Find the absolute value of the complex number z = 3 β 4i
Complex number is the sum of a real number and an imaginary number. These are the numbers that can be written in the form of a+ib, where a and b both are real numbers. It is denoted by z.
Here the value βaβ is called the real part which is denoted by Re(z), and βbβ is called the imaginary part Im(z). In complex numbers form a +bi, βiβ is an imaginary number called βiotaβ.
The value of i is (β-1) or we can write as i2 = -1.
For example:
- 7+15i is a complex number, where 7 is a real number (Re) and 15i is an imaginary number (Im).
- 8 + 5i is a complex number where 8 is a real number (Re) and 5i is an imaginary number (im)
Absolute value of a complex number
The distance between the origin and the given point on a complex plane is termed the absolute value of a complex number. The absolute value of a real number is the number itself and is represented by modulus, i.e. |x|.
Therefore the modulus of any value gives a positive value, such that;
|5| = 5
|-5| = 5
Now, finding the modulus has a different method in the case of complex numbers,
Suppose, z = a+ib is a complex number. Then, the modulus of z will be:
|z| = β(a2+b2), when we apply the Pythagorean theorem in a complex plane then this expression is obtained.
Hence, mod of complex number, z is extended from 0 to z and mod of real numbers x and y is extended from 0 to x and 0 to y respectively. Now they form a right-angled triangle, where the vertex of the acute angle is 0.
So, |z|2 = |a|2+|b|2
|z|2 = a2 + b2
|z| = β(a2+b2)
Find the absolute value of the complex number z = 3 β 4i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given : z = 3-4i
We have : |z| = β(a2+b2)
Here a = 3, b = -4
|z| = β(a2+b2)
= β(32+(-4)2)
= β(9 +16)
= β25
= 5
Hence the absolute value of complex number z = 3-4i is 5.
Similar Questions
Question 1: Find the absolute value of the following complex number. z = 5-9i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given: z = 5 β 9i
We have: |z| = β(a2+b2)
Here a = 5, b = -9
|z| = β(a2+b2)
= β(52+(-9)2)
= β(25 +81)
= β106
Hence the absolute value of complex number z = 5 β 9i is β106.
Question 2: Find the absolute value of the following complex number z = 2- 3i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given: z = 2 β 3i
We have: |z| = β(a2+b2)
here a = 2, b = -3
|z| = β(a2+b2)
= β(22+(-3)2)
= β(4 +9)
= β13
hence the absolute value of complex number z = 2 β 3i is β13.
Question 3: Perform the indicated operation and write the answer in standard form: (5 + 4i) Γ (6 β 4i).and find its absolute value?
Solution:
(5 + 4i) Γ (6 β 4i)
= (30 -20i +24i β 16i2)
= 30 + 4i +16
= 46 β 4i
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given : z = 46 β 4i
we have : |z| = β(a2+b2)
here a = 46 , b = -4
|z| = β(a2+b2)
= β(46)2+(-4)2)
= β(2116+ 16)
= β2132
Hence the absolute value of complex number. z = 46 β 4i is β2132
Question 4: Find the absolute value of the following complex number. z = 3 β 5i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given : z = 3 β 5i
We have : |z| = β(a2+b2)
here a = 3, b = -5
|z| = β(a2+b2)
= β(32+(-5)2)
= β(9 +25)
= β34
hence the absolute value of complex number z = 3 β 5i is β34
Question 5: If z1, z2 are (1 β i), (-2 + 2i) respectively, find Im(z1z2/z1).
Solution:
Given: z1 = (1 β i)
z2 = (-2 + 2i)
Now to find Im(z1z2/z1),
Put values of z1 and z2
Im(z1z2/z1) = {(1 β i) (-2 + 2i)} / (1 β i)
= {(-2 +2i +2i -2i2)} / (1-i)
= {(-2 + 4i + 2) / (1 β i)
= {( 4i) /(1 β i)}
= {(0+4i) (1 + i)} / {(1 + i)(1- i)}
= {(4i + 4i2) / (1 + 1)
= 4i -4 + 2i / 2
= (-4 + 2i) / 2
= -4/2 + 2/2 i
= -2 + i
Therefore, Im(z1z2/z1) = 1
Question 6: Perform the indicated operation and write the answer in standard form: (2 β 7i)(2 + 7i)
Solution:
Given: (2 β 7i)(2 + 7i)
= {4 + 14i β 14i β 49i2}
= (4 +49)
= 53 + 0i