Imaginary Numbers - History and Commentary

Date: 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/