Problem 11 - 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. $$ z=1+i $$

Step-by-Step Solution

Answer

Plot z at (1,1), 2z at (2,2), -z at (-1,-1), and \(\frac{1}{2}z\) at (0.5,0.5) on the complex plane.

Step 1: Understand the Complex Number

The given complex number is \( z = 1 + i \). This can be interpreted as a point on the complex plane where the real part is 1 and the imaginary part is 1.

Step 2: Plot the Complex Number z

On the complex plane, plot the point corresponding to \( z \). It is located 1 unit to the right of the origin (along the real axis) and 1 unit up (along the imaginary axis). Mark this point on the sketch.

Step 3: Calculate 2z

Multiply the complex number \( z = 1 + i \) by 2 to find \( 2z \). \[ 2z = 2(1 + i) = 2 + 2i \]The new complex number \( 2z \) has a real part of 2 and an imaginary part of 2.

Step 4: Plot 2z

On the complex plane, plot the point for \( 2z \). Move 2 units to the right of the origin and 2 units up. Mark this point on the sketch.

Step 5: Calculate -z

Multiply the complex number \( z = 1 + i \) by -1 to find \( -z \). \[ -z = -(1 + i) = -1 - i \]The new complex number \( -z \) has a real part of -1 and an imaginary part of -1.

Step 6: Plot -z

On the complex plane, plot the point for \( -z \). Move 1 unit to the left of the origin and 1 unit down. Mark this point on the sketch.

Step 7: Calculate \(\frac{1}{2}z\)

Multiply the complex number \( z = 1 + i \) by \(\frac{1}{2}\) to find \(\frac{1}{2}z\).\[ \frac{1}{2}z = \frac{1}{2}(1 + i) = \frac{1}{2} + \frac{1}{2}i \]The new complex number \(\frac{1}{2}z\) has a real part of \(\frac{1}{2}\) and an imaginary part of \(\frac{1}{2}\).

Step 8: Plot \(\frac{1}{2}z\)

On the complex plane, plot the point for \(\frac{1}{2}z\). Move \(\frac{1}{2}\) units to the right of the origin and \(\frac{1}{2}\) units up. Mark this point on the sketch.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to visually represent complex numbers. Think of it like a graph with an x-axis and a y-axis, but here, we call them the real axis and the imaginary axis respectively.
The horizontal line going from left to right is the real axis, and the vertical line running up and down is the imaginary axis. Every complex number corresponds to a single point on this plane, where its position is determined by its real and imaginary parts.

• The real part of the complex number determines its position on the real axis.
• The imaginary part determines its position on the imaginary axis.

This plotting of complex numbers makes it easier to understand and visualize complex arithmetic like addition, subtraction, and more.

Real and Imaginary Parts

Any complex number can be expressed in the form of \(z = a + bi\), where:

• \(a\) is the 'real part'. It's a real number and tells us how far, left or right, the number is placed on the complex plane.
• \(bi\) is the 'imaginary part'. Here, \(b\) is a real number, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). This part tells us how far, up or down, the number is placed on the complex plane.

For example, in the complex number \(1 + i\), 1 is the real part, and \(i\) (effectively \(1\cdot i\)) is the imaginary part.
Understanding these parts separately can help you grasp how complex numbers interact and change position and orientation on the complex plane when you perform operations on them.

Multiplying Complex Numbers

When multiplying complex numbers, it's essential to apply the distributive property, just like with binomials in algebra. Consider multiplying \(z = 1 + i\) by a real number or another complex number.
Let's look at some multiplication examples:

• By a Real Number: Multiply each part of the complex number by the real number. For \(2z\), where \(z = 1 + i\), we get: \[2z = 2(1 + i) = 2 + 2i\]
• By -1: Each part of the complex number changes sign. For \(-z\), we have: \[-z = -(1 + i) = -1 - i\]
• By a Fraction: To find half of \(z\), multiply by \(\frac{1}{2}\): \[\frac{1}{2}z = \frac{1}{2}(1 + i) = \frac{1}{2} + \frac{1}{2}i\]

This multiplication impacts where the new number is located on the complex plane, altering its distance from the origin and its direction.

Graphing Complex Numbers

Graphing complex numbers provides a visual way to understand their properties and interactions. Each complex number corresponds to a unique point on the complex plane.
For example:

• The complex number \(z = 1 + i\) is graphed as a point located 1 unit along the real axis and 1 unit up on the imaginary axis from the origin.
• For \(2z = 2 + 2i\), the point moves to 2 units right and 2 units up.
• The number \(-z = -1 - i\) is represented 1 unit left and 1 unit down.
• Finally, \(\frac{1}{2}z = \frac{1}{2} + \frac{1}{2}i\) falls halfway along both the real and imaginary axes compared to \(z\).

Visualizing these points on the plane captures their 'magnitude' (distance from the origin) and 'direction' (angle from the real axis) effectively. As you perform arithmetic operations with complex numbers, observe how their positions and relationships evolve on the complex plane.