Remainder Arithmetic for Real Numbers

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Remainder Arithmetic II [ Back ] [ Home ]

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 ????