While the quadratic formula, , has been known to give solutions to the quadratic equation, ax2 + bx + c = 0 , since the time of the ancient Babylonian civilization (around 2000 BC), the simple looking equation, x2 + 1 = 0, was an enigma until relatively recently. That is because our number concept has historically been limited to those numbers which can be graphed on the real number line. In this section, we will see how the real number system is only a part of a larger number system, call the "complex" numbers. Moreover, we will see how the nice geometric interpretations of addition, multiplication, and negation of real numbers generalize to the complex numbers. We will also learn about a new operation, which applies to complex numbers, called conjugation, and discuss its geometric significance.
For thousands of years, mathematicians considered the equation x2 + 1 = 0 to be insolvable. From a functional point of view, we know that the range of the square function, f(x) = x2, contains only positive numbers, so that x2 = -1 (and x2 + 1 = 0) has no solution in the real number system. In other words, if the solution, (usually denoted by the letter i), is a number, it is not located anywhere on the real number line. For this reason, whenever this expression arose in a calculation, mathematicians simply assumed that the original problem had no solution. However, eventually they realized that they could not dismiss so easily.
In 1545, while examining the quadratic equation x2 - 10x + 40 = 0, Girolamo Cardano noticed that, if he plugged either of the "solutions" given by the quadratic formula:
into the equation, and used the fact that i2 = -1, they did prove to "work" as valid solutions. For example:
Note: We have translated the calculation into modern notation for readability. Unfortunately, he dismissed such calculations as "mental tortures", and considered such a result to be "so subtle that it is useless". Otherwise, he could have been credited with the discovery of the complex number system.
We should not be too hard on Cardano, though. At this point, the art of algebra was still in its infancy in Europe. The "+" sign, and other algebraic symbolism had not yet been invented. In fact, even negative numbers were not yet in common use. Because he had no concrete picture for such numbers, beyond symbols on a page, he could still dismiss them by saying that the equation had no "true" solutions.
A short time later, in 1572, another mathematician, Rafael Bombelli helped to shape the nature of algebra for the next 400 years. He codified the rules of arithmetic for negative numbers and began to make common use of more modern algebraic notation. More importantly, he realized that there was no more reason to discriminate against i, since one could do algebra with it, just as with positive (and negative) "real" numbers.
Surprisingly, this discovery was based on Cardano's formula which gives a solution to the cubic equation x3 + ax + b = 0. Although it is even more complicated than the quadratic formula, since it contains both square- and cube-roots:
,
it still works quite well. For example, in the case of the equation x3 + 9x + 26 = 0, it gives the solution:
Check: (-2)3 + 9(-2) + 26 = -8 - 16 + 26 = 0.
Bombelli's process of discovery began as he noticed that when he tried to use this cubic formula to solve a very similar looking example, such as x3 - 6x - 4 = 0, the solution contained :
He too might have dismissed this result as nonsense, until he realized that (again using the fact that i2 = -1) he could carry out the following basic, algebraic calculations:
(-1 + i)3 = [(-1 + i)(-1 + i)](-1 + i) = (1 - 2i + i2)(-1 + i) = (1 - 2i + -1)(-1 + i)
= -2i(-1 + i) = 2i - 2i2 = 2i - 2(-1) = 2 + 2i,
and:
(1 + i)3 = [(1 + i)(1 + i)](1 + i) = (1 + 2i + i2)(1 + i) = (1 + 2i + -1)(1 + i)
= 2i(1 + i) = 2i + 2i2 = 2i + 2(-1) = -2 + 2i,
which show that:
and !
This meant that this "nonsensical" formula actually simplifies to x = (-1 + i) - (1 + i) = -2, which is the correct answer! Check: (-2)3 - 6(-2) - 4 = -8 + 12 - 4 = 0. In other words, doing ordinary algebra with i led to "real" solutions. Therefore, i should be accepted as a valid number.
Even after mathematicians accepted numbers, such as , as solutions to equations, they still did not feel comfortable with them. For example, in 1637 Rene Descartes wen so far as to refer to such solutions as "imaginary". Note: Such a reference appears in La Geometrie, which is the same work in which he presented his discovery of the Cartesian coordinate system (which we use to plot points) and Analytic Geometry. This prejudicial term continued to be applied to for several hundred years, which is why Euler began to denote this number as "i". It was not until Jean Robert Argand gave a concrete, geometric picture for this number, which related it to the "real" number line, that mathematicians began to accept i as more than a figment of their imagination, and started to refer to it as a "complex" number, after Gauss popularized the term. In the next section, we will show how to picture i, as well as discuss the geometric significance of arithmetic operations with complex numbers.
If i is not on the real number line, then it is not unreasonable to suggest to picture it like this:
That is, i is located off the number line, say, one unit directly above the origin. This would then suggest that other multiplies of i would be located similarly:
Recognizing the similarity with the Cartesian coordinate system, it makes sense to associate more general complex numbers, such as 2 + 3i, with points in the plane, as in the following picture:
That is, we plot the number 2 + 3i as if it were the point (2, 3). For a complex number such as 2 + 3i, we refer to 2 as its "real part", while 3 is its "complex part". In general:
For any complex number, c = x + y i, where x and y are real numbers, we refer to x as the "real part of c" and y as the "complex part of c". We plot a complex number by associating the real part with the horizontal axis and the complex part with the vertical axis.
If Cardano had come up with this simple idea, he could have been remembered as the discoverer of complex numbers. If Descartes has thought of this use of his coordinate plane, he may not have been so quick to dismiss complex numbers as "imaginary". This picture was finally conceived, in 1806 by Jean Robert Argand, and is often called an "Argand diagram". Note: In fact, another mathematician, Caspar Wessel, actually came up with the idea first in 1787; it is interesting to follow the chain of events and personalities that led to Argand's name being finally and irreversibly attached to this way of plotting complex numbers.
To summarize:
if we associate points in the plane with pairs of real numbers, (x, y), we call it the Cartesian plane
if we look at the plane as corresponding to complex numbers of the form x + y i, then it is called the Argand plane, or simply, the complex plane.
We do not need to discuss in great detail how to perform arithmetic operations with complex numbers, since:
the arithmetic of complex numbers simplify follows the ordinary rules of algebra (i.e., associative, commutative, distributive laws, etc.), simply remembering that i2 = -1.
For example, (2 + 3i) + (1 - 2i) = 2 + 3i + 1 - 2i = 2 + 1 + 3i - 2i = (2 + 1) + (3 - 2)i = 3 - i. Since we have seen how addition, multiplication, and negation correspond to geometric operations on the real number line, it will be interesting to examine how these same operations look in the complex plane.
For example, negation would take 2 + i to -(2 + i) = -2 - i. Plotting both points:
we see that we can describe this in two ways. By analogy with negation on the number line, we can view this as a reflection "through" the origin. For some, it may seem more natural to describe this as a rotation by 180º around the origin.
Algebraically, multiplication by a positive real number, such as 2, simply looks like the distributive law: 2(1 + i) = 2 + 2i. Geometrically, it is looks similar to multiplication on the real line:
in that it can be described as a stretch by a factor of 2 around the origin. In general, multiplication by a positive real number causes a scaling (i.e., stretch or shrink) of the plane around the origin.
Multiplication by i, however, is much more interesting. For example, i(2 + 3i) = 2i + 3i2 = 2i + -3 = -3 + 2i, which plots as:
Notice that the two dotted lines are the same length and form a 90º angle. That is:
Geometrically, multiplication by i causes a rotation by 90º counter-clockwise around the origin.
We can see this property of multiplication in the way in which we originally plotted i = i·1 as 90º counter-clockwise around the origin from 1. Note: It is no accident that multiplication by -1 is rotation by 180º, while multiplication by is only a a rotation by 90º; Argand thought of i in this way (that is, as a rotation by 90º), which led him to conceive of his method of plotting complex numbers.
Multiplication by an arbitrary complex number is a combination of multiplication by positive real numbers, -1 and/or i (i.e., scaling, rotation by 180º and/or 90º) together with addition. Once we discuss the geometry of addition, we will be able to understand complex multiplication more completely.
The geometric rule for addition is implicit in the very way we plot complex numbers, since a general complex number is naturally written as a sum. Notice how the plot of the number 2 + 3i is naturally related to the plots of each summand, 2 and 3i:
The numbers 2, 3i, and 2 + 3i make a rectangle with the other corner at the origin. More generally, if we plot any two complex numbers, such as 2 + 3i and 1 - 2i, together with their sum, (2 + 3i) + (1 - 2i) = 3 - i, and the origin, we obtain a parallelogram:
Note: This geometric "parallelogram rule" for addition is the same one used in Physics to add vectors, such as those used to measure force, acceleration, and velocity.
Complete the following Exercises
to learn more about the geometric properties of complex addition and multiplication.
Complex numbers have an additional operation, commonly referred to as conjugation, which takes a complex number like 2 + 3i and gives 2 - 3i; in other words:
Complex conjugation replaces i by -i. We denote this operation with a bar over the top of the expression, so that the complex conjugate of a + bi (where a and b are real numbers) would be written as .
Geometrically, this looks like a vertical reflection, taking a point like 1 + 2i to 1 - 2i:
While this is an interesting operation in its own right, it becomes even more important as we see its relationship to other algebraic and geometric operations. The key fact is that:
When we multiply a complex number, c = x + y i, by its conjugate, , we obtain a real number: .
This is useful algebraically when we want to simplify the quotient of two complex numbers. While we can easily add, subtract, and multiply complex numbers, using the ordinary rules of algebra and the fact that i2 = -1, division takes a bit more work. For example, to simplify, say, , we must first multiply the numerator and denominator by the conjugate of the denominator, so that the final result is recognizable as a complex number:
Geometrically, the expression gives the distance from c = x + y i to the origin. Since gives the distance on the number line from x to the origin, in the same way, it is common to denote as |c|.
Finally, there are two important functions, called Re and Im, respectively, which are defined in terms of complex conjugation. The domain of both functions is equal to the complex plane, while their range is the real number line. While Re and Im may be defined in terms of conjugation as:
they can easily be computed by the formulas:
and .
That is, Re(c) is the real part of c, while Im(c) is the imaginary part of c. In the next section, we will use the two functions Re and Im to define the basic trigonometric functions in terms of the exponential function base e.
Complete
the following Exercises to
learn even more useful properties of conjugation.
Go to Exponential and Trigonometric Functions
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