Natural numbers:  cardinal and ordinal

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

One, two, three, four, etc.

They answer the question How much? or  How many?.  The ordinal forms are

First, second, third, fourth, etc.

They answer the question Which one?.

Parts of natural numbers

We say that a smaller number is a part of a larger number if the larger number is a multiple of the smaller.

Here are the multiples of 5:

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

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

5 is a part of each one of those (except itself).  Since 15, for example, is the third multiple of 5, we say that 5 is the third part of 15.  We use that same ordinal number to name the part.

Similarly, 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 important to understand that we are not speaking here of proper fractions -- numbers that are less than 1, and that we use for measuring. We are explaining how the ordinal numbers -- third, fourth, fifth, etc. -- name the parts of the cardinal numbers.  We will come to those fractional symbols shortly.

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, the parts are named with ordinal numbers.  Each ordinal number tells which part.

Parts, plural

The figure shows that each 5 is a third part of 15, and so we say that 15 has been divided into thirds, that is, into three equal pieces.

5,  10,  15.

Therefore, 10 -- which is two 5's -- is two third parts of 15. One third. Two thirds.

Now, 10 is not a part of 15, because 15 is not a multiple of 10.  We say that it is parts, plural:  Two third parts, or simply two thirds.  Those words are to be taken literally.

Similarly, if we divide 15 into its fifths,

then

3 is the fifth part of 15.

6 is two fifth parts of 15.

9 is three fifth parts of 15.  (Count them!)

12 is four fifth parts of 15, or simply four fifths.

And 15 is all five of its fifth parts.

The ratio of natural numbers

Definition.  The ratio of two natural numbers is their relationship with respect to relative size which we can express in words. Specifically, it is their relationship in which one number is a multiple of the other (so many times it), a part of it, or parts of it.

The names of the fractions

In English, the proper fractions have the same names as the ratio of the

numerator to the denominator.  The number we write as 1 over 3 --

1 3

--

is called "one-third" because of the ratio of 1 to 3.  1 is one third of 3.

It is for this reason that fractions are called rational numbers. Their names are the same as the ratio of two natural numbers.

Fractional symbols may therefore be regarded as ratio symbols, in that they signify the ratio of the numerator to the denominator.

1 is to 2  as  2 is to 4  as  5 is to 10, etc.

Those are equivalent fractions;  for, each numerator is half of its denominator.

Example 4.   What ratio has 3 to 4?

Answer.  3 is three fourths of 4.  We can express the ratio of any smaller number to a larger simply by letting each number say its name.  3 says its cardinal name "three." 4 says its ordinal name "fourths."

Example 5.   What ratio has 9 to 2?

9 is four and a half times 2.

We always say that a larger number is so many times a smaller.

Proportions

The theorem of the alternate proportion

The elements in a proportion are called the terms:  the 1st, the 2nd, the 3rd, and the 4th.

1st : 2nd = 3rd : 4th

We say that the 1st and the 3rd are corresponding terms, as are the 2nd and the 4th.

The following is the theorem of the alternate proportion:

Since

1 : 3 = 5 : 15,

then alternately,

1 : 5 = 3 : 15.

The theorem of the same multiple

Let us complete this proportion,

4 : 5 = 12 : ?

12 is four fifths.

Alternately, however, 4 is a third of 12 -- or we could say that 4 has been multiplied by 3.  Therefore 5 also must be multiplied by 3,

4 : 5 = 12 : 15

That is,

4 : 5 = 3 × 4 : 3 × 5

As one 4 is to one 5, so any number of 4's will be to an equal number of 5's.  Three 4's are four fifths of three 5's.

This is the theorem of the same multiple.

Formally,

a : b = ma : mb.

In arithmetic and algebra, this appears as the principle of equivalent fractions:

Example 7.   Complete this proportion, 5 : 8 = 35 : ?

Answer.  Look at it alternately.  35 is seven times 5.  Therefore the missing term will be seven times 8, which is 56.

Problem 9.   Complete this proportion,  4 : 9 = 24 : ?

54.  The term corresponding to 4 is 6 × 4.  Therefore, the missing term must be 6 × 9.

We shall often make use of this basic property of the square root radical:

A radical multiplied by itself

produces the radicand.

Example 9.   Solve this proportion -- 8 : 12 = 2 : ?

Answer.  To produce 2,  8 has been divided by 4.  Therefore 12 also must be divided by 4.

8 : 12 = 2 : 3

Example 10.   Solve this proportion --  2 : 11 = 9 : ?

Similar figures

Geometry is the study of figures.

Similar figures are equiangular, and the sides that make the equal angles are proportional.

Thus, to say that figures ABCDE,  PQRST are similar, is to say that the angle at A is equal to the angle at P,  the angle at B is equal to the angle at Q, etc.;  and, proportionally, as AB is to BC, so PQ is to QR.

And so on, for each pair of equal angles and the sides that make them.

Example 11.   Let triangles HJK, LMN be similar, and let HJ = 2 in,
JK = 5 in, HK = 4 in, and LM = 7 in. How long are MN and LN?

Answer.  Since HJ is half of HK, then LM will also be half of LN. Therefore, LN is 14 in.