1 ## THE NATURAL NUMBERSMATHEMATICS IS NOT ONLY about numbers, it is about things that are not numbers; for example, 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 And if AB happens to be 1 centimeter, we would say that CD is 3 centimeters. That is, we can If AB is the unit of length, and CD is any length, will we always be able to AB is to CD as 1 is to That number will be called a real number, which is a number we will need for measuring rather than counting. We will see that there will be problems. At the root of the problem is the difference between arithmetic and geometry. The natural numbers A unit is that form in accordance with which each thing that exists (After Euclid, Book VII, Definition 1.) One apple, one orange, one person. A natural number is composed of the same indivisible and separate units. The people in the room, the electrons in an atom, the fingers on your hand. Our idea of each of them is that they are composed of indivisible units. You cannot take half of any one. If you do, it will not be that unit -- it will not deserve the same name -- any more. Half a person is not also a person. Every language has its names for the natural numbers. The English names are the familiar "One, two, three, four," and so on. Their numerals are "1, 2, 3, 4." Like the names, they represent the numbers, and it is with those that we count and calculate. Hence it has become common to call the numerals themselves "numbers." 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 We will see that we can always put into words how any two natural numbers are related. That relationship is called their ratio. Cardinal and ordinal The counting-names have two forms: cardinal and ordinal. The cardinal forms are One, two, three, four, and so on. They answer the question How many? The ordinal forms are First, second, third, fourth, and so on. They answer the question Which one? We will now see that the ordinal numbers express division into equal parts. They will answer the question, Parts of natural numbers If a smaller number is contained in a larger number an exact number of times, then we say that the smaller number is a part of the larger. (That is called an Consider these first few multiples of 5: 5, 10, 15, 20, 25, 30. 5 is the 5 is a part of each of its multiples 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. The ordinal number "third" names 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 It is extremely important to understand that we are not speaking here of proper fractions -- 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 (For more details, see Skill in Arithmetic, Lesson 15.) Problem 1. a) Write the first five multiples of 6. To see the answer, pass your mouse over the colored area. 6 12 18 24 30 b) 6 is 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 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 d) We say that 4 is the third part of 12, because 12 is the third multiple e) 40 is eight 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 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?
Divisors and parts
Theorem. 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
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? To see the answer, pass your mouse over the colored area. Only those numbers that are divisible by 6. They are the multiples of 6: 6, 12, 18, 24, 30, etc. The sixth part of Problem 7. a) Name all the divisors of 32. Name each 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.
If we divide 15 into three equal parts, that is, into thirds, then the third part of 15 is 5. But Those words, "two thirds," are to be taken literally, like two apples or two chairs. Now, 10 is not a Similarly, if we divide 15 into its fifths, then 3 is the fifth part of 15. 6 is 9 is 12 is And 15 is all five of its fifth parts. We can state the following theorem:
Theorem. 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 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. Next Topic: The ratio of two natural numbers Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |