulo d then  n = t  modulo d.

A whole number n is divisible by the divisor d when and only when  n = qd for some whole number q. That is when and only when  n = 0, modulo d and when and only when n is a whole or natural number  multiple of the divisor d.  The number 0 is a multiple of all divisors d. Observe, if n > m then  n = m,  modulo d when and only when  n - m is a multiple of d while if n < m then  n = m,  modulo d when and only when  m -n is a multiple of d.

Remainder Calculations for real numbers are based on the following properties or theorems. 

Theorem:  Suppose m, n, u and v are real numbers. Suppose d > 0 is a real number.   If  m = n, modolo d and   u = v, modulo d  then (i) m + u  = n +v modulo d, and (ii)  mu= nv, modulo d. 

Proof:  First, m = n, modulo d,  implies  m = a d +r and n = b d +r for some real numbers a, b and a common real remainder r with 0 < r < d. Likewise, u = v, modulo d,  implies  u = A d + s and v = B d +s for some real numbers A, B and a common real remainder r with 0 < s < d. 

Arguments for (i):  Suppose (m+ u) > (n+v)  then 

(m+ u) - (n+v) 

= (ad +r + Ad+s) - (bd+r + Bd+s) 
=  (a+A)d + (r+s) - [(b+B)d + (r+s)]
= (a+A)d-(b+B)d 
= [(a+A)-(b+b)]d

is a multiple of d, and hence (i)  m + u  = n +v modulo d holds when (m+ u) > (n+v).  The case where (n+v) > (m+u) follows similarly.

Arguments for (ii):  Suppose m u > nv  then 

mu - nv

= (ad +r)(Ad+s) - (bd+r)(Bd+s) 
=  aAd² + asd+ Ard+ rs - [bBd² + bsd+ Brd+ rs]
= [{(aA)-(bB)}d + (as-bs)]d 

is a multiple of d, and hence (i)  m  u  = n v modulo d holds when m u > nv.  The case where nv > mu follows similarly.

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Calculator Usage: For every divisor d > 0 and every number N, there is a unique integer q such that  qd < N < (q+1)d so that r = N-qd satisfies 0 < r < d.  With the aid of a calculator, if N is positive, the whole number part of the decimal representation of the computed value of  N/d gives q > 0. But if N is negative, the whole number part of the decimal representation of the computed value of  N/d gives q+1 < 0, and q is one less than the whole number part of N/d. 

Remark: Remainder arithmetic provides the base or justification for common rules for recognizing whole number multiples of 2, 3, 5, 9 and 11 from the decimal representation of those whole numbers.  Details are given [???] in the site area ????

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Number Theory
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Decimal Place Value
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Remainder Arithmetic I
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Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II

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