In the study of complex numbers we come across the following equality: i2 = – 1.
The justification for this equality is usually associated with solving 2nd degree equations with negative square roots, which is an error. The origin of the expression i2 = – 1 appears in the definition of complex numbers, another issue that also raises a lot of doubt. Let us understand the reason for such equality and how it arises.
First, let's make some definitions.
1. An ordered pair of real numbers (x, y) is called a complex number.
2. Complex numbers (x1y1) and (x2y2) are equal if and only if x1 = x2 and y1 = y2.
3. Addition and multiplication of complex numbers are defined by:
(x1y1) + (x2y2) = (x1 + x2y1 + y2)
(x1y1)*(x2y2) = (x1*x2 - y1*y2, x1*y2 + y1*x2)
Example 1. Consider z1 = (3, 4) and z2 = (2, 5), calculate z1 + z2 and z1*z2.
Solution:
z1 + z2 = (3, 4) + (2, 5) = (3+2, 4+5) = (5, 9)
z1*z2 = (3, 4)*(2, 5) = (3*2 – 4*5, 3*5 + 4*2) = (– 14, 23)
Using the third definition it is easy to show that:
(x1, 0) + (x
(x1, 0)*(x2, 0) = (x1*x2, 0)
These equalities show that with respect to addition and multiplication operations, complex numbers (x, y) behave like real numbers. In this context, we can establish the following relationship: (x, 0) = x.
Using this relationship and the symbol i to represent the complex number (0, 1), we can write any complex number (x, y) as follows:
(x, y) = (x, 0) + (0, 1)*(y, 0) = x + iy → which is the normal form call of a complex number.
Thus, the complex number (3, 4) in normal form becomes 3 + 4i.
Example 2. Write the following complex numbers in normal form.
a) (5, - 3) = 5 - 3i
b) (– 7, 11) = – 7 + 11i
c) (2, 0) = 2 + 0i = 2
d) (0, 2) = 0 + 2i = 2i
Now notice that we call i the complex number (0, 1). Let's see what happens when making i2.
We know that i = (0, 1) and that i2 = i*i. Follow that:
i2 = i*i = (0, 1)*(0, 1)
Using definition 3, we will have:
i2 = i*i = (0, 1)*(0, 1) = (0*0 – 1*1, 0*1 + 1*0) = (0 – 1, 0 + 0) = (– 1, 0 )
As we saw earlier, every complex number of the form (x, 0) = x. Thus,
i2 = i*i = (0, 1)*(0, 1) = (0*0 – 1*1, 0*1 + 1*0) = (0 – 1, 0 + 0) = (– 1, 0 ) = – 1.
We arrived at the famous equality i2 = – 1.
By Marcelo Rigonatto
Specialist in Statistics and Mathematical Modeling
Brazil School Team
Complex numbers - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/a-origem-i-ao-quadrado-igual-1.htm