Problem 12 - Algebra and Trigonometry

Question

\(11-12\) a sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$ z=-1+i \sqrt{3} $$

Step-by-Step Solution

Answer

Sketch points are at \(-1 + i \sqrt{3}\), \(-2 + 2i \sqrt{3}\), \(1 - i \sqrt{3}\), and \(-\frac{1}{2} + \frac{i \sqrt{3}}{2}\) on the complex plane.

Step 1: Identify the Complex Number

The given complex number is \( z = -1 + i \sqrt{3} \). This means it has a real part of \(-1\) and an imaginary part of \(\sqrt{3}\).

Step 2: Sketch the Complex Number

On the complex plane, plot the point corresponding to \(-1 + i \sqrt{3}\). Start at the origin, move \(-1\) unit along the real axis (to the left), and \(\sqrt{3}\) units up along the imaginary axis.

Step 3: Calculate and Sketch \(2z\)

To find \(2z\), multiply both the real and imaginary parts of \(z\) by 2:\[2(-1 + i \sqrt{3}) = -2 + 2i \sqrt{3}\]On the complex plane, plot the point \(-2 + 2i \sqrt{3}\); move \(-2\) units along the real axis and \(2\sqrt{3}\) units up the imaginary axis.

Step 4: Calculate and Sketch \(-z\)

To find \(-z\), multiply both the real and imaginary parts of \(z\) by \(-1\):\[-(-1 + i \sqrt{3}) = 1 - i \sqrt{3}\]On the complex plane, plot the point \(1 - i \sqrt{3}\); move \(1\) unit along the real axis and \(\sqrt{3}\) units down the imaginary axis.

Step 5: Calculate and Sketch \(\frac{1}{2}z\)

To find \(\frac{1}{2}z\), multiply both the real and imaginary parts of \(z\) by \(\frac{1}{2}\):\[\frac{1}{2}(-1 + i \sqrt{3}) = -\frac{1}{2} + \frac{i \sqrt{3}}{2}\]On the complex plane, plot the point \(-\frac{1}{2} + \frac{i \sqrt{3}}{2}\); move \(-\frac{1}{2}\) unit along the real axis and \(\frac{\sqrt{3}}{2}\) unit up the imaginary axis.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to visually represent complex numbers. Each complex number corresponds to a unique point in this plane. The horizontal axis is known as the "real axis," while the vertical axis is referred to as the "imaginary axis." The point where these two axes intersect is called the origin. This plane is very similar to a standard Cartesian coordinate plane but is used specifically for complex numbers.

• Each complex number, such as \( z = -1 + i \sqrt{3} \), is represented as a point with coordinates based on its real and imaginary parts.
• To plot a complex number, begin at the origin; move units along the real axis corresponding to the real part then move along the imaginary axis for the imaginary part.
• In our example, the point \( -1 + i \sqrt{3} \) is plotted by moving \(-1\) unit left (real part) and \(\sqrt{3}\) units up (imaginary part).

Real and Imaginary Parts

Understanding real and imaginary parts is crucial for dealing with complex numbers. A complex number is generally expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

• For \( z = -1 + i \sqrt{3} \), the real part is \(-1\) and the imaginary part is \( \sqrt{3} \).
• The real part affects movement along the real axis and the imaginary part along the imaginary axis on the complex plane.

Understanding these components individually helps in manipulating and transforming complex numbers, such as scaling or reflecting them.

Multiplying Complex Numbers

Multiplying a complex number by a scalar is one way to transform that number, essentially stretching or shrinking its position on the complex plane without changing its direction.

• To multiply a complex number \( z = a + bi \) by a scalar \( k \), each part of the complex number is multiplied by \( k \). This yields \( ka + kbi \).
• For instance, multiplying \( z = -1 + i \sqrt{3} \) by 2 gives \( -2 + 2i \sqrt{3} \).
• This process results in moving the point associated with \( z \) further from or closer to the origin, maintaining its direction.

This operation doesn't affect the angle with respect to the origin, only the distance from it.

Complex Number Transformations

Complex number transformations can include several operations, such as multiplication by constants, addition, or subtraction, each resulting in a predictable effect on the complex plane.

• Multiplying by -1 reflects the complex number across the origin, changing its direction but keeping its magnitude.
• Dividing by a constant, like \( \frac{1}{2} \), scales the number closer to the origin, altering the distance while preserving the direction.
• In our example, \( -z = 1 - i \sqrt{3} \) is a reflection of \( z \) across the origin.
• \( \frac{1}{2}z = -\frac{1}{2} + \frac{i \sqrt{3}}{2} \) reduces the distance to the origin by half, keeping the direction the same.

Through these transformations, one can see how versatile and dynamic complex numbers can be, providing numerous ways to visualize mathematical concepts.