Rational Mathematics

by G. P. Jelliss

Addition

From the tally symbols for numbers m and n we can form another tally symbol by putting all the tally marks within the same space, and this combined tally symbol will also represent a number. This process is known as addition. Thus, provided we can carry out this process, we can add two numbers m and n to form a number called their sum which is written using the plus sign (+) in the form m+n. For example (||)+(|||) = (|||||), i.e. 2+3 = 5.

Addition table (base ten)
+012345678910
0012345678910
11234567891011
223456789101112
3345678910111213
44567891011121314
556789101112131415
6678910111213141516
77891011121314151617
889101112131415161718
9910111213141516171819
101011121314151617181920

Any number can be represented uniquely as a sum of a sequence of ones. This statement can be expressed as n = ∑(1,1,1,...), where ∑ is the summation operator, and is true of 0 and 1 as well as of higher numbers, since 0 = ∑( ), sum of an empty array of units, and 1 = ∑(1), sum of an array of a single unit.

Addition is commutative: a+b = b+a, i.e. the result is independent of the order of the numbers. Addition is associative: a+(b+c) = (a+b)+c, i.e. the result of a series of additions is independent of the order in which they are carried out. The number 0, has the additive identity property: a+0 = a for any number a. In particular 0+0 = 0.

Order and addition are related: we have n>m if and only if there exists a number d, not zero, such that n = m+d. The number n+1 is the successor of n.

Subtraction

If n = m+d then we say that the difference of n and m is d. This number is uniquely determinate and so we are justified in introducing another operation symbol, writing n−m = d and we call this operation subtraction. We have m = n if and only if n−m = 0. Subtraction is only a partial operation on numbers, since n−m is defined only when n≥m.

Order and subtraction are related: we have n<m if and only if there exists a nonzero number d such that m−d = n. The number n−1 is the immediate predecessor of n. The number of spaces between n things in a row is n−1. This simple principle often turns up. (On the other hand, the number of spaces between n things arranged in a circle is n.)

We can also define the symmetric difference nHm to be the number d such that either n = m+d or m = n+d, one of which must be true (both are true only when n=m, then d=0).

Addition is defined by combining separate sets. More generally two sets will have a non-null intersection, and in this case the numbers of elements in the union and intersection are related to the numbers in the given sets by: |A∪B| + |A∩B| = |A| + |B|.

The number of elements in the union of two overlapping sets is given by |AuB| = (|A| + |B|) − |A∩B|. This result can be generalised to apply to a family of sets: |∪{A,B,C,D,...}| = (∑|A| + ∑|N{A,B,C}| + ∑|N{A,B,C,D,E}| + ...) (∑|N{A,B}| + ∑|N{A,B,C,D}| + ...) that is the difference between the sums of the numbers of elements common to odd and even numbers of the sets. This rule is known as the principle of inclusion and exclusion (or of 'cross-classification').

Example: Out of a set of 120 students, 42 take English, 37 take neither language, 9 take both languages, how many take French but not English?
Answer; |E∪F| = 120−37 = 83, |E| = 42, |F−E| = |E∪F|−|E| = 83−42 = 41. The information about |F∩E| is not needed.