Rational Mathematics

by G. P. Jelliss

Multiplication

The operation of multiplication is equivalent to repeated addition of the same number. Thus if a+a+ ... +a = m, where there are b occurrences of a then we say that m is the result of multiplying a by b. We can multiply any two numbers a and b to form a number called their product which we write using a dot on the line a.b read as 'a times b'. (Other notations such as a·b a×b or a*b, or simply juxtaposition, are also used for multiplication. The asterisk is used in most programming languages.)

Multiplication table (base ten)
.012345678910
000000000000
1012345678910
202468101214161820
3036912151821242730
40481216202428323640
505101520253035404550
606121824303642485460
707142128354249566370
808162432404856647280
909182736455463728190
100102030405060708090100

Like addition, multiplication is commutative: a.b = b.a, and associative: a.(b.c) = (a.b).c. The product of all the numbers in a set or sequence S of numbers is denoted by Π(S). The product of a null set or sequence is conventionally taken to be 1.

Theorem (Principle of Multiplication of Choices): The number of ways of making choices in sequence, the first from u elements, the second from v elements, the third from w, and so on, is u.v.w.... Proof: This is little more than the definition of multiplication differently expressed.

Example. The number of choices for the first move by either player in a game of chess is 20 (4 knight moves and 16 pawn moves). Thus the number of ways of playing the first pair of moves is 20.20 = 400.

Theorem (Selection Principle): The number of ways of making a selection from u things alike of one kind, v of a second kind, w of a third kind, etc, is (u+1).(v+1).(w+1).... Proof: From the set of u things alike we can choose u, u–1, u–2, ..., 3, 2, 1 or 0 a total of u+1 cases. We then apply the previous principle. This count includes the null selection where we choose none from any of the sets. If this is excluded we must deduct 1 from the total.

The number 1 has the multiplicative identity property: x.1 = x for any x. In particular 1.1 = 1. The number 0 is the multiplicative absorber or nullity: 0.n = 0. In particular 0.0 = 0. If r+n > r.n then either n = 1 or n = 0 with r ≠ 0.

Multiplication is distributive over addition: a.(b+c) = (a.b)+(a.c). This says in effect that ‘a.’ is a linear operator. It is also distributive over subtraction: a.(bHc) = (a.b)H(a.c).

Quotient and Remainder

For any numbers n and m (m not 0) there exist unique numbers q and r such that n = q.m + r where r<m. We call q the quotient of n by m, written n|m, and r the remainder, written nŽm. We thus have n = (n|m).m + (nŽr). In the exceptional case when m = 0 we get the relation: n = Q.0 + n where Q can be any number.

Quotient table (base ten)
|012345678910
0Q0000000000
1Q1000000000
2Q2100000000
3Q3110000000
4Q4211000000
5Q5211100000
6Q6321110000
7Q7321111000
8Q8422111100
9Q9432111110
10Q10532211111

It will be seen from the table that the values of n|m occur in the table in the form of wedges bounded by straight lines emanating from the top left corner, the upper line of 0s being horizontal, i.e. formed of moves with coordinates (1, 0), the line of 1s diagonally, i.e with move coordinate (1,1), the line of 2s with move coordinates (1,2), like a chess knight, the line of 3s of (1,3) moves, the line of 4s of (1,4) moves and so on.

The remainder r = nŽm is the first number less than m, found by starting with n and repeatedly subtracting m. This process, the division algorithm, gives a series of numbers: n, n – m, n – (2.m), ..., n – (q.m) each smaller than the preceding and thus the process terminates when a number is reached that is less than m. This algorithm provides a way of calculating q and r.

Remainder table (base ten)
Ž012345678910
000000000000
110111111111
220022222222
330103333333
440010444444
550121055555
660002106666
770113210777
880020321088
990101432109
10100012043210

We have the general rules:
n|n = 1, nŽn = 0
n|0 = Q, nŽ0 = n
n|1 = n, nŽ1 = 0
When n<m then n = 0.m + n so: n|m = 0 and nŽm = n. In the tables these results correspond to the figures above the main diagonal.

Some rules for manipulation of the remainder operation are:
(nŽm)Žm = nŽm
(k.a)Ža = 0
(n + k.a)Ža = nŽa
(m+n)Ža = (mŽa + nŽa)Ža
(m.n)Ža = (mŽa . nŽa)Ža

Congruences: The relation of equality of remainders, aŽm = bŽm, can be expressed in a number of alternative ways. We can say that a is congruent to b modulo m, written a≡b (mod m). Alternatively, since a = p.m + r and b = q.m + r we can say that a and b are of the same m-form, or that they belong to the same residue class mod m. Congruence is reflexive, symmetric and transitive, i.e. an equivalence relation.