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Lesson 15

PARTS
OF NATURAL NUMBERS  1


In this chapter we will learn to speak the language of arithmetic. That 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?"

To understand what that means, the student must have mastered the multiplication table, and the associated inverse question: What number times 4 equals 24?

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

Finally, fractions (Lesson 20) are the parts of number 1.


In this Lesson, we will address the following:

  1. What is a natural number?
  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 do we name the part that a smaller number is of a larger??
  5. How can we calculate a part of a number?
    Percents are parts of 100%.
    Repeated division in half.

    Section 2

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


 1.   What is a natural number?
 
 
  It is that which is countable. Hence it is a collection composed of units; of the same indivisible ones, to which we give a name and a symbol.
 

The symbols 1, 2, 3, 4, and so on, are the familiar Arabic numerals for the natural numbers.  It has become conventional to call those symbols themselves "numbers." And we shall do that. Yet a symbol is not what it symbolizes, what it stands for, which in this case is a number of units.

The people in the room, the electrons in an atom, or the idea of a group of units. Each of those is a natural number. 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.


Cardinal and ordinal

The names of the natural numbers have two forms: cardinal and ordinal. The cardinal forms are

One, two, three, four, etc.

They answer the question How many?  The ordinal forms are

First, second, third, fourth, etc.

They answer the question Which one?  We will see that the ordinal numbers express division into equal parts. They will name which part -- the third, the fourth, the fifth, and so on.

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.
 

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.)

 4.   How do we name the part that a smaller number is
of a larger?
 
  With an ordinal number. With the exception of "half," the ordinal numbers name which 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.

So with the exception of "half," an ordinal number names into which parts a number has been divided.

If we divide any number, such as 15, into three equal parts, then we say that we have divided it into thirds.  5 is the third part of 15.

If we divide a number into four equal parts,

then we have divided it into fourths; if into five equal parts,

into fifths. But if we divide into two equal parts, then we have divided it in half.

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 equal parts into which a number has been divided.  When answering the questions of this Lesson, the student should not write fractions.  We will come to those symbols in Lesson 20.

It should be clear that the names of the parts 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 the numerator 1 is one third
of the denominator 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 20.

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 number (except itself), because every 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.

 5.   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 11, 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 4, Question 4.)

Since this is money, we report the answer as $27.50. (Lesson 3.)

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

$275 ÷ 100 = $2.75

Now, in Lesson 4 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 4, 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.  5 is not any part of 12; you cannot divide 12 people into equal groups of 5.

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.  But 18 does not have a fifth part, because 5 is not a divisor of 18.

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.  Because whatever part the percent is of 100%, that is the part we mean.

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.


Repeated division in half

If we start with the whole of something, and divide it into two equal parts --

-- then we have divided the whole in Half.

If we now divide each Half in half --

-- the whole will now be in four equal parts, or Quarters.

If we divide each Quarter in half --

-- the whole will then be in twice as many, that is, eight equal parts, or Eighths.

Each time we take half, the number of equal parts doubles.  Half of a Quarter is an Eighth.  Half of an Eighth is a Sixteenth.  Half of a Sixteenth is a Thirty-second.  And so on.

Now, here are the number of equal pieces that result when we repeatedly take half:

2,  4,  8,  16,  32,  64,  and so on.

Those numbers are called the powers of 2.  Repeated division in half is very common.  The student should know the names of the sequence of those parts:

Halves, Quarters, Eighths, Sixteenths, Thirty-seconds, and so on.

*

When we say "5 is the third part of 15," we do not imply a sequence: the first part, the second part, the third, 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,
"The third part." 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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