Complex Numbers as Ordered Pairs of Reals
In mathematics, we often encounter situations where we need to extend our number systems to solve problems that cannot be solved within the existing framework. The most familiar example is the need to introduce negative numbers to solve equations like x + 5 = 0, or rational numbers to solve 2x = 1. Similarly, the equation x2 + 1 = 0 has no real solutions. To address this, we introduce the concept of complex numbers.
A complex number can be formally defined as an ordered pair of real numbers. Let 'a' and 'b' be any two real numbers. Then, the ordered pair (a, b) represents a complex number. This notation emphasizes that the order of the numbers matters, just like in coordinate geometry where (2, 3) is different from (3, 2).
Defining Complex Numbers
The set of all complex numbers is denoted by the symbol 'C'. Each complex number is represented in the form (a, b), where 'a' is called the real part and 'b' is called the imaginary part.
We can write a complex number z as z = (a, b). Here, 'a' ∈ R (real numbers) and 'b' ∈ R (real numbers).
Equality of Complex Numbers
Two complex numbers, z1 = (a, b) and z2 = (c, d), are said to be equal if and only if their corresponding real parts are equal and their corresponding imaginary parts are equal.
So, z1 = z2 if and only if a = c and b = d.
For example, if (x, 4) = (3, y), then we must have x = 3 and y = 4.
Operations on Complex Numbers
Just like with real numbers, we can define addition, subtraction, multiplication, and division for complex numbers. These operations are defined in a way that is consistent with the algebraic properties of real numbers and leads to the introduction of the imaginary unit 'i'.
Addition of Complex Numbers
The sum of two complex numbers z1 = (a, b) and z2 = (c, d) is defined as: z1 + z2 = (a + c, b + d)
This means we add the real parts together and the imaginary parts together separately.
Example: Let z1 = (2, 3) and z2 = (5, -1). Then, z1 + z2 = (2 + 5, 3 + (-1)) = (7, 2).
The addition of complex numbers is commutative (z1 + z2 = z2 + z1) and associative (z1 + (z2 + z3) = (z1 + z2) + z3).
Subtraction of Complex Numbers
The difference between two complex numbers z1 = (a, b) and z2 = (c, d) is defined as: z1 - z2 = (a - c, b - d)
We subtract the real parts and the imaginary parts separately.
Example: Let z1 = (8, 5) and z2 = (3, 2). Then, z1 - z2 = (8 - 3, 5 - 2) = (5, 3).
Multiplication of Complex Numbers
The multiplication of two complex numbers z1 = (a, b) and z2 = (c, d) is defined as: z1 * z2 = (ac - bd, ad + bc)
This definition might seem a bit arbitrary at first glance. However, it is derived from the distributive property of multiplication over addition and the fundamental property of the imaginary unit 'i'. Let's see how this leads to the standard form.
Consider the complex number (0, 1). Let's multiply it by itself: (0, 1) * (0, 1) = (0*0 - 1*1, 0*1 + 1*0) = (-1, 0).
This result, (-1, 0), is crucial. It corresponds to the real number -1. This suggests that the complex number (0, 1) plays the role of the imaginary unit, which we denote as 'i'. So, i = (0, 1).
Now, let's express a general complex number (a, b) using this 'i'. (a, b) = (a, 0) + (0, b) (a, b) = (a, 0) + (b, 0) * (0, 1) (a, b) = (a, 0) + (b, 0) * i
If we treat (a, 0) as the real number 'a' and (b, 0) as the real number 'b', then the complex number (a, b) can be written in the standard form a + bi. This is where the familiar form of complex numbers comes from.
Let's verify the multiplication formula using the standard form: z1 = a + bi z2 = c + di z1 * z2 = (a + bi)(c + di) = ac + adi + bci + bdi2 Since i2 = -1 (from (-1, 0)), we have: = ac + adi + bci - bd = (ac - bd) + (ad + bc)i This matches the ordered pair definition (ac - bd, ad + bc).
Example: Let z1 = (2, 3) and z2 = (4, 1). Using the ordered pair multiplication: z1 * z2 = (2*4 - 3*1, 2*1 + 3*4) = (8 - 3, 2 + 12) = (5, 14).
Using the standard form: z1 = 2 + 3i z2 = 4 + 1i z1 * z2 = (2 + 3i)(4 + i) = 8 + 2i + 12i + 3i2 = 8 + 14i - 3 = 5 + 14i. This corresponds to the ordered pair (5, 14).
Division of Complex Numbers
Division is defined as the inverse of multiplication. To divide z1 = (a, b) by z2 = (c, d), where z2 is not (0, 0), we need to find a complex number (x, y) such that: (c, d) * (x, y) = (a, b)
Expanding the left side using the multiplication rule: (cx - dy, cy + dx) = (a, b)
This gives us a system of two linear equations: 1) cx - dy = a 2) dx + cy = b
Solving these equations for x and y (assuming c2 + d2 ≠ 0): x = (ac + bd) / (c2 + d2) y = (bc - ad) / (c2 + d2)
So, z1 / z2 = ( (ac + bd) / (c2 + d2), (bc - ad) / (c2 + d2) )
A more practical way to perform division is by using the concept of the conjugate, which we will discuss later. In the standard a + bi form, to divide a + bi by c + di, we multiply the numerator and denominator by the conjugate of the denominator (c - di): (a + bi) / (c + di) = [(a + bi) * (c - di)] / [(c + di) * (c - di)] = [ac - adi + bci - bdi2] / [c2 - cdi + cdi - d2i2] = [ac + bd + (bc - ad)i] / [c2 + d2] = (ac + bd) / (c2 + d2) + i * (bc - ad) / (c2 + d2) This matches the ordered pair result.
Example: Let z1 = (7, 2) and z2 = (2, 3). We want to find z1 / z2. Here a=7, b=2, c=2, d=3. c2 + d2 = 22 + 32 = 4 + 9 = 13. x = (7*2 + 2*3) / 13 = (14 + 6) / 13 = 20 / 13. y = (2*3 - 7*2) / 13 = (6 - 14) / 13 = -8 / 13. So, z1 / z2 = (20/13, -8/13).
The Imaginary Unit 'i'
As we saw, the complex number (0, 1) is fundamental. We define it as the imaginary unit 'i'. i = (0, 1)
From the multiplication rule, we found that: i2 = (0, 1) * (0, 1) = (-1, 0)
The complex number (-1, 0) corresponds to the real number -1. Thus, we have the defining property of 'i': i2 = -1
This is the key that allows us to work with complex numbers and solve equations that were previously unsolvable.
Powers of 'i'
The powers of 'i' follow a cyclic pattern:
- i1 = i
- i2 = -1
- i3 = i2 * i = -1 * i = -i
- i4 = i2 * i2 = (-1) * (-1) = 1
- i5 = i4 * i = 1 * i = i
- i6 = i4 * i2 = 1 * (-1) = -1
The pattern repeats every four powers: i, -1, -i, 1, i, -1, -i, 1, ...
To find any higher power of 'i', say in, we can divide 'n' by 4 and look at the remainder.
Shortcut for Powers of 'i'
Let n be any integer. To find in: Divide n by 4. - If the remainder is 0, in = 1 (since i4k = (i4)k = 1k = 1). - If the remainder is 1, in = i (since i4k+1 = i4k * i1 = 1 * i = i). - If the remainder is 2, in = -1 (since i4k+2 = i4k * i2 = 1 * (-1) = -1). - If the remainder is 3, in = -i (since i4k+3 = i4k * i3 = 1 * (-i) = -i).
Example: What is i105? Divide 105 by 4: 105 = 4 * 26 + 1. The remainder is 1. Therefore, i105 = i.
Example: What is i200? Divide 200 by 4: 200 = 4 * 50 + 0. The remainder is 0. Therefore, i200 = 1.
Standard Form of a Complex Number
As we established, any complex number (a, b) can be written as a + bi, where 'a' is the real part and 'b' is the imaginary part. This is called the standard or Cartesian form of a complex number.
For example, the complex number (5, -3) is written as 5 - 3i. The complex number (-2, 0) is written as -2 + 0i, which is simply -2 (a real number). The complex number (0, 4) is written as 0 + 4i, which is 4i (a purely imaginary number).
Representation of Complex Numbers
Complex numbers can be represented geometrically on a plane called the Argand plane or complex plane.
The horizontal axis is called the real axis, and it represents the real part of the complex number. The vertical axis is called the imaginary axis, and it represents the imaginary part of the complex number.
A complex number z = a + bi (or (a, b)) is represented by the point P(a, b) in this plane. The distance of this point from the origin is called the modulus of the complex number, and the angle it makes with the positive real axis is called its argument.
For example, the complex number 3 + 2i is represented by the point (3, 2) in the Argand plane. The complex number -1 - i is represented by the point (-1, -1).
Modulus and Argument (Brief Introduction)
While not strictly part of defining complex numbers as ordered pairs, understanding modulus and argument is crucial for deeper comprehension.
Modulus: The modulus of a complex number z = a + bi, denoted as |z|, is the distance from the origin to the point (a, b) in the Argand plane. |z| = √(a2 + b2)
Argument: The argument of a complex number z = a + bi, denoted as arg(z) or θ, is the angle made by the line segment from the origin to the point (a, b) with the positive real axis. It is usually measured in radians. tan(θ) = b/a (with consideration for the quadrant of the point (a, b)).
Conjugate of a Complex Number
The conjugate of a complex number z = a + bi is denoted by &bar;z and is defined as: &bar;z = a - bi
In terms of ordered pairs, if z = (a, b), then &bar;z = (a, -b).
The conjugate is obtained by changing the sign of the imaginary part.
Properties of Conjugates:
- &bar;(&bar;z) = z
- z + &bar;z = 2a (twice the real part)
- z - &bar;z = 2bi (twice the imaginary part multiplied by i)
- z * &bar;z = (a + bi)(a - bi) = a2 - (bi)2 = a2 - b2i2 = a2 + b2 = |z|2
The property z * &bar;z = |z|2 is extremely useful, especially in division and in simplifying expressions.
Example: If z = (3, 4) = 3 + 4i, then &bar;z = (3, -4) = 3 - 4i. z * &bar;z = (3 + 4i)(3 - 4i) = 32 - (4i)2 = 9 - 16i2 = 9 - 16(-1) = 9 + 16 = 25. Also, |z|2 = 32 + 42 = 9 + 16 = 25.
Why Use Ordered Pairs?
While the standard form a + bi is more commonly used, understanding complex numbers as ordered pairs of reals provides a rigorous foundation. It shows that complex numbers are simply an extension of the real number system, built upon pairs of real numbers with specific rules for arithmetic. This ordered pair representation is fundamental in fields like abstract algebra and functional analysis. It also directly maps to the geometric interpretation in the complex plane.