Question

Describe geometrically the set of points in the complex plane satisfying the following equations. $$|z|=2$$

Short answer

The set of points is a circle with radius 2 centered at the origin.

Step-by-step solution

- Understand the Equation

The equation is given as \(|z| = 2\). This represents the modulus of the complex number z, which is equal to 2.

- Recall the Definition of Modulus

The modulus of a complex number \(z = a + bi/\i\) is given by \(|z| = \sqrt{a^2 + b^2}\).

- Interpret the Modulus Equation

Given \(|z| = 2\), we substitute in the modulus formula to get \(\sqrt{a^2 + b^2} = 2\).

- Square Both Sides

To simplify, square both sides of the equation: \(a^2 + b^2 = 4\).

- Geometric Interpretation

The equation \(a^2 + b^2 = 4\) represents a circle with a radius of 2 centered at the origin (0,0) in the complex plane.

Key concepts

modulus of a complex number

In complex numbers, the modulus represents the 'magnitude' of the number. Just like how we measure the length of a vector, we measure the size of a complex number using what is called the modulus.
The modulus of a complex number \( z = a + bi \) (where \( a \) and \( b \) are real numbers) is defined by the formula:
\[ |z| = \sqrt{a^2 + b^2} \] This is similar to the Pythagorean theorem. The terms \( a \) and \( b \) can be thought of as the legs of a right triangle, and the modulus \( |z| \) is the hypotenuse.
This process helps in simplifying complex numbers and in understanding their properties. If we have \ |z| = 2 \ , we know the length (distance from the origin) is 2.

geometric interpretation

The geometric interpretation of complex numbers involves visualizing these numbers in the complex plane. The complex plane is similar to a Cartesian plane but instead, the horizontal axis represents the real part and the vertical axis represents the imaginary part of the complex number.
When we plot \( z = a + bi \) on this plane, it translates to a point \( (a, b) \).
For the equation given \( |z| = 2 \), we interpret it using geometry:

• Given the modulus formula \( |z| = 2 \), we substitute into the modulus formula: \ |z| = \sqrt{a^2 + b^2} = 2 \.

Squaring both sides, we get \( a^2 + b^2 = 4 \).
This equation is the standard form of a circle in Cartesian coordinates. It describes a circle with radius 2 centered at the origin (0,0). So, every point \ z \ on this circle is 2 units away from the origin.

complex number equations

Complex number equations can sometimes be tricky to handle if not approached properly. The given problem \( |z| = 2 \) translates into an equation involving real and imaginary parts:
\