The modulus of a complex number is defined as the non-negative square root of the sum of squares of the real and imaginary parts of the complex number. That is, the modulus of the complex number z = a + bi is: | z | = √a2 + b2. The modulus of the complex number − 5 + 8i is: | − 5 + 8i | = √( − 5)2 + 82 or √89. Question: 14. How many complex numbers satisfy the equation z5=zˉ, where zˉ is the conjugate of the complex number z ? (A) 2 (B) 3 (C) 5 (D) 6 (E) 75. Usain is walking for exercise by zigzagging across a 100 -meter by 30 -meter rectangular field, beginning at point A and ending on the segment BC. He wants to increase the distance walked by i2 = − 1. If c is a real number with c ≥ 0 then √− c = i√c. Property 1 in Definition 3.4 establishes that i does act as a square root 2 of − 1, and property 2 establishes what we mean by the 'principal square root' of a negative real number. In property 2, it is important to remember the restriction on c. The complex conjugate of a complex number z = x + iy is x - iy (and vice versa) and it is represented by \(\bar{z}\) as their sum (2x) and the product x 2 + y 2 both are rational numbers. To write the complex conjugate, A complex number in polar form is written as z = r (cos θ + i sin θ), where r is the modulus of the complex number and θ is its argument. Now, the formula for multiplying complex numbers z 1 = r 1 (cos θ 1 + i sin θ 1) and z 2 = r 2 (cos θ 2 + i sin θ 2) in polar form is given as:. z 1 z 2 = [r 1 (cos θ 1 + i sin θ 1)] [r 2 (cos θ 2 + i sin θ 2)] = r 1 r 2 (cos θ 1 cos θ 2 + i cos Complex numbers are expressions of the form z= x+ iywhere x;yare real numbers, and i2 = 1 (by de nition). Complex numbers can be added by the rule (x 1 + iy 1) + (x 2 + iy 2) = (x 1 + x 2) + i(y 1 + y 2); so we can associate to a complex number a vector (x;y) in the plane R2 and the addition rule is the same as for vectors. Similarly you can asked Nov 5, 2022 in Complex Numbers by Mounindara (56.7k points) Let α and β be two fixed non-zero complex numbers and z a variable complex number. If the line \(\alpha\bar z + \bar \alpha z + 1 \) and \(\beta\bar z + \bar \beta z- 1 = 0\) are mutually perpendicular, then 6 Answers Sorted by: 6 If z2 = ˉz then taking magnitude gives | z | 2 = | z | so | z | = 0, 1. The case | z | = 0 gives z = 0. If | z | = 1, then the equation z2 = ˉz = z − 1 so z3 = 1. This gives the remaining 3 solutions which are the third roots of unity. Share Cite answered Sep 9, 2015 at 21:17 pre-kidney 29.7k 37 84 Add a comment 3 The complex number z1, z2 & z3 satisfying (z1 - z3 / z2 - z3) = 1-i√3/2 are the vertices of a triangle which is asked Dec 10, 2022 in Complex Numbers by LuciferKrish ( 54.0k points) complex numbers If z is a complex number of unit modulus and argument θ, then arg ((1 + z)/(1 + Bar z)) is equal to asked Oct 8, 2018 in Mathematics by Samantha ( 40.3k points) complex numbers 9P3oFGE.