Numbers, Complex

The set of complex numbers includes all the numbers we commonly work with in school mathematics (whole numbers, fractions, decimals, square roots, etc.), plus many more numbers that are generally not encountered until the study of higher mathematics. Complex numbers were invented centuries ago in order to provide solutions to certain equations that previously had seemed impossible to solve.

Imagine trying to find a solution to x+ 6 = 4 but being able to look for a solution only in the set of whole numbers. This is impossible. However, if we expand our domain to all integers, − 2 provides a solution. Similarly, it is impossible to find a solution to 2x = 7 using only integers, but we can expand our domain to the set of rational numbers, and or 3.5 provides a solution. Now suppose you wanted to find a solution to x² = 2 using only rational numbers. This, too, is impossible. However, the set of rational numbers can be expanded to create still another new set of numbers—the real numbers. Clearly, is one solution to the equation x² = 2 because by definition the square root of any number multiplied by itself equals the number: . Another solution is because it is also true that the negative square root of a number multiplied by itself equals the number: .

In about 50 C.E. another seemingly impossible problem emerged when Heron of Alexandria, a Greek mathematician, was trying to evaluate the square root of a negative number. Consequently, the square root of a negative number cannot be evaluated using only real numbers. To find a solution, another new number system needed to be invented. In the mid-sixteenth century an Italian mathematician named Girolamo Cardano began to do just that. He is recognized as the discoverer of the imaginary numbers that play an essential role in understanding the complex number system.

Imaginary Numbers

Contrary to their name, imaginary numbers are not imaginary at all. Imaginary numbers were invented in an effort to evaluate negative square roots, a mathematical operation that before their existence was impossible. Thus, the argument to justify the existence of imaginary numbers is similar to the argument for the existence of integers, rational numbers, and real numbers.

Leonhard Euler, an eighteenth-century Swiss mathematician known for his prolific writing in mathematics and his standardization of modern mathematics notation, chose the symbol i to stand for the square root of −1. Since that time, the imaginary number set has included i and any real number times i. So, for example, and (which we write as .

It is interesting and important to observe the behavior of i when it is multiplied by itself. To begin, i multiplied by itself is i² (also written as × , which is − 1). So i² = − 1. To continue this line of reasoning, i³ = − i (because i³ can be written as i² × i and thus − 1 × i or −i). Similarly, i⁴ = 1 (because it can be thought of as i² × i ² or − 1 ×− 1). Continuing, we find that i⁵ = i, and this is where the pattern begins again (i, − 1, − i, 1).

Incredibly, by using imaginary numbers it is possible to solve many equations that were deemed impossible for centuries. Consider x² + 4 = 0. Using algebraic manipulation (subtracting 4 from both sides of the equals sign), the equation becomes x² = − 4, and x can be either or . So x = 2i or − 2i. To check your answer, substitute the 2i or − 2i for x. So, for example, the equation becomes (2i)²+ 4 = 0, which is (2² × i²) + 4 = 0. Because i²= −1 and 2² = 4 then (4 × −1) + 4 = 0, or − 4 + 4 = 0.

Complex Number System

Carl Friedrich Gauss, a nineteenth-century German mathematician, is credited with inventing and naming the complex number system. Complex numbers are generally expressed in the form a + bi, where a and b are real numbers and i is the imaginary number described above (that is, ). The a part is considered the real part of the complex number and the bi part is the imaginary part of the complex number. Upon further inspection, we can see that the set of complex numbers includes all the pure real numbers, together with all the pure imaginary numbers, together with many more numbers that are sums of these. In other words, whenever a complex number has b = 0, it is actually a pure real number too because it is equal to a + 0i, which is just a (a real number). Whenever a complex number has a = 0, it is actually a pure imaginary number because it is equal to 0 + bi, which is just bi, an imaginary number.

The complex number system consists of all complex numbers, a + bi (where a and b are real numbers), together with the rules that define the four basic operations on this set of numbers (addition, subtraction, multiplication, and division). Indeed, in order to define any number system, there are certain rules that must be obeyed. First, addition and multiplication must be well defined (that is, it must be clear how to add and how to multiply any two numbers in the set). Addition of complex numbers occurs by adding their real parts and their imaginary parts separately. For example, (4 + 2i) + (6 + 3i) = (4 + 6) + (2i+ 3i) = 10 + 5i. In general, (a + bi) + (c + di) is equal to (a + c) + (b + d)i. Subtraction is performed similarly.

To multiply any two complex numbers, use the distributive property (in a method sometimes referred to in elementary algebra classes as the "foil" method) and then combine real terms and imaginary terms separately. Thus, (a + bi) × (c + di) = ac + (ad)i + (bc)i + (bi)(di) = (ac + bd) + (ad + bc)i. For example, (4 + 5i)(6 + 3i) = 24 + 12i+ 30i+ 15i² = 24 + 42i - 15 = 9 + 42i. Remember from previous discussions that i²= - 1. Multiplying a complex number by a constant is a simpler case of this process in which the distributive property is similarly invoked. For example, 4 (4 + 2i) = 16 + 8i. In general, c(a + bi) = ac + (bc)i.

Every number system also must have both an additive identity and a multiplicative identity—that is, numbers that when added to (or multiplied by) any number in the set produce the same number started with. For the complex numbers—just as for the whole numbers, the integers, the rational numbers, and the real numbers—the additive identity is 0 and the multiplicative identity is 1.

For any number system, there must also be both additive and multiplicative inverses—that is, numbers that when added to (or multiplied by) a number in the set produce the additive (or multiplicative) identity, respectively. For the real numbers, the opposite of any number is its additive inverse. That is, −a is the additive inverse of a (since, for example, −4 + 4 = 0, and 0 is the additive identity). For the complex numbers, we must take the opposite of both the real and imaginary parts of a number to find its inverse. Thus, (−a −bi) is the additive inverse of (a + bi) (because adding these together gives 0 + 0i, or simply 0, the additive identity).

Finally, as for the real numbers, multiplication and addition of complex numbers must be commutative, associative, and distributive. This is indeed the case because the operations in complex numbers are based on the operations in real numbers.

Dividing any complex number a + bi by a real number (say, r) is done by dividing each part of the number by r. Thus, (a + bi) ÷ r = a/r + (b/r)i. To divide a complex number by a complex number is somewhat more complicated. However, the process is similar, in some ways, to a process you probably learned for dividing by a decimal.

Consider the example, 135.5 ÷ 0.25. To perform this division, you were probably taught to multiply both dividend and divisor by 100 before proceeding with the division. Thus, the problem is transformed to 13550 ÷ 25. The purpose of performing this transformation is to create a new (easier) problem that will have the same answer as the original problem.

The process of dividing by a complex number proceeds in a similar fashion. To divide by a complex number, we choose first to multiply both dividend and divisor by something to make the divisor non-complex. In the case of division by a complex number, we choose to multiply by a number known as the "complex conjugate" of the divisor. The complex conjugate of any complex number (a + bi) is simply the complex number (a - bi). Multiplying these two numbers together produces a product that is not complex (that is, the product has imaginary part equal to 0).

For example, the complex conjugate of 2 + 3i is 2 − 3i. When you multiply these two numbers you obtain 13 because (2 + 3i)(2 − 3i) = 4 + 6i − 6i − 9i²= 4 − 9i² = 4 − 9 (−1) = 4 + 9 = 13. The product (13) has imaginary part equal to 0 because the middle (imaginary) terms cancel out.

Thus, if you want to perform (3 + 4i) ÷ (2 + 3i), you must first multiply both of these numbers by (2 − 3i) to produce an equivalent division problem: (3 + 4i)(2 − 3i) ÷ (2 + 3i)(2 − 3i). This works out to (18 − i) ÷ 13. Now the division is easily performed by dividing both real and imaginary parts by 13, producing as the answer.

Complex conjugates also arise when finding the roots of polynomials. When a polynomial is factored, the total number of roots is always equal to the degree of the polynomial, as proven by the Fundamental Theorem of Algebra. For example, x² + 5x + 6 = 0 is an equation of degree 2 (since the highest power of x is 2). By factoring this equation into (x + 3)(x + 2)= 0, we can see that it has two solutions, −3 and −2. Thus, the polynomial x² + 5x + 6 has exactly two roots.

It is quite possible, however, that some of the roots of a polynomial will not be real because (as seen above) some equations can be solved only by appealing to the set of complex numbers. For instance, a polynomial of degree 4 might have two real roots and two complex roots.

Consider, for example, the equation x⁴ − 16 = 0. This can be factored into (x²− 4)(x² + 4) = 0. Thus, we see that either x² − 4 = 0 or x² + 4 = 0. The former is true only when x = 2 or − 2. The latter is true only when x = 2i or −2i. Thus, x⁴ 16 is a polynomial of degree 4 with two real roots and two imaginary (or complex) roots. In fact, complex solutions always come in pairs of complex conjugates. That is, whenever a + bi is a root of a polynomial, then a − bi will also be a root.

Geometric Representations of Complex Numbers

In the early nineteenth century, Jean Robert Argand, an amateur French mathematician; Caspar Wessell, a Norwegian cartographer; and Carl Friedrich Gauss, a German mathematician, all worked on developing geometric representations of complex numbers in a plane. Because complex numbers have both a real part and an imaginary part, a two-dimensional plane (rather than a one-dimensional line) is needed to plot them.

The real part and the imaginary part of a complex number can be written as a coordinate pair. For example, 3 + 2i can be written as (3,2), where the first coordinate represents the real part of the complex number and the second coordinate represents the imaginary part. Then, we can use this ordered pair to produce a geometric representation of the complex number. In fact, mathematicians use two different, but related geometric representations of complex numbers.

First, the number a + bi or (a, b) can be thought of as the point P(a, b) in the complex plane (or Argand diagram) by starting at the origin and plotting a point over a units and up b units. Or, second, (a, b) can be thought of as a vector from (0,0), the origin, to P in the complex plane.

Notice that construction of the complex plane is similar to that of the Cartesian plane. However, for the purpose of plotting complex numbers, the horizontal axis is considered to be the real axis and the vertical axis to be the imaginary axis. Thus, ordered pairs for complex numbers are plotted in the same manner as plotting ordered pairs of real numbers on a Cartesian plane.

Many relationships can be defined using a geometric model of complex numbers. For example, consider the notion of absolute value. Recall that for real numbers, the absolute value of any number a is just the distance of a from 0 (or, geometrically, the length of the segment from 0 to the point a on the number line). For example, the absolute value of −3 is 3, since −3 is 3 units from 0 on the number line. For the same reason, the absolute value of ± 13 is 13.

In the complex number system, the absolute value of a number a + bi or (a,b) is defined in an analogous way as the distance from the origin (0,0) to the point P(a,b) or length of the vector that (a,b) represents. Thus, the absolute value of a + bi is determined by computing the length of the vector from the origin to P. This distance can be found by dropping a perpendicular line segment from P(a,b) to the x-axis.

Doing so forms a right triangle whose hypotenuse is the length of the vector and whose horizontal leg has length a and whose vertical leg has length b. The length of the vector, c, can be computed using the Pythagorean Theorem (c²= a²+ b²). In other words, the absolute value of a + bi is equal to the square root of (a²+ b²). For example, the absolute value of the complex number 3 + 4i is the square root of 3²+ 4², which is , or 5.

Uses of Complex Numbers

Complex numbers are used both in the study of pure mathematics and in a variety of technical, real-world applications. When we study the real number system, many of its properties are easier to illustrate by looking at the real numbers within the more inclusive set of complex numbers. In other words, complex numbers are important in the study of number theory. Another example of the utility of complex numbers in mathematical study is that some functions, which generate fractals, contain complex numbers.

Complex numbers also have practical applications in technical fields. For instance, complex numbers are used in the study of electromagnetic fields. An electromagnetic field has both an electric and magnetic component, so two measures are required: one for the intensity of the electric field and one for the intensity of the magnetic field. Complex numbers help to describe the field's strength.

Electrical engineers use complex numbers to measure electrical current and to explain how electric circuits behave. Mechanical engineers use complex numbers to analyze the stresses of beams in buildings and bridges. Complex numbers appear when the engineers look for the eigenvalues and eigenvectors of the matrix that the engineers configure to explain numerically the stresses of the beams.

Numbers, Complex

Imaginary Numbers

Complex Number System

Geometric Representations of Complex Numbers

Uses of Complex Numbers

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