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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 learn to say, for example, that 6 is the fourth part, or one quarter, of 24, and that 18 is three quarters of 24. We are not concerned here with "How do you do it?" but, rather, "What does that mean?"

What is more, to really understand Percent, one must understand parts -- because a percent is a part of 100%. 50% means half. 25% means a quarter. 20% means a fifth!


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?
  4. How can we calculate a part of a number?
    Percents are parts of 100%.

    Section 2

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


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


A natural number is a collection of indivisible ones.

5 pencils, 8 electrons, 100 people. You cannot take half of any one. If you do, it will not be that same kind of thing any more -- there is no such thing as half a person

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

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?.  We will see that the ordinal numbers name which part.

 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.

Here are the first few multiples of 5:

5,  10,  15,  20,  25,  30,  35,  40.

5 is the first multiple of 5;  10 is the second multiple;  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 larger number is a multiple of the smaller. Equivalently, the smaller is contained in the larger an exact number of times.

5, then, is a part of each one its multiples except itself.

(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 first few multiples of 5:

5,  10,  15,  20,  25,  30,  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.

Since 20 is the fourth multiple of 5, we call 5 the fourth part of 20.  5 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.

From the point of view of division into equal parts, if we divide 15

into three equal parts, then we say that 5 is the third part of 15.  The ordinal number names which part because, again, 15 is the third multiple of 5.

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 ordinal names of the parts in any case are prior to the names of the proper fractions, because the proper fractions are the parts of 1.

Why is the number we write as 1 over 3 --   1
3		  -- called

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

p

That must be understood first. We can then explain

that the number we call  1
3		  is one third of 1.

p

Lesson 19.

Example 1.   3 is which part of 18?

Answer.  The sixth part.  3 is contained in 18 six times.

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

Answer.  7.  Because 28 is made up of four sevens.

28 divided into fourths

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

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

10 divided into fifths

Every number is the fifth part of five times itself.

4 is the fifth part of 5 × 4, which is 20.

9 is the fifth part of 5 × 9, which is 45.

20 is the fifth part of 5 × 20, which is 100.

Note that 1 is a part of every natural number (except itself), because every natural number is a multiple of 1.  Which part is it?  The part that 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.

Particularly important are the numbers that are parts of 100 -- because they are percents.  Since 50 is half of 100, then 50% means one half.  See Problems 13 and 14.

 4.   How can we calculate a part of a number?
 
  Divide by the cardinal number that corresponds to the part. To take half of a number, divide by 2. To take a third, divide by 3. And so on.

See Lesson 10, Question 2, and especially Example 5.

Example 4.   How much is an eighth of $72?

Answer.   72 ÷ 8 = 9.  An eighth of $72 is $9.


Example 5.  Tenths, hundredths.   How much is a tenth of $275? How much is a hundredth?

Answer.   To find a tenth, divide by 10.

275 ÷ 10 = 27.5

(Lesson 3, Question 4.)

Since this is money, we report the answer as $27.50. (Lesson 2.)

As for a hundredth, we will separate two decimal digits:

$275 ÷ 100 = $2.75

Now, in Lesson 3 we saw that when we divide by 10, we have taken 10% of the number. And when we divide by 100, we have taken 1%. Therefore, 10% of $275 is $27.50; 1% is $2.75.

Note:  Whenever we divide by any power of 10 -- the digits do not change.

275 ÷ 100 = 2.75

Conversely, then, if two numbers have the same digits, they differ by a power of 10.

Example 6.   $85 is which part of $850?

Answer.   Apart from the 0 at the end of $850, those numbers have the same digits.  Therefore, they differ by a power of 10.  850 is in fact 10 times 85. (Lesson 3, Question 1.)  Therefore, $85 is the tenth part of $850.  To say the same thing, $85 is 10% of $850.

Example 7.   $.98 is which part of $98?

Answer.   They have the same digits.  They differ by a power of 10.

$.98 is the hundredth part of $98.  It is 1% of it.

Divisors and parts

The divisors of a number will go into the number exactly.  3 is a divisor of 12.  5 is not.  The divisors of a number (except for the number itself) are the only parts that a number has.  3 is the fourth part of 12.

Example 8.   Find all the divisors of 30 in pairs.  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:

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.

Thus divisors always come in pairs.  And that 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, for example, has a divisor 3, then 18 has a third part.  Since 18 has a divisor 6, then 18 has a sixth part.  And so on.

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

In other words, since 18 has a divisor 3, then 18 has a third part.

Example 9.   Into which parts could 12 people be divided?

Answer.   The divisors of 12 are

1, 2, 3, 4, 6, and 12.

Corresponding to each divisor (except 1), there will be a part with the ordinal name of the divisor.  12 people, therefore, could be divided into

Halves, thirds, fourths, sixths, and twelfths.

You cannot take a fifth of 12 people.  12 does not have a divisor 5.


Percent: Parts of 100%

A percent is another way of expressing a part.  Since 100% is the whole (Lesson 3), and since 50% is half of 100%, then 50% means half.  50% of 12 -- half of 12 -- is 6.

Since 25% is a quarter, or a fourth, of 100% --

-- then 25% is another way of saying a quarter.  25% of 40 -- the fourth part of 40 -- is 10.

In the next Lesson, Question 10, we will see how to take 25% by taking half of half

Since 20% is the fifth part of 100% --

(100 is made up of five 20's) -- then 20% is another way of saying a fifth. 20% of 15 -- the fifth part of 15 -- is 3.

See Problems 13 and 14.


*

When we say "5 is the third part of 15," we do not imply a sequence: the first part, the second part, the third part, and so on. When we speak of the third part, that 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." We use that ordinal number because 15 is the third multiple of 5.


At this point, please "turn" the page and do some Problems.

or

Continue on to the next Section:  Parts, plural


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