Modulus of a complex number
In mathematics, a complex number z is a number which belongs to the set C and which can be written in the form z = a + ib where a and b are real numbers and i is an imaginary value whose square is equal to -1 ( i^2 = -1).
The notation z = a + ib is the algebraic form of the complex number z where a is the real part of the complex number z and b is its pure imaginary part.
The modulus of a complex number is the square root of the sum of the squares of its real and imaginary part.
The formula for determining the modulus |z| of the complex number z is therefore:
|z| = √(a^2 + b^2).
Examples :
Find the modules of the following complex numbers:
z1 = 3 + i;
z2 =2 - 2i;
z3 = 5i.
|z1| = |3 + i| = √(3^2 + 1^2)
= √(9 + 1) = √10
So the modulus of the complex number z1 is:
|z1| = √10
|z2| = |2 - 2i| = √(2^2 + (-2)^2)
= √(4 + 4) = √(2*4) = 2√2
The modulus of the complex number z2 is then:
|z2| = 2√2
|z3| = |5i| = √(0^2 + 5^2)
= √25 = 5
The modulus of the complex number z3 is:
|z3| = 5
Argument of a complex number
To write a complex number in trigonometric form, we calculate its modulus and its argument.
The argument of a complex number is an angle that is constructed when representing a complex number in the complex plane, the standard value of the argument is the radian.
The argument of a complex number is calculated from its modulus, because the sine and cosine of the argument of a complex number are expressed from the modulus of the complex number.
To determine the argument x of a complex number z = a + ib, we calculate the sine and the cosine of the argument using the formulas cos(x) = a/|z| and sin(x) = b/|z| where |z| is the modulus of the complex number z, then we find the value of x by projecting the values of cos(x) and sin(x) onto the trigonometric circle.
The complex plane is located inside the trigonometric circle, the cosine values are on the abscissa axis (real axis) and the sine values are on the ordinate axis (imaginary axis). When the values of cos(x) and sin(x) are known, the argument x is the point of intersection of the projection of cos(x) and sin(x) on the trigonometric circle.
Exercices with answers
1)
Find the modulus of each of the following complex numbers:
z1 = -2i + 5;
z2 = 15;
z3 = i(2 + 3i)
z4 = 4(-2 + 3i)
2)
Find the modulus and argument of the complex number z5 = 1/(1 + i)
Solutions :
1)
Let's calculate the modulus of z1:
|z1| = |-2i + 5| = |5 - 2i|
= √(5^2 + (-2)^2) = √(25 + 4) = √29
=> |z1| = √29
Let's calculate the modulus of the complex number z2:
|z2| = |15| = √(15)^2 = 15
The complex number z3 is not given in its algebraic form, before calculating the modulus of z3, let's put it in the algebraic form.
z3 = i(2 + 3i) = 2i + 3i^2
= 2i - 3 = -3 + 2i
The algebraic form of z3 is:
z3 = -3 + 2i
Now let's calculate the modulus of z3:
|z3| = √((-3)^2 + 2^2) = √(9 + 4) = √13
|z3| = √13
Let's find the algebraic form of the complex number z4 before calculating its modulus
z4 = 4(-2 + 3i) = -8 + 12i
z4 = -8 + 12i
Let's calculate the modulus of z4:
|z4| = √((-8)^2 + (12)^2)
= √(64 + 144) = √208
= √(16*13) = 4√13
|z4| = 4√13
Another way to calculate the modulus of z4 is to calculate the modulus of -2 + 3i and multiply it by 4.
Let's find the algebraic form of the complex number z5 and then we can calculate its modulus:
z5 = 1/(1 + i) = (1 - i)/(1 + i)(1 - i)
= (1 - i)/(1^2 - i^2) = (1 - i)/2
then z5 = (1/2) - (1/2)i
2)
Calculation of the modulus of z5:
|z5| = √((1/2)^2 + (-1/2)^2)
= √(1/4 + 1/4) = √(2/4) = (√2)/2
|z5| = (√2)/2
Let's find the argument x5 of z5:
cos(x5) = (1/2)/(√2/2) = √2/2
sin(x5) = (-1/2)/(√2/2) = -√2/2
cos(x5) = √2/2 is located on the abscissa axis and sin(x5) = -√2/2 is located on the ordinate axis in the complex plane, their projection on the trigonometric circle allows to find the argument x5 of the complex number z5.
So we have x5 = 7π/4 + 2kπ = -π/4 + 2kπ