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.

---

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 ????

 [ Back ] [ Up ] [Top of this Page]  
Mathematics Education Consulting and Private (Online) Instruction available

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  
Mathematics Consultant/Tutor/Instructor, All Rights Reserved.