Can astronomers deal with imaginary stars or imaginary galaxies? Can physicists work with imaginary falling apples? Can a surveyor measure an imaginary field or an imaginary vacant lot? If the answer to this questions is 'no', then how come that imaginary numbers are so important in mathematics, to the point that mathematicians say that some problems cannot be solved without the aid of imaginary numbers?
To see how the imaginary numbers came about, lets see first why the negative numbers are needed. Take a look a the following equation:
Obviously, we can can solve this equation with the aid of the positive numbers only. In fact, the solution is . Now take a look at this one:
Now we have a situation where the set of the natural numbers is not enough; recall that the naturals are the numbers 1, 2, 3 ... But math is a human creation, so we can extend the set of the natural numbers as to include possible solutions to this case.
Let us denote the solution by any arbitrary symbol like the question mark (?). Then we have that the solution to the equation is simply
But at one moment or another we are going to have more instances of the same equation, like , and son on. Then, to maintain consistency in symbolism we have to say that , and so on.
However we can can make some advance if we recycle the familiar symbols of the natural numbers and say that: . At this point we realize that the question mark symbol that we initially adopted should be replaced by another, lets say the symbol: –. Then the solution to the top equation becomes , the solution to the equation becomes , and son on. Our base of integers has now been expanded, and now we can solve more equations than when we began.
Extending a number system is more complicated than the steps we took above, but, as an introduction, the reader can see the first steps to accomplish it. As far as any type of the above equations is concerned, we can solve any of them with the set made of the natural numbers, the zero and the recently invented negative numbers. The term "negative" is a matter of taste, it could have been called the "opposite" numbers, or the "anti-natural" numbers.
Now, with the set of all natural numbers, plus the zero, plus the set of all negative numbers, we are equipped to solve many more equations we not every equation.
We saw how extending the set of all natural numbers was of some help when solving some simple equations. However, this is far from being sufficient, thus, throughout the history of mathematics, more extensions like this one had been made to the number system, each time adding a new kind of number that proves adequate for unexpected needs. Its worth mentioning the extension toward the rational numbers that covers the fractions; the irrational numbers, useful for solving weird roots of numbers and numbers that cannot be expressed as the ratio of two integers, like ; and the transcendental numbers, like .
Why some numbers are called "rational" and others are called "irrational"? To put it simple, its a matter of the operation of division of numbers. The numbers that can be expressed as the quotient of other two are "rational". For example, 2.5 and 7 are rational numbers because they can be expressed as the division 5/2 and 7/1 respectively. Thus every fraction and every integer, no matter if they are positive or negative is rational.
Irrational numbers are those that cannot be expressed as the quotient of other two. The classical example is; there is no way of finding two numbers p and q such that p/q =. There are various demonstrations, some of them since antiquity, that prove thatcannot be expressed as the quotient of two numbers.
At most, the square root of 2 can be approximated by some rationals; one of the easiest to remember is 99/70.
The bar above the digits is to denote the digit string that is continually repeated in this recurring decimal.
The number (Pi), in its first 12 decimal places is 3.141592653589, and in the approximation of 22/7 is 3.142857142857. The fraction 22/7 repeats its first 6 decimal digits; as any other fraction will eventually repeat its digits, but the pure value of is never repeated. Another type of irrational numbers are the so-called transcendental numbers. The base of the natural logarithm, denoted by the symbol is an example of a transcendental irrational number.
The rational and the irrational numbers, together with their operations, is what constitutes the real number system. The real numbers are the "points" we assign in the number line, also known as the X-axis in the Cartesian co–ordinate number system.
No matter the extensions and additions made to the natural number system, eventually extending it to the refined real number system, there are still many equations that cannot be solved with the reals. A simple example will show this:
No way to solve this simple equation with the real numbers. What number x, when squared is equal to negative one, so that this equation can be solved?
The solution to this problem is not so simple. Do we have to repeat the steps we followed when extended the natural numbers to the negative integers? Say that and problem solved? This solution is not so intuitive as when we extended the natural number into the negative integers. The problem arises because is neither a positive nor a negative real: 12 = 1 and (-1)2 = 1. The second objection is that if it is not any kind of real, then there is no place for it in the number line. Then, where should we place ?
Several attempts were made before 1799 to understand negative roots, but it was the Caspar Wessel's paper of that year that the geometrical interpretation of began to clear the way for the acceptance of such a weird concept. In Nahin's2words:
More than a hundred years after Walli's valiant but flawed attempt to tame complex numbers geometrically, the problem was suddenly and quite undramatically solved by the Norwegian Caspar Wessel (1745–1818). This is quite both remarkable and, ironically, understandable, when you consider that Wessel was not a professional mathematician but a surveyor. Wessel's break through on a problem that had stumped a lot of brilliant minds was, in fact, motivated by the practical problems he faced every day in making maps.
Wessel's contribution was essentially using the y-axis as the axis of the imaginaries. Striped away from all the previous mysticism and esoteric attributes, the imaginary numbers can be considered as if they were equally real as the "real" numbers. But when the imaginary numbers are really useful is when they are handled together with the reals. This combination defines the complex plane. They are called complex because one part is real and the other is imaginary. Thus, a complex number is a pair of numbers.
The complex numbers can be added similarly as when we add real numbers. If and are complex numbers, their addition is simply . But the multiplication of A times B can look somewhat bizarre: . With this definition of complex number the set of all real numbers turn out to be a mere subset of the set of all complex numbers.
Seen closely we may notice that what most defines a number system are the operations defined on it. If the multiplication of complex numbers were defined differently then we could not say that a real number is merely a complex number with zero imaginary part, or that an imaginary number is simply a complex number with no real part.
At the end of this short expedition we have seen that it is the traditional coined terms "imaginary" and "complex" what complicates our understanding of this simple and beautiful field of math.
Everything is real and simple, even the imaginary and the complex.
Maybe Descartes never imagined how far was going to evolve his system and how much beauty is in his coordinate system. What if we add coloring to the math plots? What if we add real-life picture transformations to the mathematical equations? What are one-to-one (1-1) transformations? How can a 2-dimensional figure be transformed into a 3-dimensional surface? This book has some elementary answers to those profound questions.