Number Fields (1959)

but I know what I did miss Perkins I've rationalized the denominator of this fraction by multiplying both the numerator and denominator by 3 plus 2 times the square root of 3 right and that gave me this which led to this and finally this well then what's the trouble tom I don't see what's bothering us but why do I do this miss Perkins to get the radical the square root of 3 out of the denominator why do I want to do that well it's considered good mathematical form not to leave a radical in the denominator besides it's a lot easier to compute a decimal approximation for your final answer there than to do all that multiplying and dividing you have to do in the original fraction yes I see that but how do I know it'll always work out to a simple thing like 2 minus the square root of 3 wouldn't I sometimes still get a radical in the denominator no you'll always get an expression like that one why the answer to Tom's question was really a simple one if Miss Perkins had only known about number fields she could have answered him easily and that's what I'm here to tell you about sets closure and number fields because these are concepts you will need to know in order to teach algebra as mathematical structure most of you have probably heard these terms and possibly think of them as something very complicated in the realm of higher mathematics but that's not true the concept of set for example is no different from what we mean when we say a set of books it's merely that we recognize likenesses in a collection of objects we don't usually speak of a set of lemons but we could if we wanted to and so let's suppose we're talking about the set of all lemons if we should add an apple to our collection obviously we wouldn't have a set of lemons anymore now let's apply this concept of set to mathematics let's take the set of all integers both positive and negative and zero now suppose we add any two elements of the set for example the integers 7 & 4 the result is 11 another integer in fact we know from experience that the result of adding any two elements of this set is always another element of the set in a situation like this we say that the set is closed under addition that is if the result of an operation is always an element of the set we call the set closed under that operation now suppose we see what happens in this set when our operation is division if we divide 7 by 4 we get 1 and 3/4 our result now is not an integer like the elements it's a fraction thus the set of integers is not closed under the operation of division what about subtraction since we know from experience that the result will always be an integer one of the elements we know that the set is closed under subtraction so far as multiplication is concerned I am sure you need no discussion to convince you that the set of integers is closed under this operation - to recapitulate then when the result of performing operations on any two elements of a set is always an element of the set the set is said to be closed under that operation we need this concept of closure to describe the notion of field to which we now turn we have a set of elements called E and we'll designate them by small letters a B C and so on we have two operations that we'll call addition and multiplication now what are the requirements for a field first it must be possible to add any two elements of the set and get a unique result that is also an element of the set we'll call this property closure for addition second it must be possible to multiply any two elements and get a unique product that is itself an element of the set we call this property closure for multiplication the next five requirements are five familiar laws of algebra in fact they are so familiar that they sometimes pass unnoticed but here it is important to focus attention upon them we require that a plus B shall be the same as B plus a this is called the commutative law or addition we also require that a times B be the same as B times a this is called the commutative law for multiplication next the grouping in addition must be immaterial this is the associative law for addition similarly the associative law for multiplication must hold relating the two operations of a addition and multiplication we require that a times B plus C shall equal a b plus AC this is the distributive law now we come to a property very possibly has never been explicitly pointed out to you before it is a fact that you know but have just taken for granted in a field there must be a single element that can be added to any other element of the set and leave it unaltered we'll call this zero and we label the eighth property of a field the existence of zero there are number systems not fields of course that do not possess a zero element next there must be a single element you that when used to multiply any other element leaves that element unaltered we call this one and this property is known as the existence of unity hard as it may be to imagine there are number systems that do not possess a unit element our last two properties have to do with the inverse operations we require that for each element a there be another element that added to a produces zero we call this property the existence of the additive inverse likewise for each element a except zero there must be another element that multiplied by a produces one we call this the existence of the multiplicative inverse or reciprocal our precise definition of a field may now be stated a field is a set of elements with two operations and it must satisfy these eleven requirements it is easy to remember them because items 3 through 9 are old familiar friends on which all algebraic manipulations are based beyond these we merely require closure and the existence of both inverses let's look now at the sets of numbers used in secondary school algebra and find out which of them are fields we know that the set of integers is closed with respect to addition and multiplication that 0 and 1 exist in this set and that the five familiar laws hope moreover every number has an additive inverse but we do not always have a multiplicative inverse therefore since one of the properties of a field cannot be satisfied the set of integers is not a field but the set of rational numbers is a field although we cannot take time now to check through the 11 requirements in detail so also is the set of real numbers and the set of complex numbers in passing it is worth noting that each of these fields in the order listed is an enlargement of the sets that preceded the enlargement ends with complex numbers and these are sufficient for the purposes of algebra any algebraic equation with complex coefficients has complex roots we cannot make an analogous statement with respect to any of the other fields we have talked about so far some algebraic equations with rational coefficients do not have rational roots some algebraic equations with real coefficients do not have real roots but every algebraic equation with complex coefficients as complex roots although complex numbers constitute the most extensive field we can develop by successive enlargements of our number system there are many other fields we might take all numbers like 3 plus 2 times the square root of 3 that is all numbers of the form a plus B times the square root of 3 where both a and B are rational this set 2 is a field as you could easily show by applying the 11 requirements this film has been designed to raise your sights as teachers to show you what is meant by algebraic structure rather than to give you specific classroom techniques or applications but this background cannot help but be of value in the classroom as a specific example let's go back to Tom and his perplexity about rationalizing the denominator I don't see what's bothering you why do I do this miss Perkins to get the radical the square root of 3 out of the denominator why do I want to do that many answers could be given to Tom's question the answer Miss Perkins gave that it is not good form to leave a radical in the denominator though given by many textbooks and many teachers is a weak answer why is one form of expression better than another clearly for computing an approximate answer the form 2 minus the square root of 3 is more convenient than this miss Perkins was quite right about that but frequently it is neither necessary nor desirable to compute a rational approximation for an irrational number another and better explanation would be this the original problem involves numbers of this kind a plus B times the square root of three the answer should logically be of the same sort in that sense two minus the square root of three is better than the complicated fraction we shall see later that there is real merit in this reply another answer is this if such a manipulation is performed the radical always disappears this is a manipulative answer it shows that the manipulation tom has been performing will in fact do but the answer does not really go to the heart of the matter why does he never get a radical in the denominator why does he want to change the complicated fraction into 2 minus the square root of 3 what is the structural answer it is that the set of numbers a plus B times the square root of 3 is a field and it is closed under the four fundamental operations and specifically under division division their fourth can always be performed except for division by zero and the result will be a number of the form a plus B times the square root of 3 this tells us both why Tom wants to change the form of his fraction his computation is not complete because he hasn't actually found the quotient until he does this and also tells us that he always can do it this answer you see does not contradict any of the partial answers we gave at the beginning it includes them all in a universal statement it rests on the bedrock of the pattern of algebraic structure