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THE NATURAL NUMBERS
The ratio of a smaller number to a larger
MATHEMATICS IS NOT ONLY about numbers, it is about things that are not numbers; for example, length; the distance from here to there,
from A to B. Now a length is not a number, yet we describe lengths by saying that they are like numbers. For if CD is made up of three segments equal to AB, then we say,
AB is to CD as 1 is to 3.
And if AB happens to be 1 centimeter, we would say that CD is 3 centimeters.
That is, we can say that CD is "three times" AB, and that is called the ratio -- the relationship -- of CD to AB. The eventual question will be:
If AB, CD are any lengths, will we always be able to express their ratio in words. Will there always be a number n such that, proportionally,
AB is to CD as 1 is to n?
That number will be called a real number, which is a number required to name the length of a line, relative to the line we have called 1. We will see that there will be problems. At the root of the problem is the difference between arithmetic and geometry.
The natural numbers
Arithmetic begins with counting, and a unit is whatever we would call one. One apple, one orange, one person. We count units -- which must have the same name. One apple, two apples, three apples. Therefore we must recognize whether things are the same or different. ("Here's one; here's one; these two are the same. This is not one. These two are different.")
A natural number is a collection of these indivisible ones. 5 people,
10 chairs, 32 names. You cannot take half of any one.
The natural numbers have their everyday names and symbols: 1, 2,
3, 4, and so on. Those symbols, however, are called numerals, and they represent the numbers. They stand in for them. Throughout history there have been many ways of representing numbers. The student is surely familiar with the Roman numerals: I, V, X, and so on.
The natural number is the actual collection of units, /////, whether strokes, apples, letters, or the idea of units. For there is no '5' apart from five units, even though we do not say the word units. The natural numbers are truly natural. We find them in nature.
Nevertheless, it is conventional to refer to the numerals -- 1, 2, 3, 4, and so on -- as "numbers," which they are not.
Numerals are physical things. We write them on paper, and they obey rules of computation according to their physical appearance on the page. Mathematics at every level is concerned with physical form.
We are about to see that we can always name the ratio, the relationship, of any two natural numbers.
Cardinal and ordinal
The natural numbers have two forms, cardinal and ordinal. The cardinal forms are
One, two, three, four,
and so on. They answer the question How much? or How many?
The ordinal forms are
First, second, third, fourth,
and so on. They answer the question Which one?
We will see that the ordinal numbers express division into equal parts. They will answer the question, Which part?
Parts of natural numbers
We say that a smaller number is a part of a larger number if the smaller is contained in the larger an exact number of times. (That is called an aliquot part.) Equivalently, the larger number is a multiple of the smaller.
Consider these first few multiples of 5:
5, 10, 15, 20, 25, 30.
5 is the first multiple of 5. 10 is the second; 15, the third; and so on.
5 is a part of each multiple except itself. It is a part of 10, of 15, of 20, and so on.
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.
The ordinal number names which part of fifteen 5 is.
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 extremely 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, and so on -- name the parts of the cardinal numbers. The names of the parts in fact are prior to the names of the fractions, as we will see. (Why do we call the number we write as 1 over 3 "one-third"? Because 1 is the third part of 3
)
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, we name each part with ordinal number. The ordinal number names which part.
(For more details, see Skill in Arithmetic, Lesson 14.)
Problem 1.
a) Write the first five multiples of 6.
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6 12 18 24 30
b) 6 is which part of each one of those (except 6)?
6 is half of 12; the third part of 18; the fourth part of 24; the fifth part of 30.
Problem 2. Complete the following with the word multiple or part.
a) A larger number is a multiple of a smaller.
b) A smaller number is a part of a larger.
c) 15 is the fifth multiple of 3, is the same as saying that 3 is the fifth
c)
part of 15.
d) We say that 4 is the third part of 12, because 12 is the third multiple
d) of 4.
e) 40 is eight times 5, therefore 5 is called the eighth part of 40.
f) Every number is a certain part of each of its multiples.
g) A number is divisible by 9, is the same as saying that the number is a
g) multiple
of 9.
Problem 3.
a) Write all the divisors of 20.
1, 2, 4, 5, 10, and 20
b) Each divisor, except 20 itself, is which part of 20?
1 is the twentieth part of 20.
2 is the tenth part of 20.
4 is the fifth part of 20.
5 is the fourth part of 20.
10 is half of 20.
Problem 4. 1 is a part of every number. Which part is it of the following?
2 Half. 3 Third. 4 Fourth. 10 Tenth. 79 Seventy-ninth.
100 Hundredth.
Problem 5. What number is each of the following?
| a) The eighth part of 8. 1 | b) The tenth part of 10. 1 | |
| c) The fifth part of 5. 1 | d) The sixth part of 42. 7 | |
| e) The eighth part of 24. 3 | f) The fifth part of 10. 2 | |
| g) The tenth part of 40. 4 | h) The fourth part of 20. 5 | |
| i) The ninth part of 9. 1 | j) The third part of 21. 7 | |
| k) The fifth part of 40. 8 | l) The seventh part of 63. 9 | |
| m) The fourth part of 36. 9 | n) The sixth part of 72. 12 | |
Divisors and parts
Theorem. For every divisor that a number has (except 1), it will have a part with the ordinal name of that divisor.
That is, if a number has a divisor 3, then it will have a third part; if it has a divisor 4, it will have a fourth part; while if it has a divisor 2, then it will have a half.
Example. Into which parts could 50 people be divided?
Answer. Here are all the divisors of 50:
1, 2, 5, 10, 25, 50
Corresponding to each divisor (except 1) there will be a part with the ordinal name of that divisor. Thus, since 2 is a divisor, 50 has a half (which is 25). Since 5 is a divisor, 50 has a fifth part (which is 10). Since 10 is a divisor, 50 has a tenth part (which is 5). Finally, it has a twenty-fifth part (which is 2), and a fiftieth part (which is 1).
These are the only parts into which 50 people could be divided. You cannot take a third of 50 people. 50 does not have a divisor 3.
Problem 6. Which numbers have a sixth part?
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Only those numbers that are divisible by 6. They are the multiples of 6:
6, 12, 18, 24, 30, etc.
The sixth part of 6 is 1; the sixth part of 12 is 2; the sixth part of 18 is 3; and so on.
Problem 7.
a) Name all the divisors of 32. Name each part that 32 has.
Divisors: 1 2 4 8 16 32
32 has a half, a fourth part, an eighth part, a sixteenth part, and a thirty-second part.
b) Name each part that 13 has.
Only a thirteenth part, which is 1.
c) 10 people could be divided into which parts?
Half, fifths, and tenths.
d) 7 pencils could be divided into which parts?
Sevenths.
Parts, plural
If we divide 15 into three equal parts, that is, into thirds, then the third part of 15 is 5.
But each 5 is a third part of 15. Therefore, two 5's -- 10 --are two third parts of 15. Or simply two thirds.
Those words, "two thirds," are to be taken literally, like two apples or two chairs. Two thirds are twice as much as one third. Count them:
One Third, two Thirds. 5, 10.
Now, 10 is not a part of 15 because 15 is not a multiple of 10. We say, rather, that it is parts, plural. Two third parts.
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.
We can state the following theorem:
Theorem. Each number is either a part of a larger number or parts of it.
The following problem will illustrate this. It will illustrate that each number less than 9 is either a part of 9 or parts of 9.
Problem 8.
a) Into which parts can 9 be divided?
Ninths and Thirds.
b) Each number less than 9 is which part of 9, or which parts of 9?
1 is the ninth part of 9.
2 is two ninths of 9.
3 is three ninths -- and also the third part -- of 9.
4 is four ninths of 9.
5 is five ninths of 9.
6 is six ninths -- and also two thirds -- of 9.
7 is seven ninths of 9.
8 is eight ninths of 9.
Each number less 9, then, is either a part of 9 or parts of it. We can therefore express in words how each of those numbers is related to 9. We can say that 7, for example, is "seven ninths" of 9.
Notice how each number says its name. 7 says its cardinal name "seven." 9 says its ordinal name "ninth."
Problem 9. What relationship has 9 to 10?
9 is nine tenths of 10.
The ratio of natural numbers
The student can now begin to appreciate that we can always express in words how any two natural numbers are related. 7 is half of 14. 24 is three times 8. 10 is two thirds of 15. This way of expressing the relationship of any two natural numbers is called their ratio.
Definition. The ratio of two natural numbers is their relationship with respect to relative size that we can express verbally in a sentence. 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.
(Skill in Arithmetic, Lesson 16. Cf. Euclid, VII. Def. 20.)
Example 1. Multiple What ratio has 15 to 5?
Answer. 15 is three times 5.
That is the ratio -- the relationship -- of 15 to 5.
We do not answer "3 to 1," because we want to name the ratio of 15 to 5 explicitly. It is true that 15 is to 5 in the same ratio as 3 is to 1. 3 is three times 1, just as 15 is three times 5.
The two numbers in a ratio are called the terms; the first term and the second.
Notice that we answer with a complete sentence beginning with the first term and ending with the second: "15 is three times 5." For, a ratio is a relationship.
Example 2. Part What ratio has 5 to 15?
Answer. 5 is the third part of 15.
That is called inverse ratio of 15 to 5. The terms are exchanged.
Example 3. Parts What ratio has 10 to 15?
Answer. 10 is two thirds of 15.
These are the three types of ratio: One number is a multiple of the other (so many times it), a part of it, or parts of it.
Problem 10. What ratio have the following? Answer with a complete sentence beginning with the first term.
a) 2 to 10? 2 is the fifth part of 10.
| b) 10 to 2? | 10 is five times 2. A larger number is always so many times a smaller. |
c) 7 to 1? 7 is seven times 1.
d) 1 to 7? 1 is the seventh part of 7.
e) 25 to 100? 25 is the fourth part of 100.
f) 75 to 100? 75 is three fourths of 100.
g) 12 to 6? 12 is two times 6, or twice as much as 6, or double 6.
h) 6 to 12? 6 is half of 12.
i) 40 to 8? 40 is five times 8.
j) 24 to 6? 24 is four times 6.
k) 6 to 24? 6 is the fourth part of 24.
l) 10 to 1? 10 is ten times 1.
m) 1 to 10? 1 is the tenth part of 10.
The ratio of a smaller number to a larger
2 to 3
We can always express the ratio of a smaller number to a larger simply by letting each number say its name. Let the smaller number say its cardinal name -- One, two, three, . . . Let the larger number say its ordinal name -- third, fourth, fifth . . . .
Example 4. What ratio has 2 to 3?
Answer. "2 is two thirds of 3."
2 says its cardinal name, "two." 3 says its ordinal name, "third."
Example 5. What ratio has 4 to 5?
Answer. "4 is four fifths of 5." Each number says its name.
To see this, consider that 1 is one fifth of 5:
2 is two fifths of 5.
3 is three fifths of 5.
4 is four fifths of 5.
Each number says its name.
Problem 11. What ratio has
a) 5 to 8? 5 is five eighths of 8.
b) 3 to 4? 3 is three fourths of 4.
c) 2 to 9? 2 is two ninths of 9.
c) 99 to 100? 99 is ninety-nine hundredths of 100.
What ratio has 8 to 12? While it is correct to say that 8 is eight twelfths of 12, we will see how to express that ratio with the smallest numbers that have that ratio, that is, with the lowest terms. (Lesson 2: The theorem of the common divisor.)
A mixed number of times
By a mixed number of times, we mean a whole number of times plus a part.
Example 6. How much is two and a half times 8?
Answer. "Two and a half times 8" means
Two times 8 plus half of 8.
Two times 8 is 16. Half of 8 is 4. 16 plus 4 is 20.
Example 7. A cheese sells for $6 a pound, and you buy three and a half pounds. How much do you pay?
| Answer. | Three pounds cost $18. |
| Half a pound costs $3. | |
| You pay $21. | |
That is, "Three and a half times 6" means
Three times 6 plus half of 6.
18 + 3 = 21.
This is a mixed number of times: A whole number of times plus a part.
Example 8. How much is five and a quarter times 8?
| Answer. | "Five times 8 is 40. |
| "A quarter (or a fourth) of 8 is 2. | |
| "40 + 2 = 42." | |
Problem 12. How much is
a) Two and a half times 40? 80 + 20 = 100
b) One and a half times 12? 12 + 6 = 18
c) One and a quarter times 20? 20 + 5 = 25
d) Two and a quarter times 8? 16 + 2 = 18
e) Three and a half times 10? 30 + 5 = 35
f) Five and a third times 6? 30 + 2 = 32
g) Two and a quarter times 100? 200 + 25 = 225
h) Two and three quarters times 100? 200 + 75 = 275
Mixed ratio
Ratio and division
We just saw that "20 is two and a half times 8." That statement expresses the ratio of 20 to 8. It is called a mixed ratio. In a mixed ratio, the larger number is a multiple of the smaller number, plus a part or parts of the smaller number.
Example 9. What ratio has 25 to 10?
Answer. We can decompose 25 into a multiple of 10 plus a remainder:
25 = 20 + 5.
25 is made up of two 10's, plus a remainder of 5. The remainder 5 is a part of 10, namely half. Therefore we say,
"25 is two and a half times 10."
Two times 10 is 20; half of 10 is 5; 20 plus 5 is 25.
We always say that a larger number is so many times a smaller number. 25 is two and a half times 10.
Example 10. What ratio has 13 to 3? That is, 13 is how many times 3?
To answer, we can divide 13 by 3.
13 ÷ 3 = 4 R 1.
13 is made up of four 3's with remainder 1.
The remainder 1 is a part of 3 -- it is the third part. We say,
"13 is four and a third times 3."
Notice again: We always say that a larger number is so many times a smaller.
Example 11. What ratio has 50 to 40?
Answer. 50 is one and a quarter times 40.
For, 50 contains 40 one time with remainder 10.
The remainder 10 is a quarter of 40. Therefore,
50 is one and a quarter times 40.
We now see that we can always express in words the relationship -- the ratio -- of any two natural numbers.
We also see the relationship between ratio and division. The quotient of two numbers indicates the ratio of those numbers. The ratio of 15 to 5, for example, is indicated by 15 ÷ 5 = 3. This implies:
15 = 3 × 5.
"15 is three times 5."
The traditional notation for ratio is 15 : 3, which is the divison sign ÷ but without the bar.
Problem 13. Express each ratio. (The larger number is how many times the smaller number?)
a) 45 to 10? 45 is four and a half times 10.
b) 20 to 8? 20 is two and a half times 8.
c) 22 to 4? 22 is five and a half times 4.
d) 5 to 2? 5 is two and a half times 2.
e) 7 to 2? 7 is three and a half times 2.
f) 13 to 2? 13 is six and a half times 2.
g) 5 to 4? 5 is one and a quarter times 4.
h) 9 to 4? 9 is two and a quarter times 4.
i) 11 to 4? 11 is two and three quarters times 4.
j) 11 to 3? 11 is three and two thirds times 3.
k) 44 to 6? 44 is seven and a third times 6.
Finally, then, we see that we can always express in words the ratio of any two natural numbers.
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