Complex numbers and their applications form a crucial foundation in ... Show more
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Subjects
Careers
Energy transfers (a2 only)
Infection and response
Homeostasis and response
Responding to change (a2 only)
Cell biology
Organisms respond to changes in their internal and external environments (a-level only)
Biological molecules
Organisation
Substance exchange
Bioenergetics
Genetic information & variation
Inheritance, variation and evolution
Genetics & ecosystems (a2 only)
Ecology
Cells
Show all topics
World war two & the holocaust
Medieval period: 1066 -1509
1f industrialisation and the people: britain, c1783-1885
2m wars and welfare: britain in transition, 1906-1957
1l the quest for political stability: germany, 1871-1991
Britain & the wider world: 1745 -1901
2d religious conflict and the church in england, c1529-c1570
The cold war
1c the tudors: england, 1485-1603
Inter-war germany
2j america: a nation divided, c1845-1877
2n revolution and dictatorship: russia, 1917-1953
2s the making of modern britain, 1951-2007
World war one
Britain: 1509 -1745
Show all topics
4,462
•
30 Jun 2022
•
Complex numbers and their applications form a crucial foundation in ... Show more
The study of Complex numbers A Level further Maths builds upon familiar number systems by introducing imaginary components. Starting from natural numbers and progressing through integers, rational numbers, and real numbers, complex numbers emerge when we encounter equations like x² = -4 that have no real solutions.
A complex number takes the form z = x + iy, where x represents the real part, y represents the imaginary part, and i is defined as √-1. This structure allows us to solve previously "impossible" equations and opens up new mathematical possibilities. For example, the quadratic equation x² - 2x + 10 = 0 has complex solutions x = 1 ± 3i, demonstrating how complex numbers extend beyond the real number line.
Definition: A complex number z = x + iy consists of a real part x and an imaginary part y, where i² = -1. The real and imaginary parts are both real numbers.
When working with A level Further Maths quadratic equations, complex solutions often appear when the discriminant is negative. This is a key indicator that the solutions will involve imaginary numbers. For instance, in the equation 2z² + 4z + 13 = 0, the solutions are -2 ± 3i, showing how complex numbers naturally arise in quadratic problems.
Understanding complex conjugates is crucial for Complex numbers A Level notes. The complex conjugate of a number z = x + iy, written as z*, is obtained by keeping the real part the same while negating the imaginary part, giving z* = x - iy.
Example: For z = 2 + 7i, its complex conjugate z* = 2 - 7i
Arithmetic operations with complex numbers follow specific rules. Addition and subtraction are performed by combining like terms. Multiplication requires careful attention to the fact that i² = -1, which leads to interesting patterns in powers of i:
Highlight: When multiplying complex numbers, remember that i² = -1 is the fundamental relationship that simplifies all calculations.
Division of complex numbers requires a special technique involving complex conjugates. To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator. This eliminates the imaginary terms in the denominator and simplifies the division.
When solving equations involving complex numbers, it's often helpful to equate the real and imaginary parts separately. This technique creates a system of equations that can be solved simultaneously. For example, when solving iz + 2i = 1 - 4i, we let z = x + iy and separate the equation into its real and imaginary components.
Vocabulary: The process of equating real and imaginary parts is called "comparing coefficients" and is a fundamental technique in complex number manipulation.
Argand diagram plotter tools help visualize complex numbers on a two-dimensional plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. This geometric representation allows us to understand complex numbers as points or position vectors in a plane.
The modulus |z| represents the magnitude of a complex number - the distance from the origin to the point representing the complex number. The argument is the angle between the positive real axis and the line from the origin to the point, measured counterclockwise.
Definition: The modulus of a complex number z = x + iy is given by |z| = √, while the argument is the angle θ = tan⁻¹ with appropriate adjustments for different quadrants.
Complex conjugates have a beautiful geometric interpretation on the Argand diagram - they are reflections of each other across the real axis. This visual representation helps in understanding the relationship between a complex number and its conjugate.
Complex numbers and matrices form essential components of A Level Further Maths complex numbers studies. This comprehensive guide breaks down key concepts and provides detailed explanations with practical applications.
The modulus-argument form of complex numbers allows us to represent numbers in polar coordinates, combining magnitude and direction. When expressing complex numbers in this format, we use the formula z = r, where r represents the modulus and θ represents the argument.
Definition: The modulus-argument form expresses a complex number using its magnitude and angle , written as z = r
For example, when converting √3 + i into modulus-argument form, we follow these steps:
Matrix operations form another crucial topic in further mathematics. When working with matrices, understanding multiplication rules and special matrices like the identity matrix is essential for solving complex problems.
Example: Matrix Multiplication
(3 2) × (4 9) = (3×4 + 2×8 3×9 + 2×6)
(5 7) (8 6) (5×4 + 7×8 5×9 + 7×6)
Matrix transformations play a vital role in A Level Further Maths quadratic equations and geometric applications. Understanding how matrices affect vectors and points in space helps visualize mathematical concepts.
The transpose operation flips a matrix across its leading diagonal, converting rows into columns and vice versa. This operation is crucial for many matrix applications and properties.
Highlight: The leading diagonal remains unchanged during matrix transposition, while all other elements swap positions across this line.
Matrix multiplication dimensions must be compatible - the number of columns in the first matrix must equal the number of rows in the second matrix. This rule is fundamental for all matrix operations and transformations.
Vocabulary: The identity matrix is a square matrix with 1s on the leading diagonal and 0s elsewhere, satisfying the property AI = IA = A for any matrix A.
When working with matrices in Complex numbers A Level further Maths, understanding their geometric interpretations becomes crucial. Matrix transformations can represent various geometric operations like reflections, rotations, and scaling.
The identity matrix serves as the multiplicative identity for matrices, similar to how 1 functions for regular numbers. For any matrix A, multiplying by the identity matrix results in the original matrix: AI = IA = A.
Example: Identity Matrix Properties
I₂ = (1 0) For any 2×2 matrix A:
(0 1) AI₂ = I₂A = A
Matrix operations with column vectors allow us to represent and transform points in space efficiently. These transformations are particularly useful in computer graphics and engineering applications.
Matrices can represent various geometric transformations in the plane, making them essential tools for Argand diagram applications and coordinate geometry. Understanding how matrices transform points helps visualize complex mathematical operations.
For example, the matrix represents a reflection in the x-axis, changing the sign of all y-coordinates while leaving x-coordinates unchanged. This transformation is particularly useful in solving geometric problems and understanding symmetry.
Definition: A transformation matrix is a matrix that, when multiplied by a position vector, produces a new vector representing the transformed position.
The study of matrix transformations connects abstract algebra with geometric visualization, providing powerful tools for solving complex mathematical problems and understanding spatial relationships.
In Complex numbers A Level Further Maths, matrix transformations play a crucial role in representing geometric operations. When dealing with rotations in the complex plane, we can express these transformations using 2×2 matrices that preserve the essential properties of the original vectors.
Definition: A rotation matrix is a transformation that rotates points in a plane by a specified angle θ about the origin. For any angle θ, the standard rotation matrix is:
[cos θ -sin θ]
[sin θ cos θ]
For a 180-degree rotation specifically, which is commonly encountered in A Level Further Maths complex numbers questions, the transformation becomes particularly elegant. Since cos = -1 and sin = 0, the rotation matrix simplifies to:
[-1 0]
[0 -1]
When applying these transformations, it's essential to understand how they affect vectors in the complex plane. The original vector's length remains unchanged during rotation - a property known as isometry. This concept is fundamental in Complex numbers A Level further Maths and appears frequently in examination scenarios.
Example: Consider rotating the point by 180 degrees:
[-1 0] [1] = [-1]
[0 -1] [0] [0]
This transformation maps the point to , exactly as expected for a half-turn about the origin.
Understanding matrix transformations in Complex numbers A Level notes extends beyond simple rotations. These transformations form the foundation for more advanced topics like linear mappings and geometric transformations in the complex plane.
The relationship between complex numbers and matrices becomes particularly evident when working with Argand diagram representations. Each point in the complex plane can be transformed using matrix operations, allowing us to visualize how geometric transformations affect entire regions of the complex plane.
Highlight: Key properties of rotation matrices:
When solving problems involving Locus of complex numbers solutions, understanding these matrix transformations becomes invaluable. They provide a systematic approach to tracking how points move under various geometric operations and help in visualizing complex loci problems.
Vocabulary: Important terms:
App Store
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Complex numbers and their applications form a crucial foundation in A Level Further Maths complex numbers studies, extending beyond basic real number operations into a richer mathematical landscape.
The study begins with understanding the fundamental concept of i = √-1... Show more
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The study of Complex numbers A Level further Maths builds upon familiar number systems by introducing imaginary components. Starting from natural numbers and progressing through integers, rational numbers, and real numbers, complex numbers emerge when we encounter equations like x² = -4 that have no real solutions.
A complex number takes the form z = x + iy, where x represents the real part, y represents the imaginary part, and i is defined as √-1. This structure allows us to solve previously "impossible" equations and opens up new mathematical possibilities. For example, the quadratic equation x² - 2x + 10 = 0 has complex solutions x = 1 ± 3i, demonstrating how complex numbers extend beyond the real number line.
Definition: A complex number z = x + iy consists of a real part x and an imaginary part y, where i² = -1. The real and imaginary parts are both real numbers.
When working with A level Further Maths quadratic equations, complex solutions often appear when the discriminant is negative. This is a key indicator that the solutions will involve imaginary numbers. For instance, in the equation 2z² + 4z + 13 = 0, the solutions are -2 ± 3i, showing how complex numbers naturally arise in quadratic problems.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Understanding complex conjugates is crucial for Complex numbers A Level notes. The complex conjugate of a number z = x + iy, written as z*, is obtained by keeping the real part the same while negating the imaginary part, giving z* = x - iy.
Example: For z = 2 + 7i, its complex conjugate z* = 2 - 7i
Arithmetic operations with complex numbers follow specific rules. Addition and subtraction are performed by combining like terms. Multiplication requires careful attention to the fact that i² = -1, which leads to interesting patterns in powers of i:
Highlight: When multiplying complex numbers, remember that i² = -1 is the fundamental relationship that simplifies all calculations.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Division of complex numbers requires a special technique involving complex conjugates. To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator. This eliminates the imaginary terms in the denominator and simplifies the division.
When solving equations involving complex numbers, it's often helpful to equate the real and imaginary parts separately. This technique creates a system of equations that can be solved simultaneously. For example, when solving iz + 2i = 1 - 4i, we let z = x + iy and separate the equation into its real and imaginary components.
Vocabulary: The process of equating real and imaginary parts is called "comparing coefficients" and is a fundamental technique in complex number manipulation.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Argand diagram plotter tools help visualize complex numbers on a two-dimensional plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. This geometric representation allows us to understand complex numbers as points or position vectors in a plane.
The modulus |z| represents the magnitude of a complex number - the distance from the origin to the point representing the complex number. The argument is the angle between the positive real axis and the line from the origin to the point, measured counterclockwise.
Definition: The modulus of a complex number z = x + iy is given by |z| = √, while the argument is the angle θ = tan⁻¹ with appropriate adjustments for different quadrants.
Complex conjugates have a beautiful geometric interpretation on the Argand diagram - they are reflections of each other across the real axis. This visual representation helps in understanding the relationship between a complex number and its conjugate.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Complex numbers and matrices form essential components of A Level Further Maths complex numbers studies. This comprehensive guide breaks down key concepts and provides detailed explanations with practical applications.
The modulus-argument form of complex numbers allows us to represent numbers in polar coordinates, combining magnitude and direction. When expressing complex numbers in this format, we use the formula z = r, where r represents the modulus and θ represents the argument.
Definition: The modulus-argument form expresses a complex number using its magnitude and angle , written as z = r
For example, when converting √3 + i into modulus-argument form, we follow these steps:
Matrix operations form another crucial topic in further mathematics. When working with matrices, understanding multiplication rules and special matrices like the identity matrix is essential for solving complex problems.
Example: Matrix Multiplication
(3 2) × (4 9) = (3×4 + 2×8 3×9 + 2×6)
(5 7) (8 6) (5×4 + 7×8 5×9 + 7×6)
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Matrix transformations play a vital role in A Level Further Maths quadratic equations and geometric applications. Understanding how matrices affect vectors and points in space helps visualize mathematical concepts.
The transpose operation flips a matrix across its leading diagonal, converting rows into columns and vice versa. This operation is crucial for many matrix applications and properties.
Highlight: The leading diagonal remains unchanged during matrix transposition, while all other elements swap positions across this line.
Matrix multiplication dimensions must be compatible - the number of columns in the first matrix must equal the number of rows in the second matrix. This rule is fundamental for all matrix operations and transformations.
Vocabulary: The identity matrix is a square matrix with 1s on the leading diagonal and 0s elsewhere, satisfying the property AI = IA = A for any matrix A.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When working with matrices in Complex numbers A Level further Maths, understanding their geometric interpretations becomes crucial. Matrix transformations can represent various geometric operations like reflections, rotations, and scaling.
The identity matrix serves as the multiplicative identity for matrices, similar to how 1 functions for regular numbers. For any matrix A, multiplying by the identity matrix results in the original matrix: AI = IA = A.
Example: Identity Matrix Properties
I₂ = (1 0) For any 2×2 matrix A:
(0 1) AI₂ = I₂A = A
Matrix operations with column vectors allow us to represent and transform points in space efficiently. These transformations are particularly useful in computer graphics and engineering applications.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Matrices can represent various geometric transformations in the plane, making them essential tools for Argand diagram applications and coordinate geometry. Understanding how matrices transform points helps visualize complex mathematical operations.
For example, the matrix represents a reflection in the x-axis, changing the sign of all y-coordinates while leaving x-coordinates unchanged. This transformation is particularly useful in solving geometric problems and understanding symmetry.
Definition: A transformation matrix is a matrix that, when multiplied by a position vector, produces a new vector representing the transformed position.
The study of matrix transformations connects abstract algebra with geometric visualization, providing powerful tools for solving complex mathematical problems and understanding spatial relationships.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
In Complex numbers A Level Further Maths, matrix transformations play a crucial role in representing geometric operations. When dealing with rotations in the complex plane, we can express these transformations using 2×2 matrices that preserve the essential properties of the original vectors.
Definition: A rotation matrix is a transformation that rotates points in a plane by a specified angle θ about the origin. For any angle θ, the standard rotation matrix is:
[cos θ -sin θ]
[sin θ cos θ]
For a 180-degree rotation specifically, which is commonly encountered in A Level Further Maths complex numbers questions, the transformation becomes particularly elegant. Since cos = -1 and sin = 0, the rotation matrix simplifies to:
[-1 0]
[0 -1]
When applying these transformations, it's essential to understand how they affect vectors in the complex plane. The original vector's length remains unchanged during rotation - a property known as isometry. This concept is fundamental in Complex numbers A Level further Maths and appears frequently in examination scenarios.
Example: Consider rotating the point by 180 degrees:
[-1 0] [1] = [-1]
[0 -1] [0] [0]
This transformation maps the point to , exactly as expected for a half-turn about the origin.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Understanding matrix transformations in Complex numbers A Level notes extends beyond simple rotations. These transformations form the foundation for more advanced topics like linear mappings and geometric transformations in the complex plane.
The relationship between complex numbers and matrices becomes particularly evident when working with Argand diagram representations. Each point in the complex plane can be transformed using matrix operations, allowing us to visualize how geometric transformations affect entire regions of the complex plane.
Highlight: Key properties of rotation matrices:
When solving problems involving Locus of complex numbers solutions, understanding these matrix transformations becomes invaluable. They provide a systematic approach to tracking how points move under various geometric operations and help in visualizing complex loci problems.
Vocabulary: Important terms:
App Store
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user