Question

Express the complex number z=5-5i in modulus amplitude form.​

Answer 1

Answer:

Step-by-step explanation:

Let z = x + iy where x and y are real numbers and i = √(-1). Then the non-negative square root of (x2 + y2) is known as the modulus or absolute value of z. Modulus or absolute value of z is denoted by |z| and read as mod z.

Hence if z = x + iy, then |z| = |x+iy| = +√x2 + y2.

For example, if z = -3 + 4i then, |z| = |-3 + 4i |= √(-3)2 + 42 = 5.

AMPLITUDE (OR ARGUMENT) OF A COMPLEX NUMBER:

Let z = x + iy where x and y are real numbers and i = √(-1) and x2 + y2 ≠ 0, then the value of θ for which the equations x = |z| cosθ ........(1) and y = |z| sin θ .......(2) are concurrently satisfied is named as the amplitude or argument of z and is denoted by Amp z or Arg z.

Equations (1) and (2) are satisfied for infinitely many values of θ, any of these infinite values of θ is the value of amp z. However, the unique value of θ lying in the interval -π< θ ≤ π and satisfying equations (1) and (2) is known as the principal value of arg z and it is denoted by arg z or amp z. Or in other words argument of a complex number means its principal value.

Since, cos(2nπ + θ)= cos θ and sin(2nπ + θ)= sin θ

Answer 2

Step-by-step explanation:

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