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A R I T H M E T I C

Lesson 14

# PARTS OF NATURAL NUMBERS  1

In this chapter we will learn to speak the language of arithmetic. This will allow us to relate any two numbers. We will be able to say, for example, that 12 is half of 24, or that 18 is three fourths of 24. We are not concerned here with "How do you do it?" but, rather, "What does that mean?"

In this Lesson, we will address the following:

1. Which numbers are called the natural numbers?
2. What do we mean by the multiples of a number?
3. What does it mean to say that a smaller number is a part of a larger number?

Section 2

4. What does it mean to say that a smaller number is parts (plural) of a larger number?
5. How can we calculate parts of a number?
6. Percent: Parts of 100%.

 1. Which numbers are called the natural numbers? The counting numbers, starting with "One" or "First."

The natural numbers 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?.

A natural number is a collection of indivisible ones.

5 pencils, 8 electrons, 100 people. You cannot take half of any one.

By a "number" in what follows, we will mean a natural number.

 2. What do we mean by the multiples of a number? They are the numbers produced when that number is repeatedly added.

The natural numbers, then, are the multiples of 1:

1,  2,  3,  4,  5,  6,

and so on, for as far as we please.

Here are the multiples of 5:

5,  10,  15,  20,  25,  30,  35,  40, . . .

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

 3. What does it mean to say that a smaller number is a part of a larger number? It means that the smaller number is contained in the larger an exact number of times. Equivalently, the larger number is a multiple of the smaller.

5, then, is a certain part of each its multiples, except itself.  5 is a certain part of 10, a part of 15, of 20, etc.

(We do not call 5 a part of 12, because 12 is not a multiple of 5.  We are speaking throughout of what is called an aliquot part.)

Here again are the multiples of 5:

5,  1015202530,  35,  40, . . .

Now, since 15 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.

5 is contained in 15 three times.  Therefore we say "5 is the third part of 15."  That is which part of fifteen 5 is.

Similarly, 5 is the fourth part of 20.  It is the fifth part of 25, the sixth part of 30; and so on.  But, 5 is half of 10.  (We do not say the second part.)  And 5 is not a part of itself; there is no such thing as the first part.

It is 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, etc. -- name the parts of the cardinal numbers.  When answering the questions of this Lesson, the student should not write fractions.  We will come to those symbols in Lesson 19.

The parts in any case are prior to the fractions, because
the names of the fractions come from the names
of the parts.  Why is the number we write as 1 over 3

 -- 13 -- called "one-third"?  Because 1 is one third of 3.

That must be understood first. We can then explain that

 the number 13 is one third of 1.

In Lesson 10 we learned about dividing a number into equal parts. We can now say into which parts the number has been divided:  into thirds, fourths, fifths, sixths -- or halves.

Example 1.   3 is which part of 18?

Answer.  3 is the sixth part of 18.  We say that because 18 is made up of six 3's.

Example 2.   What number is the fourth part, or a quarter, of 28?

Example 3.   2 is the fifth part of what number?

Answer.  10.  Because five 2's are 10.

Every number is the fifth part of 5 times itself.

4 is the fifth part of 5 × 4 = 20.

9 is the fifth part of 5 × 9 = 45.

20 is the fifth part of 5 × 20 = 100.

Example 4.   Find all the divisors of 30 in pairs. That is, find all the numbers that go into 30 exactly.  Each divisor (except 30) is which part of 30?

Answer.  Here are all the divisors of 30 in pairs:

1  and  30.   (Because 1 × 30 = 30.)

2  and  15.   (Because 2 × 15 = 30.)

3  and  10.   (Because 3 × 10 = 30.)

5  and  6.     (Because 5 × 6 = 30.)

On naming which part of 30, each divisor will say the ordinal name of its partner:

 (thirtieth)  1  and 30 (fifteenth)  2  and 15 (half) (tenth)   3  and 10 (third) (sixth)   5  and 6   (fifth)

1 is the thirtieth part of 30.

2 is the fifteenth part of 30.  15 is half of 30.

3 is the tenth part of 30.  10 is the third part of 30.

5 is the sixth part of 30.  6 is the fifth part of 30.

Note that 1 is a part of every natural number (except 1 itself), because every natural number is a multiple of 1.  That is what a natural number is. 1 is that part which says the number's name.

1 is the third part of 3, the fourth part of 4, the fifth part of 5, the hundredth part of 100.  1 is half of 2.

Thus divisors always come in pairs.  3 and 6, for example, are a pair of divisors of 18, because 3 × 6 = 18.  What is more, 3 is the sixth part of 18, while 6 is the third part.

This implies the following:

Theorem.  For every divisor (except 1) that a number has, it will have a part with the ordinal name of that divisor.

(Euclid, VII. 37.)

Since 18 has a divisor 3, then 18 has a third part.  And since 18 has a divisor 6, then 18 has a sixth part.

Here is an illustration that 18 has a divisor 3:

18 = 6 × 3.

But according to the order property of multiplication:

18 = 3 × 6.

This means that 6 -- the partner of 3 -- is the third part of 18.

Therefore since 18 has a divisor 3, then 18 has a third part.

*

When we say "5 is the third part of 15," we do not imply a sequence of parts: the first part, the second part, the third part, and so on. This is a different meaning for the word "third." It means each one of three equal parts that together make up the whole.

We say that we have divided 15 into thirds.

Yet "third" still retains an ordinal character. Because to the question, "Which part of 15 is 5?", we answer,
"5 is the third part of 15."

or

Continue on to the next Section:  Parts, plural

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