Question

find the modules of and conjugate of the following complex number z= -2+i7​

Answer 1

Answer:

If z = a + ib, then the complex conjugate of z is a - ib.

Given Complex Number (z): - 2 + 7i

Conjugate of a complex number is ( -z ).

⇒ ( -2 - 7i )

⇒ -2 - 7i

Therefore the conjugate of the given complex number -2 - 7i.

Modulus of a complex number z which is represented as a + bi is given as:

$|z| = \sqrt{a^2 + b^2$

where, 'a' is the numerical value of real part and 'b' is the numerical value of complex part .

According to the given complex number:

a = -2 , b = 7

⇒ |z| = √(-2)² + (7)²

⇒ |z| = √(4) + (49)

⇒ |z| = √53

Hence the modulus of the complex number is √53.

Answer 2

Topic : Complex number

Explanation:

We are asked to find the conjugate and modulus of the given complex number.

Conjugate of a complex number is equal to the complex number with same real part but imaginary part of opposite sign.

For a complex number $\sf z = a + ib$, it's conjugate is given by $\sf\overline{z} = a - ib$. (Conjugate of a complex number z is denoted by z bar.)

We have,

z = -2 + 7i

So it's conjugate is given by,

$\boxed{\red{\sf \overline{z} = -2 - 7i}}$

$\rule{280}{1}$

Modulus of a complex number is it's distance in the argànd plane from origin.

In order to find the modulus (or magnitude) of a complex number z = a + ib, we may use the formula $\sf|z| = \sqrt{a^2 + b^2}$where |z| is the magnitude of complex number.

We have,

z = -2 + 7i

Here,

• Re(z) = a = -2
• Im(z) = b = 7

Therefore the modulus of this complex number is given by,

$\sf |z| = \sqrt{(-2)^2 + (7)^2}$

$\sf |z| = \sqrt{4 + 49}$

$\boxed{\pink{\sf |z| = \sqrt{53}}}$

So we can conclude that conjugate of complex number is -2 - 7i and it's modulus is √53.

Additional information:

There is one more method to find the magnitude of complex number.

$\sf |z|^2 = \overline{z}\cdot z$

Therefore the magnitude of given complex number can also be given by,

$\sf |z|^2 = (-2+7i)(-2-7i)$

$\sf |z|^2 = (-2)^2 - (7i)^2$

$\sf |z|^2 = 4 - 49i^2$

$\sf |z|^2 = 4 + 49$

$\boxed{\green{\sf |z| = \sqrt{53}}}$

Refer to the attachment to know graphical presentation of conjugate of complex number. [Picture source - Wikipedia]