find the modulus argument and principal argument of each of the following
a) i
b) -√5
a) modulus 1, argument PI/2 (in radians)
b) modulus √5 , argument PI
To find the modulus, argument, and principal argument of a complex number, we need to convert the complex number into its polar form.
a) i:
To find the modulus of i, we can use the formula for the magnitude (modulus) of a complex number:
|a + bi| = √(a^2 + b^2)
In this case, a = 0 and b = 1. So the modulus of i is:
|0 + 1i| = √(0^2 + 1^2) = √1 = 1
The argument (also known as the angle or phase) of a complex number can be found using the arctan function:
arg(a + bi) = arctan(b/a)
In this case, a = 0 and b = 1. So the argument of i is:
arg(0 + 1i) = arctan(1/0) = undefined
Since the real part (a) is 0, we cannot apply the arctan formula directly. However, we can say that the argument of the imaginary unit i is π/2 (90 degrees) or its equivalent rotations (π/2 + 2πk, where k is an integer).
The principal argument is the argument within the range (-π, π]. In this case, the principal argument of i is π/2.
b) -√5:
To find the modulus of -√5, we follow the same formula as before:
|-√5| = √((-√5)^2) = √(5) = √5
The argument of -√5 can be found using the arctan function:
arg(-√5) = arctan(0/(-√5)) = arctan(0) = 0
Since the real part (a) is 0, the argument is 0. The principal argument is also 0.
So, the modulus, argument, and principal argument for each of the given complex numbers are:
a) i: Modulus = 1, Argument = undefined, Principal Argument = π/2
b) -√5: Modulus = √5, Argument = 0, Principal Argument = 0