When a complex number is purely imaginary?

When a complex number is purely imaginary?

A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero.

What is a purely imaginary root?

A simple observation is that if p ( x ) = ∑ a i x i is a polynomial with real coefficients and b ∈ R ( b ≠ 0 ), then has a root at if and only if p e v e n ( x ) = ∑ a 2 i x i and p o d d ( x ) = ∑ a 2 i + 1 x i both have roots at .

When z is purely real?

So, z=a+ib and z=a-ib. Adding ,2z=2a . Therefore z=a , which is real.

What numbers are purely imaginary?

A complex number is said to be purely imaginary if it’s real part is zero. Zero is purely imaginary, as it’s real part is zero. 0 is both purely real and purely imaginary.

What is purely real and purely imaginary?

A complex number z is said to be: Purely real, if Im(z) = 0, Purely imaginary, if Re(z) = 0.

Is a purely imaginary number?

A pure imaginary number is any complex number whose real part is equal to 0. A complex number is a number with both real and imaginary parts written…

How do you know if a number is purely imaginary?

If z≠0, this means that z is purely imaginary. There is also a nice geometric interpretation: |z−1| is the distance from z to 1, and |z+1| is the distance to −1. If these are equal, then z must lie on the perpendicular bisector of the line segment connecting 1 and −1, which is the line x=0.

What is the value of z * conjugate of Z?

Let z=a+ib. Then conjugate of z’ =a-ib. As given ,z=z’. So, z=a+ib and z=a-ib.

What are the real and imaginary parts of z?

In general, the x part of a complex number z = x + yi is called the real part of z, while y is called the imaginary part of z. (Sometimes yi is called the imaginary part.) When we use the xy-plane for the complex plane C, we’ll call the x-axis by the name real axis, and the y-axis we’ll call the imaginary axis.

What is the value of z – 1/z + 1?

z – 1/z + 1 = k*i , where k is a constant real number. upon simplification. z = 1 + k*i/1 – k*i. or. z = 1 – k^2 + 2*k*i/1 + k^2. so Z can be any complex number written in the above form.

What is the mod of z+1?

If A Equal Z 1 Z Plus 1 Is Purely Imaginary Number If a = (z-1)/ (z+1) is purely imaginary number (z not equal -1), then mod z is (1) 1 (2) 2 (3) 3 (4) 4 Let z = x+iy (z-1)/ (z+1) = (x+iy-1)/ (x+iy+1), z ≠-1

What is the argument for a complex number on imaginary axis?

If a complex number is purely imaginary, then its argument must be either [math]\\dfrac {\\pi} {2} [/math] or [math]\\dfrac {-\\pi} {2}, [/math] i.e., it lies on the imaginary axis. Put z-1 and z+1, in place of z and w. z-1 and z+1, denote line segments (much like vectors), joining z to 1 and -1 respectively.

What is the locus of Z-1?

z-1 and z+1, denote line segments(much like vectors), joining z to 1 and -1 respectively. arg(z-1)-arg(z+1), denotes the angle between the line segments. Therefore, The locus of z, is a circle with the points 1 and -1 as the ends of one of the diameters.

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