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Ahmed Nour ✓™
12/6/2025
Math (ACT®)
Complex numbers
77
•
Dec 6, 2025
•
Understanding Complex Numbers: Notes and Insights
Ahmed Nour ✓™
@ahmednour
Understanding Complex Numbers
Complex numbers combine real and imaginary parts in the form a + bi, where i represents the square root of -1. For any complex number z = a + bi, the real component (Re) is a, and the imaginary component (Im) is b. Every complex number has a complex conjugate , found by changing the sign of the imaginary part .
Visually, we represent complex numbers on an Argand diagram, plotting the real component on the x-axis and the imaginary component on the y-axis. Complex conjugates appear as reflections across the real axis, while the modulus (distance from the origin) remains the same for both z and z*.
💡 The modulus-argument form offers a powerful alternative way to express complex numbers. Instead of using rectangular coordinates , we can use polar coordinates with modulus r and argument θ.
Complex numbers can be written in polar form as z = r, where r is the modulus and θ is the argument (angle). This form connects to Euler's formula, which gives us the elegant expression z = re^(iθ). This relationship makes complex multiplication and division much simpler than in Cartesian form.
When performing operations with complex numbers, remember:
- Addition/subtraction: Easier in Cartesian form
- Multiplication/division: Easier in polar form
De Moivre's Theorem states that ^n = cos(nθ) + isin(nθ), providing a powerful tool for finding powers of complex numbers. This theorem works for all integers and can even be extended to rational powers, allowing us to find complex roots of numbers.
Complex Number Applications & Advanced Topics
Complex numbers create fascinating geometric transformations on the Argand diagram. Multiplying by a real number causes an enlargement, while multiplying by i creates a 90° rotation. The complex conjugate reflects points across the real axis, giving us powerful tools for analyzing geometric problems.
When working with complex loci, remember these key patterns:
- |z - a| = c describes a circle
- |z - a| + |z - b| = c gives an ellipse
- |z - a| - |z - b| = c produces a hyperbola
- arg = θ creates a half-line from point a
Complex roots of unity are solutions to equations like z^n = 1. These roots form regular polygons on the Argand diagram. For example, the cube roots of 1 form an equilateral triangle. An important property: the sum of all n roots of unity equals zero.
💡 Looking for the roots of a complex number? Convert to polar form, take the nth root of the modulus, and divide the argument by n. Don't forget to add 2π/n to the argument to find all n distinct roots!
De Moivre's Theorem provides a powerful method for proving trigonometric identities. By expressing trigonometric functions in terms of complex exponentials, we can use algebraic manipulations to derive relationships like cos^4θ = /8.
Complex numbers also appear in infinite series. The expression e^x = Σ extends naturally into the complex plane. This connection allows us to use complex number techniques to evaluate sums like:
S = sinθ + sin2θ + sin3θ + ... + sinnθ
By combining complex exponentials with geometric series, we can transform seemingly difficult trigonometric sums into elegant closed-form expressions. These techniques are essential for advanced calculus, physics, and engineering applications.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
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4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Understanding Complex Numbers: Notes and Insights
Ahmed Nour ✓™
@ahmednour
Complex numbers blend real and imaginary components, opening up a whole new dimension in mathematics. They allow us to solve equations that have no real-number solutions and provide powerful tools for analyzing problems in physics, engineering, and advanced mathematics. Let's... Show more
Understanding Complex Numbers
Complex numbers combine real and imaginary parts in the form a + bi, where i represents the square root of -1. For any complex number z = a + bi, the real component (Re) is a, and the imaginary component (Im) is b. Every complex number has a complex conjugate , found by changing the sign of the imaginary part .
Visually, we represent complex numbers on an Argand diagram, plotting the real component on the x-axis and the imaginary component on the y-axis. Complex conjugates appear as reflections across the real axis, while the modulus (distance from the origin) remains the same for both z and z*.
💡 The modulus-argument form offers a powerful alternative way to express complex numbers. Instead of using rectangular coordinates , we can use polar coordinates with modulus r and argument θ.
Complex numbers can be written in polar form as z = r, where r is the modulus and θ is the argument (angle). This form connects to Euler's formula, which gives us the elegant expression z = re^(iθ). This relationship makes complex multiplication and division much simpler than in Cartesian form.
When performing operations with complex numbers, remember:
- Addition/subtraction: Easier in Cartesian form
- Multiplication/division: Easier in polar form
De Moivre's Theorem states that ^n = cos(nθ) + isin(nθ), providing a powerful tool for finding powers of complex numbers. This theorem works for all integers and can even be extended to rational powers, allowing us to find complex roots of numbers.
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Access to all documents
Improve your grades
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Complex Number Applications & Advanced Topics
Complex numbers create fascinating geometric transformations on the Argand diagram. Multiplying by a real number causes an enlargement, while multiplying by i creates a 90° rotation. The complex conjugate reflects points across the real axis, giving us powerful tools for analyzing geometric problems.
When working with complex loci, remember these key patterns:
- |z - a| = c describes a circle
- |z - a| + |z - b| = c gives an ellipse
- |z - a| - |z - b| = c produces a hyperbola
- arg = θ creates a half-line from point a
Complex roots of unity are solutions to equations like z^n = 1. These roots form regular polygons on the Argand diagram. For example, the cube roots of 1 form an equilateral triangle. An important property: the sum of all n roots of unity equals zero.
💡 Looking for the roots of a complex number? Convert to polar form, take the nth root of the modulus, and divide the argument by n. Don't forget to add 2π/n to the argument to find all n distinct roots!
De Moivre's Theorem provides a powerful method for proving trigonometric identities. By expressing trigonometric functions in terms of complex exponentials, we can use algebraic manipulations to derive relationships like cos^4θ = /8.
Complex numbers also appear in infinite series. The expression e^x = Σ extends naturally into the complex plane. This connection allows us to use complex number techniques to evaluate sums like:
S = sinθ + sin2θ + sin3θ + ... + sinnθ
By combining complex exponentials with geometric series, we can transform seemingly difficult trigonometric sums into elegant closed-form expressions. These techniques are essential for advanced calculus, physics, and engineering applications.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user