Modulus of complex number means the distance of the complex number from the origin of the argand plane. It also helps in finding the distance between two points of complex numbers. The modulus of a complex number is also called the absolute value of the complex number.
In this article, we will cover the concept of the modulus of a complex number. This concept falls under the broader category of complex numbers. It is essential not only for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), as well as other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of seventeen questions have been asked on this concept, including two in 2013, two in 2014, one in 2015, one in 2019, three in 2020, three in 2021, three in 2022, and two in 2023.
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If z = x + iy is a complex number. Then, the modulus of z, denoted by | z |, is the distance of z from the origin in the Argand plane, and it is a non-negative real number equal to
i.e. |z| =
Every complex number can be represented as a point in the argand plane with the x-axis as the real axis and the y-axis as the imaginary axis.
The modulus of a complex number
The distance between the complex number's coordinates and the origin on a complex plane, as shown in a graph of a complex number
Moreover, this can be interpreted in terms of the Pythagorean theorem, which states that the hypotenuse, base, and height of a right-angled triangle reflect the modulus, real portion, and imaginary part, respectively, of a complex number. The magnitude (or length) of the vector that represents
Properties of Modulus
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
iх)
We concluded that the modulus is an important term in complex numbers that are used to find the distance from the origin. Like the distance formula in coordinate geometry, it seems to be similar to this for calculating distance. It is also applied in real-life scenarios to find the distance between two points in a complex plane or argand plane.
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Example 1: The complex number
1)
2)
3)
4)
Solution:
As we learned in
Definition of Modulus of z(Complex Number) -
- wherein
Real part of
Now,
if
if
if
if
Hence, the answer is the option (2).
Example 2: Let
1)
2)
3)
4)
Solution
As we learned in
Definition of Modulus of z(Complex Number) -
- wherein
Real part of
Hence, the answer is
Example 3: Let
1)
2)
3)
4)
Solution
As we have learned
Property of Modulus of z(Complex Number) -
- where |.| denotes the modulus of a complex number
Now,
And
Hence, the answer is the option 4.
Example 4: If a>0 and
Solution:
Multiplication of Complex Numbers -
(a+ib)(c+id)=(ac-bd)+i(bc+ad)
Definition of Modulus of z(Complex Number) -
- wherein
Real part of z = Re (z) = a & Imaginary part of z = Im (z) = b
given that
Hence, the answer is
Example 5:The equation
1) a circle of radius
2) a circle of radius
3) the line through the origin with slope
4) the line through the origin with slope
Solution:
Definition of Complex Number -
- wherein
Real part of
Definition of Modulus of z(Complex Number) -
- wherein
Real part of
The line through the origin with slope 1
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