A summer program and resource for middle school students showing high promise in mathematics


MATHEMATICS CONCEPTS

What is Number?

The concept of number is the most basic and fundamental in the world of science and mathematics. Yet a satisfactory answer was attained only in 1884 A.D. by Fregé [2], the founder of modern mathematical logic. His answer remained unknown to the world until Bertrand Russell, the English mathematician and logician, in his attempt to base all of mathematics in terms of the concept of sets, rediscovered the concept of number.

The concept of number is associated with the concept of a "set." By a set is meant a collection of definite and separate objects for which we can decide whether or not a given object belongs. So to exhibit a set, you show all the objects - popularly called elements - in the collection by either exhibiting each individual element in the collection or precisely describing which elements belong. An instance of the former is the set {Bob, Mary and Marg}. Yes, the set is exhibited by providing a list of its elements inside two curly brackets. In showing a set by describing its members, rather than exhibiting the individual members, we can use ordinary language when that will do - for example, the set of all people on earth. Or, we can also use curly brackets thus: {x | x is a person of the Earth}; where the vertical line means "satisfying the condition that," or simply "such that."

Equivalent Sets

Let A be the name of the set comprised of Bob, Mary and Marg. Let B be the set {Brian, chair, Mississipi river} . Consider a correspondence between A and B:
Bob --> chair
Mary --> Brian
Marg --> Mississipi river
In fact, we can write down five more such correspondences! Let us use the one above. Such a correspondence is called a one-to-one (often written 1-1) correspondence from A on to (often written "onto")B. A member of each set is accounted for exactly once in a 1-1 onto correspondence. Such a correspondence from A to B has the property that you can reverse the arrows and get a 1-1 onto correspondence from B to A. (It is not always easy to show that a correspondence is on-to, especially when dealing with infinite sets. But if we can show that a correspondence is 1-1 and that its reverse correspondence is also 1-1, then each correspondence is 1-1 and on-to; see the link below called "Schroder-Bernstein Theorem.")

Mathematicians call a 1-1 onto correpondence also a "bijection." They say A is "bijective" with B. Now, there are all kinds of sets that are bijective with A. So we might as well continue with A and B. The sets A and B share a quality. That quality is called the "cardinal" number. Cardinal means important. The cardinal number of our A or B is written as "3". We will call this important number just "number."

So what is a number? It is that property of sets which is common to all sets that are bijective with each other. Two sets that are bijective with each other are also called "equivalent" sets. So our A and B are equivalent sets. A number is the property common to equivalent sets.

Numeral

Numeral is the symbol for the idea called number. Put another way, the number is the idea we think of when we see the numeral or when we see or hear the word for a numeral.

Suppose there is a person named Jim. This person has the name Jim because he was named so. It is very convenient! A numeral is like the name Jim.

Now, if someone says number 3, we know what really is meant. 3 is the numeral for the number the person wishes to communicate to us. Since this is to be always understood, we just say "number 3."

An alien coming to earth might be amused to note that we have given this number the name 3. A computer on earth would have to be told that this number is 11, because 11 is 3 in binary notation. 11 in binary and 3 in decimal notation are called "numerals." As you know, III is the Roman numeral for 3. These are all names for the quality shared by all sets that are bijective with the set A above. Or the set B above!

On the web page "Numeration systems," we discuss the various systems like the binary, decimal and others for writing numerals.

Cardinal Numbers and Ordinal Numbers

There are mainly two kinds of numbers:cardinals and ordinals. Cardinal numbers tell how many things there are in a set, as in "There are four people in my family." If you consider a set and a copy of it, you get two identical sets which are clearly equivalent. The number here is often referred to as the "cardinality" of the set; it is still the cardinal number of the set. The cardinality of a set is the size of the set; it is the number of things - or elements - in the set.

Ordinal numbers have to do with the order or priority of the elements in a set that has such order or priority for all its elements. So ordinal numbers give the positions of elements in an ordered set. For example, "I am fourth in the queue for lunch." The word "fourth" denotes an ordinal number.

Zero

0 is the quality shared by all sets that contain nothing. Mathematicians call such sets, the empty sets. So 0 is not nothing; it is a numeral or let us be lazy like everybody else and call it a number! So if you are asked if zero is a number, you could say "well, it is." "Well" because it is a number in the sense of being a numeral. Well, it should be really called a numeral. But number will do.

Why use equivalent sets to define number?

First, it makes sense that we consider these objects called sets; for, afterall, a use of numbers is for counting the elements in a set. Second, bijection does make sense. Consider this. Suppose there was a shepherd in 5000 B.C. who had a large flock of sheep. How does he make sure that at the end of the day, all the sheep are back? Since, probably, counting to numbers as large as the cardinal number of the large herd did not exist at that time, he could still account for all the sheep as follows. He will take a heap of stones and when the herd goes out in the morning, pass a stone for each sheep to form a new heap of stones - and throw away the stones left in the old heap. The new heap is his SET of stones. When the herd returns in the evening, he can pass on one stone for each sheep that comes in. If the last stone is passed on to form a new heap and a sheep has indeed gone in to correspond to that stone, the SET of sheep is equivalent to the set of stones and all the sheep are indeed back.

A third reason for not abandoning the concept of equivalent sets to define number is that it enables us to also handle sets that are not finite. Georg Cantor (1845-1918) is the creator[1] of the theory of sets and the use of equivalence to define number. Cantor went farther with equivalence. He used it to introduce a notion called the "power set" of a set, and discovered infinitely many distinct and successively larger infinities!

A more precise definition of 1-1 correspondence and Number

Let us revisit the definition of a 1-1 correspondence from set A to set B. There is a problem with our earlier definition, namely, that the numeral 1 is used in it. Notice that we used the 1-1 correspondence to define Number. So, using the numeral 1 to lead to a definition of number is circular - or self-referential - and needs to be avoided.

Here then is the precise definition. A correspondence from A to B is "one-one" if an element x in A corresponds to an element y in B and no other element x' in A corresponds to y, and x does not correspond to any element y' other than y. When only the first of these conditions holds, the correspondence is called "one-many"; when only the second holds, it is called "many-one." You might still object that we have used the word "one" which is representing the numeral 1. This is not a problem, for you could use another word, say, "banana' and the definition is not affected. The use of the word "one" only serves as a pnemonic; that is, the word "one" reminds us what we need to accomplish.

We can make the definition even more precise as follows: A correspondence from A to B is "one-one" if an element x in A corresponds to an element y in B and an element x' in A corresponds to y, then x = x', and if x also corresponds to an element y' then y = y'.

Let us now revisit our definition of number. I stated: It is that property of sets which is common to equivalent sets. The problem with this definition is that we have not proved that equivalent sets have a common property and that there is no other common property. In the language of mathematics, we need to show that there exists a common property and that it is unique. A proof of this will take us too far into the field of mathematical logic. Therefore I will give you the precise definition of number that does not explicitly mention the common property. Here is the definition. A number is the set of all equivalent sets. While this definition is precise, it lacks the feel one would get referring to the definition using the common property of equivalent sets - which was why we resorted to that definition earlier.

A fundamental question might occur to you: How do you add two numbers? The answer is this. Let A be the first number; that is, A is the set of equivalent sets. Let B be the second number. Then the sum of the two numbers is the union of the sets A and B. Think about it! Once addition is taken care of, multiplication is similarly accomplished. Subtraction and division are harder to formulate. But our task was to define number. For division, subtraction and other matters, please see the section below.

Arithmetic - The mathematics of counting

There is an alternate route that can be taken to the concept of number. That is to just say "let it be." In other words, leave it as an undefined or "primitive" concept. This is the approach taken in the method called the AXIOMATIC method. In this method, it is best to leave the basic objects undefined, or problems might arise later when proofs are sought. This is exemplified by Euclid. Having the undefined concepts also settles questions about the concept embodied in the undefined term.

For plane geometry that now bears his name, Euclid started with the concept of a point. But he defined it: "A point is that which has no part." After also defining "line," "angle," and the like, he described the relationship between these quantities, and wrote down five axioms to deduce the satements (theorems) of geometry. But some of his proofs had uncorrectable logical blemishes which were corrected after David Hilbert (1862-1943)[3] wrote down five groups of axioms using the following undefined concepts: point, line, plane, lie on (a relation between a point and line or a point and plane), betweenness, congruence of pairs of points, and congruence of angles. Hilbert also gave an arithmetic interpretation of geometry. Such a connection forged betwen geometry and arithmetic is today considered as proof that Hilbert's axioms would never produce two theorems that contradicted each other, provided arithmetic possessed that quality.

Giuseppe Peano[4] started with the undefined concepts of "set," "natural number," "successor," and "belongs to", and proceeded to provide a system of axioms for arithmetic. His first five axioms deal with the natural numbers:
(1) 1 is a natural number.
(2) 1 is not a successor of any other natural number.
(3) Each natural number a has a successor.
(4) If the successors of a and b are equal then so are a and b.
(5) If a set S of natural numbers contains 1 and if when S contains any number a it also contains the successor of a, then S contains all the natural numbers.

This last axiom is the axiom of mathematical induction which is a method of proof used for proving statements directly or indirectly involving natural numbers. Peano wrote down a few more axioms to compare numbers and to add and multiply them, and proved the properties we take for granted of numbers - like 1 + 1 = 2. From these, others defined and established the properties of positive and negative integers and rational numbers in terms of pairs of natural numbers. Then the logical foundation of all FINITE numbers including irrational numbers - which are precisely the nonperiodic decimals - was attained.

Cantor's concept of the number helps us to visualize the abstract; Peano's axioms helped to provide the foundation for calculating with the finite ones of those numbers. Leopold Kronecker has said, "God made the integers, all else is the work of man."

HISTORICAL SOURCES
[1] Georg Cantor, Mathematische Annalen (1874)
[2] Gottlob Fregé, Die Grundlagen der Arithmetik (1884)
J.L. Austin (1953 English trans.) The Foundations of Arithmetic: A Logico-Mathematical Enquiry in to the Concept of Number
[3] David Hilbert, Grundlagen der Geometrie (Seventh Edition), 1930 - Foundations of Geometry
[4] Giuseppe Peano, Arithmetices Principia Nova Methodo Exposita (1889)

(G.R.T.)