Question

Describe geometrically the set of points in the complex plane satisfying the following equations. Re \(z=0\)

Short answer

The set of points forms a vertical line along the imaginary axis in the complex plane.

Step-by-step solution

Understand the Complex Number

A complex number is generally written as \(z = x + iy\), where \(x\) is the real part \(\text{Re}(z)\) and \(y\) is the imaginary part \(\text{Im}(z)\).

Identify the Given Condition

The given condition is \(\text{Re}(z) = 0\). This means that the real part of the complex number is zero.

Express the Condition Geometrically

If \(\text{Re}(z) = 0\), then \(z\) can be written as \(iy\), where \(y\) is any real number. This implies that the complex number lies on the imaginary axis (the y-axis in the complex plane).

Visualize the Set of Points

In the complex plane, the set of points where the real part is zero forms a vertical line passing through the origin, aligned with the imaginary axis. Every point on this line represents a complex number with a zero real part.

Key concepts

Complex Numbers

Complex numbers are an extension of the real numbers, consisting of a real part and an imaginary part. A complex number is typically written as \(z = x + iy\), where \(x\) represents the real part \(\text{Re}(z)\) and \(y\) represents the imaginary part \(\text{Im}(z)\).

The imaginary unit is denoted by \(i\), satisfying the property that \(i^2 = -1\).

Complex numbers allow us to solve equations that do not have solutions in the real numbers alone, such as \(x^2 + 1 = 0\). This wider range of solutions enhances our understanding of algebra and opens up new areas in mathematics.

Imaginary Axis

In the complex plane, complex numbers are represented as points with the form \(x + iy\).
The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

The condition \(\text{Re}(z) = 0\) indicates that \(x = 0\), meaning the complex number lies entirely on the imaginary axis. In this case, the complex number can be written simply as \(iy\).

Any point on the imaginary axis can be described by an imaginary number, where the real part is zero and the imaginary part can be any real number. Visualizing this, we see a vertical line intersecting the origin and running up and down along the imaginary axis.

Geometric Representation

Geometric representation helps in visualizing complex numbers and understanding their properties.

When we plot complex numbers on the complex plane:

• The real part \(\text{Re}(z)\) is plotted along the x-axis
• The imaginary part \(\text{Im}(z)\) is plotted along the y-axis.

For our condition \(\text{Re}(z) = 0\), every point \(iy\) on the imaginary axis corresponds to a complex number with only an imaginary part. Thus, the set of points satisfying \(\text{Re}(z) = 0\) forms a vertical line along the imaginary axis.

Geometrically, this helps us understand that any point on this line represents a complex number whose real part is zero, reinforcing the connection between algebraic conditions and their geometric implications.