Imaginary Numbers Are Real [Part 7: Complex Multiplication]

Summary

The video script explores the relationship between complex multiplication and the complex plane, using four examples to illustrate the process. It reveals that multiplying complex numbers involves adding their angles and multiplying their distances from the origin. This insight leads to the introduction of polar form, a representation of complex numbers using magnitude and angle, simplifying multiplication and division in the complex plane. The script emphasizes the beauty of math, showing how different perspectives can reveal the same underlying truth.

Takeaways

• 🔍 The connection between complex multiplication and the complex plane is explored through examples, focusing on the angles and distances of complex numbers.
• 📐 The angle of the result in complex multiplication is the sum of the angles of the numbers being multiplied.
• 🌐 Tracking the angle a complex number makes with the real axis is crucial for understanding its behavior on the complex plane.
• 🔄 The magnitude (distance from the origin) of the result in complex multiplication is the product of the magnitudes of the numbers being multiplied.
• 🔄 The script highlights the importance of considering both angle and magnitude in complex multiplication, as they provide a complete picture of the operation.
• 📈 The script introduces the concept of polar form for complex numbers, which expresses them in terms of magnitude and angle from the real axis.
• 📖 Multiplication and division of complex numbers in polar form are simplified, involving only the addition or subtraction of angles and the multiplication or division of magnitudes.
• 🤔 The script encourages pondering and pattern recognition to understand the deeper connections between algebraic rules and geometric interpretations.
• 🌟 The script emphasizes the beauty of mathematics by showing how different approaches (algebraic and geometric) can lead to the same results.
• 🎯 The script concludes by suggesting that the understanding of complex multiplication on the complex plane can simplify and enhance problem-solving in algebra.

Q & A

What is the main topic discussed in the script?

The main topic discussed in the script is the connection between complex multiplication and the complex plane, specifically how angles and distances are involved in this process.

What mathematical concept is used to determine the angle of complex numbers with the real axis?

The script mentions the use of the arctangent function to determine the angle that complex numbers make with the real axis.

What pattern was observed when multiplying complex numbers in terms of their angles?

The pattern observed was that the angle of the result of complex multiplication is equal to the sum of the angles of the numbers being multiplied.

Why is it insufficient to only track angles when describing complex multiplication in the complex plane?

It is insufficient because, as seen in the script's examples, two complex numbers can have the same angle but different results, indicating that another factor, such as distance from the origin, is also important.

What geometric principle is used to measure the distance between the origin and a complex number?

The script refers to the use of the Pythagorean theorem to measure the distance from the origin to a complex number by forming right triangles.

What is the complete picture of complex multiplication on the complex plane?

The complete picture is that when multiplying complex numbers, their angles from the real axis add together, and their distances from the origin multiply.

What alternative way to write complex numbers is introduced in the script?

The script introduces polar form as an alternative way to write complex numbers, using their distance from the origin (magnitude) and the angle they make with the real axis.

How is complex multiplication performed in polar form?

In polar form, complex multiplication is performed by multiplying the magnitudes of the complex numbers and adding their angles.

What is the process for dividing complex numbers in polar form?

To divide complex numbers in polar form, you divide the magnitudes and subtract the angles.

What deeper insight does the script suggest about the nature of math?

The script suggests that math can reveal deeper truths embedded in the universe, and that there are multiple vantage points or interpretations for the same underlying process.

What is the script's promise for the next part of the discussion?

The script promises to show that the understanding of complex multiplication on the complex plane is not only interesting but also useful for making difficult algebra problems easier, faster, and more intuitive.

Outlines

00:00

🧩 Complex Multiplication and the Complex Plane

This paragraph delves into the relationship between complex multiplication and the geometry of the complex plane. It introduces the concept of multiplying two complex numbers and visually representing them on the complex plane. The focus is on identifying patterns, particularly the sum of angles made by the complex numbers with the real axis, and how this relates to the angle of the resulting complex number. The paragraph also hints at the importance of the magnitude, or distance from the origin, in understanding complex multiplication.

📏 The Geometry of Complex Multiplication

The second paragraph explores the geometric interpretation of complex multiplication further. It discusses the observation that the angles of the resulting complex numbers are the sum of the angles of the numbers being multiplied, but also notes that the magnitudes, or distances from the origin, are crucial for a complete understanding. The paragraph emphasizes that while angles alone are insufficient, the product of the magnitudes equals the magnitude of the result, thus revealing the full geometric connection between complex multiplication and the complex plane.

🔍 Deeper Insights into Complex Multiplication

This paragraph provides a deeper insight into the process of complex multiplication, highlighting the discovery that multiplying complex numbers involves both the addition of angles and the multiplication of distances from the origin. It emphasizes the equivalence of algebraic rules and the geometric interpretation on the complex plane, showcasing the beauty of mathematics in expressing underlying truths from different perspectives.

📚 Polar Form and Its Simplifications

The final paragraph introduces the polar form of complex numbers, which is an alternative representation using the magnitude (distance from the origin) and the angle from the real axis. It explains that multiplying and dividing complex numbers in polar form is more straightforward than in rectangular form, offering a simplified approach to complex arithmetic. The paragraph concludes with a teaser for the next session, where the utility of this discovery in simplifying algebra problems will be demonstrated.

Keywords

💡Complex Multiplication

Complex multiplication refers to the process of multiplying two complex numbers. In the video, this concept is explored through the lens of geometric representation on the complex plane, where multiplication results in a new complex number whose angle and magnitude are determined by the sum and product of the angles and magnitudes of the original numbers, respectively.

💡Complex Plane

The complex plane is a two-dimensional coordinate system used to represent complex numbers, with the horizontal axis representing the real part and the vertical axis representing the imaginary part. The video uses the complex plane to visually demonstrate the effects of complex multiplication, showing how the multiplication of two complex numbers affects their positions on the plane.

💡Angle

In the context of the video, the angle of a complex number is the angle it makes with the real axis on the complex plane. This angle is crucial for understanding the geometric interpretation of complex multiplication, as the angle of the product of two complex numbers is the sum of their individual angles.

💡Arctangent Function

The arctangent function, often denoted as atan or tan^(-1), is used in trigonometry to find the angle whose tangent is a given number. In the video, it is mentioned as a method to determine the angles of complex numbers on the complex plane, which is essential for understanding the geometric implications of complex multiplication.

💡Magnitude

The magnitude of a complex number is its distance from the origin on the complex plane. The video explains that when multiplying complex numbers, their magnitudes (distances from the origin) are multiplied together to form the magnitude of the resulting complex number.

💡Polar Form

Polar form is an alternative way to represent complex numbers, using their magnitude and angle from the real axis instead of their real and imaginary parts. The video introduces polar form as a simplified method for multiplying and dividing complex numbers, highlighting its advantages over the traditional algebraic approach.

💡Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states the relationship between the sides of a right triangle. In the video, it is used to calculate the distance of complex numbers from the origin on the complex plane, which is necessary for understanding how the magnitudes of complex numbers interact during multiplication.

💡Algebraic Interpretation

Algebraic interpretation refers to the traditional method of performing operations on complex numbers using their real and imaginary components. The video contrasts this approach with the geometric interpretation on the complex plane, showing that both methods yield the same result but through different processes.

💡Geometric Interpretation

Geometric interpretation in the context of the video refers to understanding complex number operations through their representation on the complex plane. This approach allows for a visual understanding of how operations like multiplication and division affect the position and properties of complex numbers.

💡Decomposition

Decomposition, as used in the video, refers to breaking down complex numbers into their constituent parts—magnitude and angle—for the purpose of simplifying multiplication and division in the complex plane. This method leverages the geometric properties of complex numbers to make calculations more intuitive and straightforward.

💡Intuition

Intuition, in the context of the video, refers to the ability to understand or know something immediately, without the need for conscious reasoning. The video suggests that using the complex plane to visualize complex number operations can enhance intuition by providing a more concrete and visual representation of abstract mathematical concepts.

Highlights

Exploration of the connection between complex multiplication and the complex plane.

Use of four examples to illustrate complex multiplication graphically and algebraically.

Introduction of the concept that 'i' is related to rotation on the complex plane.

Utilization of the arctangent function to determine angles in complex numbers.

Observation that the angle of the multiplication result equals the sum of the angles of the multiplicands.

Identification of the need to track both angle and magnitude for a complete description of complex multiplication.

Realization that multiplying by '2i' affects the distance from the origin differently than multiplying by 'i'.

Measurement of distances from the origin using right triangles and the Pythagorean theorem.

Discovery that the product of distances equals the distance of the result from the origin.

Conclusion that complex multiplication involves adding angles and multiplying distances.

Introduction of polar form as an alternative way to represent complex numbers.

Explanation of the polar form's magnitude and angle components.

Simplification of complex number multiplication in polar form by multiplying magnitudes and adding angles.

Simplification of complex number division in polar form by dividing magnitudes and subtracting angles.

Anticipation of using the complex plane to simplify difficult algebra problems in future discussions.

Reflection on the deeper truths in mathematics and its role in expressing universal principles.

Transcripts

Last time, we left off with a real math problem: what is the connection between complex multiplication and the complex plane?

To get to the bottom of this, we'll use the 4 examples we mentioned last time. For each example, we'll plot the two numbers are multiplying together. We'll also compute the result algebraically and add it to each plot.

Our job now is to look for patterns. Back in part 5, we learned that 'i' had something to do with rotation on the complex plane.

So a good thing to keep track of here will be the angle our complex numbers make with the real axis.

We can determine our angles using a little trigonometry, specifically the arctangent function.

Now, let's look for a connection between our three angles.

After a little pondering, we see that the angle of the result is exactly equal to the angles of the numbers are multiplying added together.

This is the first half of the connection we're looking for: when multiplying on the complex plane, the angle of our result is equal to the sum of the angles of the numbers we're multiplying.

Let's now have a closer look at our first 2 examples. Notice that the angles are identical, but the resulting complex numbers are not.

This means that just keeping track of angles alone is not enough to sufficiently describe complex multiplication in the complex plane. There is something else going on.

So what is the difference between these examples?

It looks like multiplying by 2i has pushed our results further from the origin than multiplying by i. A good follow-up question is, "How much further?"

We can measure the distance between the origin and our complex numbers by forming right triangles and using the Pythagorean theorem.

Just as before, let's compute our measurement for each example and look for patterns.

After some more pondering, we see that if we multiply our distances, we obtain the distance from the origin of the result.

We now have the complete picture. When we multiply complex numbers on the complex plane, their angles from the real axis add and their distance from the origin multiply. This is the connection we were looking for between complex multiplication and the complex plane.

We now have completely separate but completely equivalent interpretations of complex multiplication.

To multiply two complex numbers together, we can follow the rules of algebra, or we can find each numbers distance from the origin and angle to the real axis on the complex plane and multiply and add each.

And what's really cool here is that although these approaches look and are totally different, they do the same exact thing.

What we're seeing here is the same underlying process from two separate vantage points.

I really like this idea because it reminds me that there's more to math than what we see on the page. There are deeper truths embedded in our universe, and math is one way of expressing them.

Now that we've made our discovery, let's formalize our results a bit.

We found that the quantities we should keep track of when multiplying complex numbers in the complex plane are the distance from the origin and the angle from the real axis.

These quantities turn out to be so important that we use them as another way to write complex numbers.

Instead of writing complex numbers as the sum of their real and imaginary parts, we instead write them as their distance from the origin and the angle they make with the real axis.

This is called polar form, and the distance from the origin gets a special name 'magnitude.'

Multiplying complex numbers in polar form is super easy. We just multiply the magnitudes and add the angles.

Division is pretty simple too, especially compared to dividing in rectangular form. To divide in polar form, we divide the magnitudes and subtract our angles.

Next time, we'll show that this discovery is not only cool, but useful. We'll use the complex plane to make hard algebra problems easier, faster, and more intuitive.