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 fractions are named the same 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.

Answer.  5 is the third part of 15, and 8 is the third part of 24.

63 is seven times 9.  And 14 is seven times 2.

Proportions

A proportion is a statement that two ratios are the same.

5 is to 15  as  8 is to 24.

5 is the third part of 15, just as 8 is the third part of 24.

We will now introduce this symbol  5 : 15  to signify the ratio of 5 to 15.  A proportion will then appear as follows:

5 : 15 = 8 : 24

"5 is to 15  as  8 is to 24."

Example 5.   12 : 2 = 42 : 7.  ("12 is to 2  as  42 is to 7.")

Why is this a proportion?

Answer.  Because 12 is six times 2, just as 42 is six times 7.

Problem 8.   Complete this proportion,  3 : 12 = 7 : ?

28.  3 is the fourth part of 12, and 7 is the fourth part of 28.

The theorem of the alternate proportion

The numbers 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 6.   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 : 9 = 5 : ?

Answer.  What must we do to 8 to make it into 5?

We must divide it by 8, and multiply by 5 -- that is, we must multiply

And so,

Problem 11.   Multiply a so that it becomes b.

Problem 12.   Complete this proportion:  p : q = r : ?

(As for the method of cross-multiplying, that is an algebraic method which is taught in order to avoid teaching ratio and proportion verbally.  It stems from a 19th century resistance to language in mathematics.)

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

Answer.  Since HJ is half of HK, then LM will also be half of LN. Therefore, LN is 16 in.

Next, since HJ is two fifths of JK, then LM is also two fifths of MN: 8 is two fifths of 20.

How to know that?  8 is 4 × 2.  And 4 × 5 is 20.

Example 11.   In the similar triangles below, let HJ = 3 cm,  JK = 5 cm,  KH = 4 cm, and LM = 6 cm.  How long are MN and LN?

Answer.  We have

HJ : JK = LM : MN

and therefore, alternately,

HJ : LM = JK : MN.

That is

3 : 6 = 5 : 10.

MN = 10 cm.

Similarly, LN will be twice as long as HK.   LN will be 8 cm.