Imaginary Numbers - History and CommentaryDate: 09/04/97 at 23:19:43 From: Howard Engel Subject: "i" I have just discovered Dr. Math, as the result of a mention of the page in the Los Angeles Times this week. I think it useful to youngsters through grade 12. I have some comments to add to your presentation on imaginary numbers. The ancient Greeks once believed that all numbers were rational numbers; that is, that every number could be expressed as the ratio of two integers, and they were very disturbed when it was demonstrated that the measure of the hypotenuse of an isosceles right triangle, having arms of unit measure, was not a rational number. I omit the simple proof here. The new numbers, of which I have given only one example, are now called irrational numbers to distinguish them from rational numbers. (Whether irrational numbers, or negative numbers, or the transcendental numbers yet to come were invented or discovered is a philosophical question I choose to avoid.) The point I wish to make is that irrational numbers were a kind of number new to the experience of mathematicians. Prior to the proof of existence of irrational numbers, it was not necessary to distinguish between rational and irrational numbers; all numbers were expected to be rational. Mathematicians for a long time were unwilling to accept as solutions to equations numbers that were less than zero. Eventually numbers of this sort were accepted as solutions. Today we call them negative numbers, another kind of number once new to mathematicians, and requiring a revision of beliefs. Prior to the acceptance of negative numbers, it was not necessary to refer to positive and negative numbers; only positive numbers were believed to exist. For centuries there were quadratic equations that were deemed not to have solutions. Equations like x^2 = -1 and x^2 -2x + 2 = 0 have no solutions among the positive and negative numbers. The problem in seeking solutions to equations like these two is that the squares of positive and negative numbers are both positive. Solutions for equations like these can be found, however, if we decide to invent a completely new number whose square is -1; of course, it is not a number that we have seen before. We name this number "i". The square of -i is also -1. By multiplying i by positive and negative numbers (in other words, all the non-zero numbers we had before we added i) we can obtain a whole set of new numbers that have the property that their squares are negative numbers. These new numbers, for better or worse, were called "imaginary" numbers, and the old positive and negative numbers (and zero) were called "real" numbers. Still further, letting a and b be positive or negative real numbers, we can construct infinitely many numbers of the form a+ib. We then find that we can write the solutions to the equation x^2 = -1 as x = i or x = -i, and the solutions to the equation x^2 - 2x + 2 = 0 as x = 1+i or x = 1-i. Unfortunately, because the word "imaginary" is associated with the make-believe, there has been a lot of confusion over the concept of this new number i. The term "imaginary", when used to refer to multiples of i, is a technical term and because of its pervasive use amongst scientists and mathematicians, it helps to learn the term for the sake of communication. Furthermore, numbers of the form a+ib, in which a and b are real numbers, were then called "complex" numbers. If only mathematicians had waited a while before assigning these names! Hamilton, a few years later, found another way to express complex numbers where he never had to introduce the word "imaginary". Hamilton's solution was to expand the definition of number, just as other mathematicians in the past had expanded the definition of number in the following way: Hamilton decided that our ordinary "real" numbers are a subset of a larger set of numbers that are referred to as "ordered number pairs", and written (a,b), in which a and b are positive or negative numbers, including zero (in other words, in which a and b are real numbers). The rules of arithmetic must be altered for ordered number pairs. Letting letters represent real numbers, we have: (a,b) + (c,d) = (a+c,b+d) (a,b) - (c,d) = (a-c,b-d) (a,b) * (c,d) = (ac-bd,ad+bc) (a,b) / (c,d) = ((ac+bd)/(c^2+d^2),(bc-ad)/(c^2+d^2)) These rules are considerably more complicated than those learned in elementary school for the elementary operations of arithmetic, but ordered number pairs continue to obey the laws of associativity, distributivity, and commutativity. The ordered number pair (a,b) is equivalent to the complex number a+ ib. That is, if b is zero, then (a,0) and a+i0 behave algebraically as the same real numbers. If a is zero, then (0,b) and 0+ib behave algebraically as the same "imaginary" numbers. Finally, if neither a nor b is zero, (a,b) and a+ib behave algebraically as the same complex numbers. By my argument and exposition, I do not mean to imply that ordered number pairs should be used to the exclusion of representations of the form a+ib. Once ordered number pairs and their algebra have been introduced, and used to express the roots of equations such as x^2 = -1 and x^2 - 2x + 2 = 0, the equivalent representation a+ib for (a,b) may be introduced, together with the simpler rules for manipulation, and it may be mentioned in passing that i may be treated as if it were a square root of the ordinary number -1 -- but do not dwell on the term "imaginary number". - Howard Engel Date: 09/05/97 at 12:03:00 From: Doctor Ceeks Subject: Re: "i" Hi, Thank you very much for your thoughtful comments. I think you have some very good points. One of the problems with the concept of "i" as a number is that most people associate the word "number" with the concept of a measure of the magnitude of some set... such as the number of people in a stadium. Since one cannot say there are "2+i" slices of bread in a loaf, people have a bad reaction to calling "i" some sort of number. Mathematicians view the complex numbers as a construction which, as you point out, allows for the complete factorization of any polynomial with real (or complex) coefficients. It's wonderful that it is possible to construct a system of numbers which contain a number whose square is -1, and deduce that such a system exists with many favorable properties! For the sake of communication, mathematicians gave a name to some of the new objects relevant to the construction, and, unfortunately, the term "imaginary number" was introduced. The reason this is unfortunate is because people have a natural tendency to want to reconcile the name with the old meanings of the words that make up the name. Since most people learn the words "imaginary" and "number" in a completely different context from that used by mathematicians, there is trouble. But then this suggests that pedagogically, it helps if we can convince the student to accept the idea that there are new concepts and that it is misplaced to try to force the new concept into the mold of the old concept. -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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