Question

Describe the set of all complex numbers that are at a distance of 2 units from the origin.

Answer 1

Answer:

The description of the set is $A = \{s\in \mathbb{C}|\|s\| = 2\}$.

Step-by-step explanation:

From Complex Analysis, we remember that complex numbers are numbers whose form is:

$s = a+i\, b$, $a,b \in \mathbb{R}$ (1)

Where $i = \sqrt{-1}$.

In addition, the distance from the origin is defined by the following Pythagorean identity:

$\|s\| = \sqrt{a^{2}+b^{2}}$ (2)

The following condition must be satisfied:

$\|s\| = 2$

Then, the set of all complex numbers that are at a distance of 2 units from the origin is described below:

$A = \{s\in \mathbb{C}|\|s\| = 2\}$ (3)

Answer 2

The set of all complex numbers that are at a distance of 2 units from the origin is $\{z\in \mathbb{C} :\text{ }|z|=2\}$

A complex number, written in rectangular form is

$z=a+ib\\a,b\in \mathbb{R}$

The distance of a complex number from the origin (the modulus of the complex number, denoted by $|z|$) is given by the formula

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

Since we want all the complex numbers to be at the distance of 2 units from the origin, they must satisfy

$|z|=2$

thus, the set we are looking for is

$D=\{z\in \mathbb{C} :\text{ }|z|=2\}$

where $\mathbb{C}$ is the set of all complex numbers

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