WHAT IS THE NATURAL NUMBERS?

Math Adda                           WHAT IS THE NATURAL NUMBERS?

MATHEMATICS IS NOT ONLY about numbers, it is about things that are not numbers; for example, length;  the distance from here to there,

A--------------------------B

C--------I------------------I-------------------D

from A to B.  Now a length is not a number,  yet we describe lengths by saying that they are like numbers.  For if CD is made up of three segments equal to AB, then we say,

AB is to CD  as  1 is to 3.

And if AB happens to be 1 centimeter, we would say that CD is 3 centimeters.

That is, we can say that CD is "three times" AB, and that is called theratio -- the relationship -- of CD to AB.  The eventual question will be:
                  1
A--------------------------B
                                 n
C---------------------------------------------D

If AB, CD are any lengths, will we always be able to express their ratio in words.  Will there always be a number n such that, proportionally,

AB is to CD  as  1 is to n?

That number will be called a real number, which is a number required to name the length of a line, relative to the line we have called 1.  We will see that there will be problems.  At the root of the problem is the difference between arithmetic and geometry.

The natural numbers

Arithmetic begins with counting, and a unit is whatever we would call one.  One apple, one orange, one person.  We count units -- which must have the same name. One apple, two apples, three apples. Therefore we must recognize whether things are the same or different. ("Here's one; here's one; these two are the same.  This is not one. Those two are different.")

A natural number is a collection of these indivisible ones.  5 people,
10 chairs, 32 names.  You cannot take half of any one.

The natural numbers have their everyday names and symbols:  1, 2,
3, 4, and so on.  Those symbols, however, are called numerals, and they represent the numbers.  They stand in for them.  Throughout history there have been many ways of representing numbers.  The student is surely familiar with the Roman numerals:  I, V, X, and so on.

The natural number is the actual collection of units, /////, whether strokes, apples, letters, or the idea of units.  For there is no '5' apart from five units, even though we do not say the word units.  The natural numbers are truly natural.  We find them in nature.

Nevertheless, it is conventional to refer to the numerals -- 1, 2, 3, 4, and so on -- as "numbers," which they are not.

Numerals are physical things. We write them on paper, and they obey rules of computation according to their physical appearance on the page. Mathematics at every level is concerned with physical form.

We are about to see that we can always name the ratio, the relationship, of any two natural numbers.

Cardinal and ordinal

The natural numbers have two forms, cardinal and ordinal.  The cardinal forms are

One, two, three, four,

and so on.  They answer the question How much? or How many?

The ordinal forms are

First, second, third, fourth,

and so on.  They answer the question Which one?

We will see that the ordinal numbers express division into equal parts. They will answer the question, Which part?

Parts of natural numbers

We say that a smaller number is a part of a larger number  if the smaller is contained in the larger an exact number of times. (That is called analiquot part.)  Equivalently, the larger number is a multiple of the smaller.

Consider these first few multiples of 5:

5, 10, 15, 20, 25, 30.

5 is the first multiple of 5.  10 is the second;  15, the third; and so on.

5 is a part of each multiple except itself.  It is a part of 10, of 15, of 20, and so on.

Now, since 15 is the third multiple of 5, we say that 5 is the third partof 15.  We use that same ordinal number to name the part.

The ordinal number names which part of fifteen 5 is.

5 is the fourth part of 20; it is the fifth part of 25; the sixth part of 30.  And so on.

5 is which part of 10?  We do not say the second part. We say half.  5 is half of 10.

It is extremely important to understand that we are not speaking here of proper fractions -- numbers that are less than 1 and that we need for measuring.  We are explaining how the ordinal numbers -- third, fourth, fifth, and so on -- name the parts of the cardinal numbers.  The names of the parts in fact are prior to the names of the fractions, as we will see. (Why do we call the number we write as 1 over 3 "one-third"? Because 1 is the third part of 3)

Note that 5 is not a part of itself.  There is no such thing as the first part.

So, with the exception of the name half, we name each part with ordinal number.  The ordinal number names which part.