Log In Start studying!

Select your language

Suggested languages for you:
Vaia - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

Array C

In the realm of computer programming, understanding the concept and applications of Array C is crucial for effective software development. As a fundamental data structure in Computer Science, Array C plays a significant role in various fields and has several advantages. This article will provide you with comprehensive knowledge on the importance, advantages, and applications of Array C, along with…

Content verified by subject matter experts
Free Vaia App with over 20 million students
Mockup Schule

Explore our app and discover over 50 million learning materials for free.

Array C
Illustration

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmelden

Nie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmelden
Illustration

In the realm of computer programming, understanding the concept and applications of Array C is crucial for effective software development. As a fundamental data structure in Computer Science, Array C plays a significant role in various fields and has several advantages. This article will provide you with comprehensive knowledge on the importance, advantages, and applications of Array C, along with guiding you through basic array types and operations. Additionally, it delves into advanced array topics such as 2D arrays, arrays of pointers, and efficient ways to define arrays in C programming. This knowledge will empower you to excel in your programming endeavours and enhance your software development skills.

Importance of Array C in computer programming

Array C is an essential concept in computer programming that every programmer should be familiar with. It provides a framework for organising and managing data, which is crucial to the smooth functioning of any programming language. Array C is widely used for storing multiple values in a single variable, which can significantly reduce the workload and complexity of code. In this section, we will discuss the importance of arrays in computer programming.

An Array C is a collection of elements, each identified by at least one array index or key, where elements are the same data type.

There are numerous reasons that Array C plays such an important role in computer programming:
  • Organisation: Arrays help organise data in a structured and easy-to-understand manner. They can represent data in multiple dimensions, allowing programmers to easily manage and manipulate information.
  • Efficient storage and retrieval: Arrays allow for efficient storage of data in memory. Since elements are stored in contiguous memory locations, it's easier and faster to access values within the array. This makes data retrieval and manipulation quite efficient and reduces the access times significantly.
  • Code optimisation and readability: Using arrays can lead to shorter and simpler code. By storing multiple data values in a single variable, programmers can avoid creating multiple separate variables. This optimisation results in increased code readability.
  • Flexibility: Arrays can be used in various ways to meet the needs of different programming needs. They can be resized, reshaped and used in conjunction with other data structures or algorithms.

Advantages of using Array C

There are several advantages associated with using Array C in computer programming, which can make it an essential tool for developers. Some of these benefits include:
  • Memory management: Arrays enable efficient allocation and deallocation of memory, which helps in managing memory resources effectively in the program.
  • Time complexity: Array C data structures have a lower time complexity in the case of certain operations such as random access, making them ideal for data processing tasks.
  • Better cache performance: Contiguous memory allocation in arrays can lead to better cache performance and reduced cache misses, providing faster access to the data.
  • Reusability: Array C is a versatile data structure that can be easily reused in various applications and algorithms, making it an invaluable tool for programmers.

Applications of Array C in different fields

Array C is not only limited to programming languages but also finds applications in various fields and industries due to its simplicity and efficiency. Some of these applications include:
  • Scientific modelling: Arrays are widely used in scientific modelling and simulations to represent complex structures and data sets, enabling efficient processing of the data.
  • Data mining: Array C is used in data mining algorithms to efficiently analyze large volumes of data by organising it into specific structures.
  • Computer graphics: In computer graphics, arrays are utilised for managing image data and pixel values, making image processing tasks more manageable.
  • Financial services: Array C finds applications in the financial services industry for tasks such as financial modelling, risk management and portfolio optimisation, among others.
  • Machine learning: In machine learning, arrays are used for representing large data sets and managing complex neural networks effectively.

It's worth noting that Array C is not always the optimal data structure in all scenarios. For instance, growing the size of an array can be a time-consuming task, leading to slower performance. Alternative data structures, such as linked lists, can sometimes offer better solutions depending on the requirements of the specific task.

Basic Array Types and Operations in C

In C programming language, an array of integers is a data structure that can store a fixed number of integer values. This section will explore how to declare, initialise, access, and manipulate arrays of integers in C.

Declaring and initializing an array of integers

To declare an array of integers in C, you need to specify the data type (int), the array name, and the size within square brackets. Here is the syntax for declaring an array of integers:
int array_name[size];
For example, to declare an array of five integers, you can use the following code:
int numbers[5];
To initialise an array of integers, there are a few different methods. The most common method is to assign values during declaration:
int numbers[5] = {1, 2, 3, 4, 5};
You can also assign values individually, using the index:
numbers[0] = 1;
numbers[1] = 2;
numbers[2] = 3;
numbers[3] = 4;
numbers[4] = 5;

Accessing and manipulating integers in an array

To access and manipulate elements in an integer array, use the array name and an index inside square brackets. The index starts at 0 and goes up to the size of the array minus 1. For example, to access the first element in the array, use the following syntax:
int first_number = numbers[0];
You can also perform operations on the elements in the array. For example, to add the first two elements of the array, use the following code:
int sum = numbers[0] + numbers[1];
A common operation is to use loops to iterate and manipulate the elements within the array. For example, to print all elements in the array:
for (int i = 0; i < 5; i++) {
  printf("%d ", numbers[i]);
}

Array string in C

An array string in C is an array of characters, grouped together to represent a sequence of characters or text. In this section, we will discuss how to declare, initialise, and work with string arrays in C.

Declaring and initializing a string array

To declare a string array, you need to specify the data type (char), the array name, and the size within square brackets. Here is the syntax for declaring a string array:
char array_name[size];
For example, to declare a string array of 10 characters:
char greeting[10];
To initialise a string array, you can either assign values during declaration or assign values individually using the index:
char greeting[10] = "Hello";
Or:
greeting[0] = 'H';
greeting[1] = 'e';
greeting[2] = 'l';
greeting[3] = 'l';
greeting[4] = 'o';
greeting[5] = '\0'; // null terminator to indicate the end of the string

Working with strings in an array

Working with strings in an array involves various operations, including accessing individual characters, iterating through the characters, and manipulating them. To access a character in the string array, use the array name and an index within square brackets:
char first_character = greeting[0];
Iteration through the characters in a string array can be done using loops. For example, to print each character on a separate line:
for (int i = 0; greeting[i] != '\0'; i++) {
  printf("%c\n", greeting[i]);
}
Various functions can help work with strings in C, such as strcat (concatenate strings), strlen (length of the string), and strcmp (compare two strings). To use these functions, you will need to include the

library in your program. For example, to find the length of the string:

#include 

int len = strlen(greeting);
printf("Length: %d", len);

Advanced Array Topics in C Programming

A 2D array in C, also known as a "matrix" or a "two-dimensional array", is an array of arrays. It can be visualised as a table with rows and columns, where each cell contains an element. 2D arrays are useful in representing and processing multidimensional data, such as storing data in a grid or matrix format. In this section, we will delve deep into declaring, initializing, and using 2D arrays in various programming tasks.

Declaring and initializing a 2D array

To declare a 2D array in C, you need to specify the data type, the array name, and the size of both dimensions within square brackets. Here is the syntax:

data_type array_name[row_size][column_size];

For example, to declare a 3x4 integer array, you would use the following code:

int matrix[3][4];

To initialize a 2D array, you can assign values during declaration using nested curly braces:

int matrix[3][4] = {
  {1, 2, 3, 4},
  {5, 6, 7, 8},
  {9, 10, 11, 12}
};

Or assign values individually using indices:

matrix[0][0] = 1;
matrix[0][1] = 2;
matrix[0][2] = 3;
matrix[0][3] = 4;
// And so on...

Common use cases of 2D arrays in programming

2D arrays find various applications in computer programming due to their ability to represent and manipulate multidimensional data. Some common use cases include:

  • Image processing: A 2D array can represent and store pixel data for image processing tasks such as filtering, scaling, and transformation.
  • Graphs and networks: Many graph algorithms utilise 2D arrays as adjacency matrices to represent connections between vertices in a graph or a network.
  • Spatial data: In geospatial applications, 2D arrays are commonly utilised to store elevation, temperature, or other spatially distributed data.
  • Matrix operations and linear algebra: 2D arrays are well-suited for representing matrices and performing operations such as addition, subtraction, multiplication, and inversion.
  • Game development: Grid-based games, such as chess or tic-tac-toe, often use 2D arrays to represent the game state and manage game logic.

Array of pointers in C

An array of pointers in C is an array where each element is a pointer to a specific data type. Using arrays of pointers can provide improved efficiency and flexibility in certain situations, such as dynamically allocating memory or when working with multidimensional arrays. In this section, we will explore the concept of pointers in an array and how they can be utilised in programming tasks.

Understanding pointers in an array

To declare an array of pointers, you need to specify the data type followed by an asterisk (*), the array name, and the size within square brackets. Here is the syntax:

data_type *array_name[size];

For example, to declare an array of pointers to integers with a size of 5:

int *ptr_array[5];

Assigning values to an array of pointers can be done by using the address-of operator (&) to assign the address of a variable to each element:

int var1 = 1, var2 = 2, var3 = 3, var4 = 4, var5 = 5;

ptr_array[0] = &var1
ptr_array[1] = &var2
ptr_array[2] = &var3
ptr_array[3] = &var4
ptr_array[4] = &var5

Dereferencing the pointers in an array can access the actual values:

int value1 = *ptr_array[0]; // value1 is now 1

Advantages and applications of using array pointers

Array pointers offer several advantages and can be employed in various situations, including:

  • Dynamic memory allocation: Array pointers can manage memory effectively by allocating and deallocating memory as required during runtime.
  • Flexible array size: Using an array of pointers enables resizing the array during runtime, providing flexibility for various programming tasks.
  • Efficient memory usage: Pointers can reduce memory overhead, especially in large arrays or multidimensional arrays, by only storing the memory addresses of the data elements.
  • Function parameters: Array pointers can be used as parameters in functions, enabling the passing of large arrays efficiently by passing only the memory address.
  • Data structure adaptation: Arrays of pointers are commonly used to create and manipulate advanced data structures such as linked lists and trees.

Define array in C

An array in C is defined as a contiguous block of memory, containing elements of the same data type. Defining an array involves specifying the data type, array name, and size. This section will discuss various methods of defining arrays in C programming and provide tips for efficient array definition and usage.

Methods of defining arrays in C programming

There are several ways to define an array in C:

  • Static allocation: The size and elements of the array are fixed at the time of declaration. Memory for the array is allocated during the compile time.
  • Dynamic allocation: The size of the array can be specified during runtime, allowing for more flexibility. Memory for the array is allocated during the execution of the program.
  • Initialization at declaration: Assigning values to the elements of the array during the declaration process using curly braces and a comma-separated list of values.
  • Value assignment after declaration: Assigning values to the elements of the array individually or using loops after the array has been declared.

Tips for efficient array definition and usage

Here are some tips for defining and using arrays efficiently in C programming:

  • Choose the appropriate array definition method based on your program's requirements. Static allocation is more suitable when you know the size and elements of the array in advance, while dynamic allocation allows for greater flexibility during runtime.
  • Be mindful of array indices, which start at 0 and go up to the size of the array minus 1. Using an index outside the array's bounds can lead to undefined behaviour.
  • Make use of loops for efficient value assignment and manipulation within the array. For loops and while loops are particularly useful for iterating through elements.
  • Familiarise yourself with C standard library functions for array manipulation, such as memcpy(), memset(), and memmove(), which can provide efficient and high-performance solutions for common operations.
  • When working with strings in arrays, always remember to include the null terminator to indicate the end of the string. Neglecting to include a null terminator can result in unpredictable outcomes.

Array C - Key takeaways

  • Array C: Fundamental data structure in computer programming; provides a framework for organising and managing data, essential for efficient software development.

  • Array types: Includes array of integers, array string (array of characters), 2D arrays, and arrays of pointers in C.

  • 2D array in C: Also known as a matrix, it's an array of arrays that stores data in rows and columns, useful for representing multidimensional data.

  • Array of pointers in C: Each array element is a pointer to a specific data type, useful for dynamically allocating memory, efficient storage/retrieval, and working with multidimensional arrays.

  • Defining arrays in C: Can be done using static or dynamic allocation, initialization at declaration, and value assignment after declaration; efficient definition and usage are important for effective programming.

Frequently Asked Questions about Array C

An array in C is a collection of elements of the same data type, stored in contiguous memory locations. Each element can be accessed by its index, which represents its position within the array. Arrays provide a way to group and manage multiple variables with a single identifier. They are useful for organising data and performing repeated operations more efficiently.

To declare an array in C, specify the data type, followed by the array name and its size in square brackets. For example, to declare an integer array of size 5, use the syntax: `int arrayName[5];`.

To declare an array in C, specify the data type, followed by the array name and the size in square brackets. For example, to declare an integer array of size 5, write: `int myArray[5];`. The size should be a constant integer expression, and the array indexing starts from 0.

To create an array in C programming, define the type of the elements, the array name, and the size within square brackets. For example, to create an integer array named "numbers" with size 5, write: int numbers[5]; Then, assign values using the index within square brackets, such as: numbers[0] = 10;

The benefits of arrays in C include efficient memory usage, easy access to elements via indexing, ability to store multiple values of the same data type in one variable, and improved readability and organisation of code.

Final Array C Quiz

Array C Quiz - Teste dein Wissen

Question

What is an Array C?

Show answer

Answer

An Array C is a collection of elements, each identified by at least one array index or key, where elements are the same data type.

Show question

Question

What are the main advantages of using Array C in computer programming?

Show answer

Answer

The main advantages are memory management, lower time complexity, better cache performance, and reusability.

Show question

Question

Why is Array C important for code optimisation and readability?

Show answer

Answer

Using arrays can lead to shorter and simpler code by storing multiple data values in a single variable, which results in increased code readability.

Show question

Question

In which industries and fields can Array C be applied?

Show answer

Answer

Some applications include scientific modelling, data mining, computer graphics, financial services, and machine learning.

Show question

Question

What is one limitation of Array C as a data structure?

Show answer

Answer

Growing the size of an array can be a time-consuming task, leading to slower performance.

Show question

Question

How to declare an array of integers in C?

Show answer

Answer

Specify the data type (int), the array name, and the size within square brackets: int array_name[size];

Show question

Question

How do you initialise an array of integers in C during declaration?

Show answer

Answer

Assign values within curly braces following the declaration: int numbers[5] = {1, 2, 3, 4, 5};

Show question

Question

How do you access and manipulate elements in an integer array in C?

Show answer

Answer

Use the array name and an index inside square brackets: int first_number = numbers[0]; int sum = numbers[0] + numbers[1];

Show question

Question

How to declare a string array in C?

Show answer

Answer

Specify the data type (char), array name, and size within square brackets: char array_name[size];

Show question

Question

How do you initialise a string array in C during declaration?

Show answer

Answer

Assign values within double quotes following the declaration: char greeting[10] = "Hello";

Show question

Question

How do you declare a 2D array in C?

Show answer

Answer

To declare a 2D array in C, you need to specify the data type, the array name, and the size of both dimensions within square brackets. The syntax is `data_type array_name[row_size][column_size];`. For example: `int matrix[3][4];`.

Show question

Question

How are one-dimensional arrays defined in C?

Show answer

Answer

data_type array_name[array_size];

Show question

Question

What is the starting index for an array in C?

Show answer

Answer

The starting index is 0.

Show question

Question

What are some advantages of using one-dimensional arrays in C?

Show answer

Answer

Organization of data, managing large data sets, memory efficiency, and reduced time complexity.

Show question

Question

What are some applications of one-dimensional arrays in C programming?

Show answer

Answer

Sorting algorithms, mathematical computations, searching algorithms, and string manipulation.

Show question

Question

How is an integer array of size 5 declared in C?

Show answer

Answer

int numbers[5];

Show question

Question

What is the first step in implementing a one-dimensional array?

Show answer

Answer

Declare an array by specifying its data type, name, and size within square brackets. Example: int myArray[10];

Show question

Question

What are some best practices for initializing one-dimensional arrays in C?

Show answer

Answer

Specify the data type, choose unique array names, define a constant array size, choose between dynamic or static initialization, use partial initialization, and omit the array size during declaration if initial values are provided.

Show question

Question

How do you add two one-dimensional arrays in C and store the result in a separate array?

Show answer

Answer

Declare and initialize two arrays of the same data type and size, declare a separate result array, use a loop to iterate through all elements, add corresponding elements of each array, and store the sum in the result array.

Show question

Question

How do you perform a linear search for an element in a one-dimensional array in C?

Show answer

Answer

Declare a search value and index variable, use a loop to iterate through the array elements, compare each element against the search value, and update the index variable if a match is found. Break the loop when a match is found.

Show question

Question

What is bubble sort, and how can it be used to sort a one-dimensional array in C?

Show answer

Answer

Bubble sort is a simple sorting algorithm that compares adjacent elements in the array and swaps them if they are in the wrong order. This is done using nested loops, iterating over the array to compare and swap elements as needed, until the array is sorted.

Show question

Question

How can you modify the value of an element in a one-dimensional array in C?

Show answer

Answer

Access the array element using its index and update the value with a new assigned value.

Show question

Question

What is a 2D array in C programming?

Show answer

Answer

A 2D array is a data structure that stores data in a grid format with rows and columns. Each element can be accessed using two indices: the row number and the column number.

Show question

Question

How do you declare a 2D array in C?

Show answer

Answer

Use the syntax `data_type array_name[row_size][column_size];` where `data_type` is the type of data, `array_name` is the name of the 2D array, and `row_size` and `column_size` are the number of rows and columns, respectively.

Show question

Question

What is row-major ordering in C?

Show answer

Answer

Row-major ordering, also known as row-wise storage, is a method for representing 2D arrays in memory such that each row is stored as a contiguous block of memory. C language uses row-major ordering by default.

Show question

Question

What is column-major ordering?

Show answer

Answer

Column-major ordering, also known as column-wise storage, is a method for representing 2D arrays in memory such that each column is stored as a contiguous block of memory.

Show question

Question

How does memory allocation for a 2D array work in C?

Show answer

Answer

Memory allocation for a 2D array in C is contiguous, meaning it is stored in consecutive memory locations. The layout of the 2D array in memory depends on the ordering used, which can be either row-major or column-major.

Show question

Question

What is the formula for matrix multiplication using 2D arrays in C?

Show answer

Answer

\[c_{ij} = \sum_{k=1}^{n} a_{ik} * b_{kj}\]

Show question

Question

How do you declare a 2D array using double pointers in C?

Show answer

Answer

data_type** array_name;

Show question

Question

How do you allocate memory for rows in a dynamically allocated 2D array in C?

Show answer

Answer

array_name = (data_type**)malloc(sizeof(data_type*) * row_size);

Show question

Question

How do you allocate memory for columns in a dynamically allocated 2D array in C?

Show answer

Answer

array_name[i] = (data_type*)malloc(sizeof(data_type) * column_size);

Show question

Question

How do you deallocate memory for a dynamically allocated 2D array in C?

Show answer

Answer

Loop through rows, call free(array_name[i]) for each row, then call free(array_name).

Show question

Question

How to perform matrix addition using 2D arrays in C?

Show answer

Answer

1. Declare and initialize matrices A and B, and their dimensions (m, n). 2. Declare matrix C with dimensions m×n to store the result. 3. Loop through each element in matrix A using variable i ∈ [0, m - 1] and in matrix B using variable j ∈ [0, n - 1]. 4. Compute sum c[i][j] = a[i][j] + b[i][j]. 5. Output matrix C as the result.

Show question

Question

What are the three main techniques for searching and sorting 2D arrays in C?

Show answer

Answer

Linear search, binary search, and selection sort.

Show question

Question

How is linear search implemented in a 2D array?

Show answer

Answer

1. Declare and initialize the 2D array and the target element. 2. Loop through the array using nested loops with variables i ∈ [0, row_size - 1] and j ∈ [0, column_size - 1]. 3. Check if the current element matches the target. If so, return position (i, j). 4. If target is not found, return an appropriate message or value to indicate failure.

Show question

Question

How to implement binary search in a 2D row-sorted array?

Show answer

Answer

1. Declare and initialize 2D row-sorted array and target element. 2. Loop through each row using variable i ∈ [0, row_size - 1]. 3. Perform binary search for each row, keeping track of starting index left (0) and ending index right (column_size - 1). 4. Compute middle index mid = (left + right) / 2. Return position (i, mid) if target is found; update left or right as necessary.

Show question

Question

What are the four common methods to declare and initialize 2D arrays in C?

Show answer

Answer

Static allocation, static allocation with initialization, variable-length array (C99), and dynamic memory allocation with pointers.

Show question

Question

What is an array in C?

Show answer

Answer

An array in C is a collection of elements of the same type, stored in contiguous memory locations, with each element accessed using its index starting from zero.

Show question

Question

What is the difference between an array and an array pointer in C?

Show answer

Answer

An array refers directly to memory locations containing its elements and is immutable, whereas an array pointer is a mutable pointer variable that points to the first element of the array and can be reassigned.

Show question

Question

How is an array passed as a function argument in C?

Show answer

Answer

In C, when passing an array as a function argument, you pass a pointer to the first element of the array, as C does not support passing arrays directly to functions. The array name without square brackets or index is used as the argument.

Show question

Question

What steps should one follow to pass an array as a function argument in C?

Show answer

Answer

1. Specify the array element type with an asterisk (*) or empty square brackets in the function prototype. 2. Match parameter type in the function definition. 3. Pass the array name without square brackets or index when calling. 4. Access array elements using pointer arithmetic and dereferencing inside the function.

Show question

Question

How to pass a 2D array as a function argument in C?

Show answer

Answer

Use a pointer to a pointer (**) in function prototype, pass array name without brackets or index, access elements using pointer arithmetic and dereferencing. Optionally, pass the number of rows and columns as separate arguments.

Show question

Question

What is essential to remember when passing a character array as a function argument in C programming?

Show answer

Answer

It's essential to remember that strings are typically terminated with a NULL character ('\0') that marks the end of the string when passing a character array as a function argument in C programming.

Show question

Question

In C, what do you need to pass instead of the entire character array as a function argument?

Show answer

Answer

In C, you need to pass a pointer to the first character of the array instead of the entire character array as a function argument.

Show question

Question

How do you access the characters in a character array passed as a function argument in C?

Show answer

Answer

To access the characters in a character array passed as a function argument in C, you can dereference the pointer and use pointer arithmetic.

Show question

Question

In the function prototype, how do you specify that the function will receive a pointer to the character array's first element?

Show answer

Answer

In the function prototype, specify the 'char' type and provide an asterisk (*) to indicate that the function will receive a pointer to the character array's first element.

Show question

Question

What happens to an array's size information when it is passed as a function argument in C?

Show answer

Answer

When an array is passed as a function argument in C, it loses its size information. To work with functions that require knowing the size of the array, pass its size as a separate argument.

Show question

Question

How is a character array (string) terminated in C?

Show answer

Answer

A character array (string) in C is terminated with a NULL-terminating character ('\0') which marks the end of the string.

Show question

Question

What's the difference between an array name and a pointer in C?

Show answer

Answer

Array names correspond to the address of the first element and cannot be reassigned, while pointer variables can store any address and can be reassigned. They have different meanings and behaviors in C.

Show question

Question

Which function argument is required when working with 2D arrays?

Show answer

Answer

When working with 2D arrays, you need to use a pointer to a pointer in your function prototype and definition and provide the number of rows and columns as additional arguments.

Show question

60%

of the users don't pass the Array C quiz! Will you pass the quiz?

Start Quiz

How would you like to learn this content?

Creating flashcards
Studying with content from your peer
Taking a short quiz

94% of StudySmarter users achieve better grades.

Sign up for free!

94% of StudySmarter users achieve better grades.

Sign up for free!

How would you like to learn this content?

Creating flashcards
Studying with content from your peer
Taking a short quiz

Free computer-science cheat sheet!

Everything you need to know on . A perfect summary so you can easily remember everything.

Access cheat sheet

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.

Start learning with Vaia, the only learning app you need.

Sign up now for free
Illustration