Negative Imaginary Number
The concept of imaginary numbers:
We know that on squaring any number, whether that number is negative, positive, or zero, the result will be a positive number or zero. Thus, for any real number '`x` ', `x^2 >= 0` . Thus, equations such as `x^2 = -1` cannot have any solution. All similar equations like `x^2 = -25` , `x^2 + 3 = 0` are not solvable in the real number system. To solve these unreal equations, we need to extend our concept of the real numbers to the complex number system. A complex number includes a real number and an imaginary number.
Many mathematicians' works show the formation of the concept of imaginary numbers, but the imaginary number theory was first introduced by Euler.Definition of Imaginary Numbers
The imaginary number `i` is defined as `i = sqrt (-1)` . Thus, `i^2 = -1` .Negative imaginary numbers
Imaginary number `i` equals `sqrt (-1)` . In general, all pure imaginary numbers are multiples of `i` . Some examples of imaginary numbers are `2i` , `3i` , `0.12i` , `sqrt (-2)` , etc.
Thus, negative imaginary number equals `-i`` = - sqrt (-1)` .Facts about Negative Imaginary Numbers:A negative imaginary number is equal to the reciprocal of a positive imaginary number:-
The reciprocal of a positive imaginary number is `1/i` .
Rationalizing the fraction, `1/i * i/i = i/(i^2)`
Now we know that `i = sqrt (-1)`
Thus, `i^2 = -1`
Thus `i/(i^2) = i/(-1) = -i` .
Thus `-i =1/i`Negative imaginary number is equal to the cube of a positive imaginary number
The cube of a positive imaginary number is `i^3` .
Which can aslo be written as `i^2*i`
Now, we know that `i = sqrt (-1)`
Thus, `i^2 = -1`
Thus, `i^2*i = -1*i = -i`
Thus, `i^3 = -i`