The equation x² - 1 = 0 can be solved by factorising it to (x - 1)(x + 1) = 0, or rearranging it to x² = 1, giving solutions of ±1.
However, x² + 1 = 0 cannot be factorised using the currently known numbers (real numbers), and rearranging it to x² = -1 requires the square root of a negative number to be found. No real number, when squared, gives a negative number. Squaring a real number always gives a non-negative number (positive or zero).
Hence, this equation cannot be solved using real numbers, so a new number must be defined. The new number is often given the symbol i (or j), and it is defined to equal -1.
The equation can now be solved, x = ± i, and now may be factorised to (x + i)(x - i) = 0.
(Since i² = -1, (-i)²= -1 also, as it can be considered as (-1 x i)² = (-1)² x i² = i² = -1)
The number i is called an imaginary number. Quadratic equations can be solved using this number. As an example, x² + x - 1 = 0 will be solved, using the quadratic formula.
The equation to be solved is x² + x - 1 = 0
Using x = (-b ± sqr(b²- 4ac) / 2a where a = 1, b=1, c= - 1
x = (-1 ± sqr(1-4)) / 2
x = (-1 ± sqr(-3)) / 2
But what is the square root of -3?
The identity sqr(ab) = sqr(a) * sqr(b) can be used.
sqr(-3) = sqr(-1) * sqr(3) = sqr(3) * i
The solutions to the equation are therefore:
x = (-1 ± sqr(3) * I) / 2
The previous example gave solutions, x = (-1 ± sqr(-3)*i) / 2.
This can be rearranged to - (1 / 2) ± (sqr(3) / 2) * I
Numbers of the form a + bi which contain both a real and imaginary component (where a and b are both real numbers) are called complex numbers.
a is the real part, and b is the imaginary part of a complex number a + bi.
The four basic operations of addition, subtraction, multiplication and division must be defined for complex numbers, just as they must be for matrices, and vectors.
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) - (c + di) = (a - c) + (b - d)i
As i = sqr(-1), i² = -1; this is useful in defining multiplication
(a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
Division is a little more complex, but multiplying a complex number a + bi by a - bi will give a real number (a² + b²), allowing the denominator to be rationalised (in a similar way to rationalising the denominator when surds are involved).
The value a - bi is called the conjugate of a complex number a + bi.
The conjugate of a complex number, z is written as z with a bar over it.
(a + bi) / (c + di) = [(a + bi) / (c + di)] * [(c-di) / (c-di)] = [(ac + bd) + (bc - ad) * i] / (c² + d²)
Figure 1 - an Argand diagram
Just as real numbers can be represented on a number line, complex numbers can be similarly represented on a complex number plane. The real component gives the x co-ordinate, and the imaginary component the y co-ordinate. The resulting diagram is called an Argand diagram, and an example is shown in Figure 1.
The magnitude of a complex number is given by the length from the origin to the point on the plane representing the number. This is the modulus of the complex number (written |x|), and can be calculated using Pythagoras' theorem on the triangle shown.
In general, |a + bi| = sqr(a² + b²)
The angle ß between the line joining the point representing the number and the origin, and the real number axis and the modulus line is the argument, and by convention is chosen to lie between -pi and pi, since there are an infinite number of possible values. It can be calculated using tan-¹, and is written arg (a + bi)
The conjugate of a complex number a + bi, has already been defined as a - bi, and is equivalent to a reflection in the x (real) axis on the Argand diagram.
Let x be the complex number, a + bi
x(bar) = a - bi
|x| = sqr(a² + b²)
x * x(bar) = (a + bi)(a - bi) = (a² + b²) = |x|²
In general,
x * x(bar) = |x|²