Problem 15 - Precalculus Mathematics for Calculus

Question

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 points on the complex plane: \((1,1)\), \((2,2)\), \((-1,-1)\), and \(\left(\frac{1}{2},\frac{1}{2}\right)\).

Step 1: Identify the Original Complex Number

The given complex number is \(z = 1 + i\). This can be written as \(z = a + bi\), where \(a = 1\) and \(b = 1\). In the complex plane, the point corresponding to this complex number is \((1,1)\).

Step 2: Plot the Original Complex Number

Plot the point \((1,1)\) on the complex plane. This represents the complex number \(z\). The x-axis represents the real part, while the y-axis represents the imaginary part.

Step 3: Determine and Plot the Complex Number \(2z\)

Multiply the original complex number by 2: \(2z = 2(1+i) = 2 + 2i\). The point \((2, 2)\) represents \(2z\) on the complex plane. Plot this point.

Step 4: Determine and Plot the Complex Number \(-z\)

Negate the original complex number: \(-z = -(1+i) = -1 - i\). The point \((-1, -1)\) represents \(-z\) on the complex plane. Plot this point.

Step 5: Determine and Plot the Complex Number \(\frac{1}{2}z\)

Divide the original complex number by 2: \(\frac{1}{2} z = \frac{1}{2}(1+i) = \frac{1}{2} + \frac{1}{2}i\). The point \(\left(\frac{1}{2}, \frac{1}{2}\right)\) represents \(\frac{1}{2} z\) on the complex plane. Plot this point.

Key Concepts

Complex Plane

The complex plane is a visual tool used to understand complex numbers. It looks much like a regular graph, but it has a special twist. Rather than showing just numbers, it shows the two parts of a complex number: the real part and the imaginary part.

• The horizontal axis (like the x-axis) is used for the real part.
• The vertical axis (like the y-axis) is used for the imaginary part.

When you plot a complex number, you're actually plotting a point in this plane. For example, the number \( z = 1 + i \) is plotted as the point \( (1, 1) \) because 1 is the real part and the real number 1 is also the imaginary part. This makes it easy to see where complex numbers "live" both visually and mathematically.

Multiplication of Complex Numbers

When you multiply complex numbers, it changes both their size and direction on the complex plane. This is because complex numbers can be thought of as vectors. Multiplying by a real number scales their length, while the imaginary part affects direction.
Let's consider the example \( z = 1 + i \):

• Multiplying by 2, yields: \( 2z = 2(1 + i) = 2 + 2i \). It doubles the length of the vector \((1,1)\) and results in the point \((2, 2)\).
• Multiplying by \(-1\), means \(-z = -(1 + i) = -1 - i\). This flips the vector across both axes, changing its direction to \((-1, -1)\).
• Multiplying by \(\frac{1}{2}\) scales the vector down by half, producing \( \frac{1}{2}z = \frac{1}{2} + \frac{1}{2}i \) which is point \((0.5, 0.5)\).

Each multiplication operation alters how far the point is from the origin and its angle relative to the real axis.

Graphing Complex Numbers

Graphing complex numbers is a straightforward yet powerful method to visualize their properties and behavior. It makes mathematical operations like addition, subtraction, and multiplication much more intuitive as they can be seen as transformations on the plane.
To graph a complex number, treat it as a point or a vector:

• For \( z = 1 + i \), plot \( (1, 1) \) in the complex plane.
• For \( 2z = 2 + 2i \), plot \( (2, 2) \).
• For \( -z = -1 - i \), plot \( (-1, -1) \).
• For \( \frac{1}{2}z = \frac{1}{2} + \frac{1}{2}i \), plot \( (0.5, 0.5) \).

Each point is connected to its corresponding operation, visually representing how complex numbers interact with each other. Graphing these points makes it easier to spot patterns and understand complex number operations, which is an essential skill in various fields of math and engineering.