Kansas Academy of Science. 



WHAT, THEN, IS NUMBER? 



If, as some allege, no formal and perfect definition can be given, yet by means of careful 

 study we should be able to talk intelligently about it, and be able to say confidently, 

 "thvi" is number, or "that" is not number. 



Mill says in his System of Logic: "The fact asserted in the definition of number is a 

 physical fact. Each of the numbers two, three, four, etc., denotes physical phenomena 

 and connotes a physical property of those phenomena. Two, for instance, denotes all 

 pairs of things, twelve all dozens of things, connoting what makes them pairs or dozens; 

 and that which makes them so is something physical, since it caimot be denied that two 

 apples are physically distinguishable from three apples, two horses from one horse, and 

 BO forth ; that they are a different, visible, and tangible phenomena." 



NUMBER USED TO DESIGNATE HOW MANY. 



Perhaps it will be admitted that one of the first and most obvious uses of number is to 

 distinguish or define how many things are thought of. It may or may not be in answer to 

 a question. In either case, the term called number, completely fulfills the function of 

 telling how many. 



Does number do anything more than tell how many ? It appears not. If not, let 

 number be defined as that which expresses how many things are thought of, and let this defini- 

 tion be accepted until some defect be found, or until some better definition be invented. 



TWO CLASSES OF NUMBERS. 



If there be a group. of oranges, consisting of three whole ones and a half of one, that 

 is, one of two equal parts of a whole one, then to describe how many are in this group 

 one would say three-and-a-half, and this term obviously performs the same kind of oQice 

 which in other instances is performed by the term three, or by four; in other words, it 

 completely fulfills the olfice of number, and should be included under the name. The 

 same may evidently be aflirmed of one-half or three-fourths. Either may be used to express 

 how many, as one may say "I have ten dollars," while another states "I have three-fourths 

 of a dollar," or still anotli^r says "I have one-half a dollar." It appears then that number 

 may be aflirmed of whole things or parts of things, and this suggests two classes of numbers, 

 one denoting whole things, the otlier denoting pai'ts of things, called respectively integi-al 

 and fractional numbers. 



To many, no doubt mucii of the foregoing seems so obvious as scarcely to merit a 

 statement, yet recently an advanced college class in mathematics were puzzled by the 

 quesion, " is one-half a number ? " and the opinions were divided, though a majority 

 thought not. The sarne question propounded to a large convention of teachers, caused a 

 nearly equal division in the opinions expressed, while afterwards an able mathematician, 

 widely known and admitted to be such, expressed serious doubt in regard to the matter, 

 though finally admitting the view here expressed. 



The facts just stated, indicate not so much a superficial acquaintance with the subject, 

 as the imperfection of the current nietliods of presenting the elements of numbers, which 

 should leave sucli confused notions in the mind, even after subsequent progress had made 

 the problems of the higher mathematics quite familiar. One objected to the existence 

 of fractional number, because " there cannot be such things as three-and-a-half ifien, or 

 four-and-one-fourth sheep." To this it may be answered, that fact simply indicates a fea- 

 tuTe in the constitution of men and of living beings, which does not permit the existence 

 of life in fractional parts, but in no wise does it impair the flexibility of number which 

 is applicable either to wholes or parts. 



In this connection, a practical suggestion occurs, which seems worthy of consideration. 

 Since integral and fractional numbers are two classes of number in general, the expo.si- 

 tlon of the elementary operations should be applicable alike to both these classes; 

 for instance, the definitions of multiplication and of division should be such as to apply- 

 to either integral or fractional numbers. By such means it is believed that the subject of 



