Math | Margarita Andrianova

Modular arithmetic
1. Divisibility Rules
  1. Pascal's Divisibility Rule
    1. for 11
      1. Simply put, if you split all the digits of a number into two groups – every other digit (one group will have all the digits in odd positions, the other – in even positions), sum up all the digits in each group and subtract one sum from the other, then the remainder when dividing the result by 11 will be the same as for the original number.
2. Theory of Divisibility
  1. basics
    1. trivial divisor
      1. A divisor of n is called a trivial divisor of n if it is either 1 or n itself.
    2. nontrivial divisor
      1. A divisor of n is called a nontrivial divisor if it is a divisor of n, but is neither 1, nor n.
    3. theorems
      1. a - dividend
      2. q - quotient
      3. r - reminder
      4. The Sieve of Eratosthenes
3. Rules
  1. positive integers
  2. addition
  3. multiplication
  4. exponential
4. conguent
  1. /equiv - latex
5. example
  1. (1 * 1) % 5 = (2 * 3) % 5 = (3 * 2) % 5 = = (4 * 4) % 5 = 1
  2. (2 ** (-1)) % 5 = 3 % 5 = 3
  3. (3 ** (-1)) % 5 = 2 % 5 = 2
  4. (4 ** (-1)) % 5 = 4 % 5 = 4
  5. (23 ** (-1)) % 5 = (3 ** (-1)) % 5 = 2
  6. 23 % 5 = 3
  7. (163 ** 42) % 49 = 1
  8. (163 ** (-1)) % 42 = 46
  9. 3 ** 203 % 7 = 5
  10. inverse mod k
  11. A modular inverse of a number is such a natural number that, when multiplied modulo a given number, results in one. The modular inverse modulo m can be computed using the so-called Extended Euclidean Algorithm.
6. tricks
  1. (11 * 4) % 12 = ((-1) * 4 % 12 = (-4) % 12 = 8
  2. (11 ** 7) % 12 = ((-1) ** 7) % 12 = 11
7. p doesn't divide a
8. inverse mod k
9. encription
  1. M - message
  2. p - prime number
  3. e - encryption key
  4. C - scramble message
  5. d - decryption key
    1. private key
    2. Шаги
    3. 1. Два простых числа - 7 и 11
    4. 2. Вычисляем модуль p=7*11=77
    5. 3. Вычисляем функцию Эйлера: k = (7-1) * (11-1) = 6 * 10 = 60
    6. 4. Выбираем число e: простое, меньше k, оно должно быть взаимно простое с k. Выберем e = 43 (открытая экспонента)
    7. Взаимно простые числа — целые числа, не имеющие никаких общих делителей, кроме ±1. Равносильное определение: целые числа взаимно просты, если их наибольший общий делитель равен 1
    8. 5. Вычислить d, обратно по модулю k: (d * e) % k = 1 (d * 43) % 60 = 1 d = 7 или 67
  6. public key
  7. Algorithms
    1. DSA
      1. Digital Signature Algorithm
      2. The Digital Signature Algorithm (DSA) is a widely used asymmetric cryptographic algorithm for creating and verifying digital signatures. It was proposed by the National Institute of Standards and Technology (NIST) as a standard for digital signatures.
      3. DSA algorithm
      4. Key Generation:
      5. Generate a large prime number p, typically 1024 or 2048 bits long, and a smaller prime number q, such that q divides p-1. These primes are public parameters shared by the participants.
      6. Select an integer g, where g is a generator of the multiplicative group of integers modulo p. This group is denoted as Zp*.
      7. Choose a private key x, a randomly selected integer from the range [1, q-1].
      8. Compute the corresponding public key y, where y = g^x mod p.
      9. Signing Process:
      10. Hash the message to be signed using a secure hash function, such as SHA-256, to obtain a fixed-size message digest.
      11. Generate a random number k from the range [1, q-1].
      12. Compute r, where r = (g^k mod p) mod q.
      13. Compute s, where s = ((hash + x*r) * k^(-1) mod q), and hash represents the message digest.
      14. The signature for the message is the pair (r, s).
      15. Signature Verification:
      16. Obtain the public key (y) of the signer from a trusted source.
      17. Hash the received message to obtain the message digest.
      18. Compute w, where w = s^(-1) mod q.
      19. Compute u1, where u1 = (hash * w) mod q.
      20. Compute u2, where u2 = (r * w) mod q.
      21. Compute v, where v = ((g^u1 * y^u2) mod p) mod q.
      22. The signature is valid if and only if v is equal to r.
      23. The DSA algorithm relies on the computational difficulty of computing discrete logarithms to provide security. The private key x must be kept secret, while the public key y can be freely distributed.
      24. DSA is widely used in various cryptographic applications, such as secure communication, digital certificates, and authentication. However, it has been largely replaced by the more efficient and secure Elliptic Curve Digital Signature Algorithm (ECDSA) in many modern systems and protocols.
    2. ECDSA
      1. Elliptic Curve Digital Signature Algorithm
      2. The Elliptic Curve Digital Signature Algorithm (ECDSA) is an asymmetric cryptographic algorithm used for creating and verifying digital signatures. It is based on the mathematical properties of elliptic curves over finite fields. ECDSA provides a high level of security with shorter key lengths compared to other algorithms like RSA.
      3. ECDSA algorithm
      4. Key Generation:
      5. Select an elliptic curve defined over a finite field. The curve parameters, including the equation and base point, must be agreed upon by the communicating parties.
      6. Choose a private key d, a random integer from a specific range defined by the curve's order.
      7. Compute the corresponding public key Q, where Q = d * G, and G is the base point on the elliptic curve.
      8. Signing Process:
      9. Hash the message to be signed using a secure hash function, such as SHA-256, to obtain a fixed-size message digest.
      10. Generate a random number k within the specified range.
      11. Compute the point R = k * G on the elliptic curve, where G is the base point.
      12. Derive the x-coordinate of R as r, where r = R.x mod n, and n is the order of the base point.
      13. Compute the value s, where s = (k^(-1) * (hash + r * d)) mod n.
      14. The signature for the message is the pair (r, s).
      15. Signature Verification:
      16. Obtain the public key Q of the signer from a trusted source.
      17. Hash the received message to obtain the message digest.
      18. Compute the value w, where w = s^(-1) mod n.
      19. Compute the value u1, where u1 = (hash * w) mod n.
      20. Compute the value u2, where u2 = (r * w) mod n.
      21. Compute the point R' = u1 * G + u2 * Q on the elliptic curve.
      22. The signature is valid if and only if the x-coordinate of R' is equal to r.
      23. ECDSA offers strong security and performance advantages over traditional digital signature algorithms. It requires shorter key lengths, making it more efficient in terms of computation and storage. The use of elliptic curves enhances the security of the algorithm.
      24. ECDSA is widely used in various cryptographic applications, such as secure communication protocols, digital certificates, and blockchain technologies. It provides a reliable method for verifying the authenticity and integrity of digital data.
    3. RSA
      1. Rivest-Shamir-Adleman
      2. Rivest-Shamir-Adleman (RSA) is an asymmetric cryptographic algorithm widely used for secure communication, digital signatures, and encryption. It was introduced in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman. RSA relies on the computational difficulty of factoring large integers into their prime factors.
      3. RSA algorithm
      4. Key Generation:
      5. Select two large prime numbers, p and q.
      6. Compute the modulus N as N = p * q.
      7. Compute Euler's totient function ϕ(N) as ϕ(N) = (p-1) * (q-1).
      8. Choose a public exponent e, which is typically a small prime number, coprime to ϕ(N) (i.e., gcd(e, ϕ(N)) = 1).
      9. Compute the private exponent d as the modular multiplicative inverse of e modulo ϕ(N) (i.e., d ≡ e^(-1) (mod ϕ(N))).
      10. The public key is (N, e), while the private key is (N, d).
      11. Encryption:
      12. Convert the plaintext message into a numerical representation (e.g., using ASCII or Unicode encoding).
      13. Split the message into blocks if necessary.
      14. For each block m, compute the ciphertext c as c ≡ m^e (mod N).
      15. The resulting ciphertext represents the encrypted message.
      16. Decryption:
      17. Obtain the private key (N, d).
      18. For each ciphertext block c, compute the plaintext block m as m ≡ c^d (mod N).
      19. If necessary, combine the decrypted blocks to obtain the original plaintext message.
      20. Digital Signatures:
      21. To create a digital signature for a message, the signer uses their private key (N, d).
      22. The signer computes the hash of the message using a secure hash function.
      23. The hash is then encrypted using the private key: s ≡ hash^d (mod N).
      24. The resulting value s represents the digital signature.
      25. Signature Verification:
      26. To verify the signature, the recipient uses the public key (N, e) and the received signature s.
      27. The recipient decrypts the signature using the public key: hash ≡ s^e (mod N).
      28. The recipient computes the hash of the original message using the same secure hash function.
      29. If the computed hash matches the decrypted hash, the signature is valid.
      30. RSA is widely used in various applications due to its security, efficiency, and versatility. However, the security of RSA relies on the difficulty of factoring large integers, and as computing power advances, longer key lengths are required to maintain security against attacks.
  8. Standarts
    1. DES
      1. Data Encryption Standard
      2. The Data Encryption Standard (DES) is a symmetric-key block cipher algorithm that was widely used for encryption and decryption of electronic data. It was developed in the 1970s by IBM and later standardized by the National Institute of Standards and Technology (NIST) in the United States.
      3. DES algorithm
      4. Key Generation
      5. The DES algorithm uses a 56-bit key, which is generated by the user or system.
      6. The key undergoes a process called key schedule, involving permutations and transformations, to generate 16 subkeys of 48 bits each.
      7. Encryption Process
      8. The data to be encrypted is divided into blocks of 64 bits.
      9. Each block goes through an initial permutation (IP) stage.
      10. The permutation rearranges the bits according to a fixed table.
      11. The permuted block is then divided into two halves, left and right, each consisting of 32 bits.
      12. Rounds
      13. The DES encryption process consists of 16 rounds of similar operations.
      14. In each round, the right half of the data block is expanded from 32 bits to 48 bits using an expansion permutation.
      15. The expanded right half is then XORed (bitwise exclusive OR) with a subkey generated during the key schedule.
      16. The XOR result is passed through a series of S-boxes (substitution boxes).
      17. The S-boxes perform a non-linear substitution, replacing each 6-bit input with a 4-bit output.
      18. The outputs of the S-boxes are combined and passed through a permutation called a P-box.
      19. The result of the P-box permutation is XORed with the left half of the data block.
      20. The updated left half becomes the new right half, and the previous right half becomes the new left half.
      21. This process is repeated for 16 rounds, with the subkeys used in a predetermined order.
      22. Final Permutation and Output
      23. After 16 rounds, the left and right halves of the data block are swapped.
      24. The combined block goes through a final permutation (IP inverse).
      25. The output of the final permutation is the encrypted data block.
      26. Decryption
      27. Decryption in DES is essentially the reverse of the encryption process.
      28. The subkeys are used in the reverse order during decryption.
      29. The same operations of expansion, XOR, S-boxes, and permutation are applied, but in the reverse order.
      30. After 16 rounds, the final permutation is applied to obtain the decrypted data block.
      31. DES was widely used for many years as a standard encryption algorithm, but its key size of 56 bits is now considered too short for secure encryption. Consequently, the Advanced Encryption Standard (AES) has largely replaced DES in modern cryptographic applications due to its stronger security and larger key sizes.
    2. AES
      1. Advanced Encryption Standard
      2. The Advanced Encryption Standard (AES) is a symmetric-key encryption algorithm that was selected by the National Institute of Standards and Technology (NIST) in 2001 as a replacement for the Data Encryption Standard (DES). AES has become the most widely used encryption algorithm worldwide due to its strong security and efficiency.
      3. AES algorithm
      4. Key Generation:
      5. AES supports key sizes of 128, 192, or 256 bits.
      6. The key is generated by the user or system and must match the chosen key size.
      7. Encryption Process:
      8. The data to be encrypted is divided into blocks of 128 bits.
      9. AES operates on a fixed number of rounds depending on the key size: 10 rounds for AES-128, 12 rounds for AES-192, and 14 rounds for AES-256.
      10. Each round consists of several transformation stages, including SubBytes, ShiftRows, MixColumns, and AddRoundKey.
      11. SubBytes: Each byte of the data block is substituted using a predefined substitution box (S-box), which provides non-linearity and confusion.
      12. ShiftRows: The bytes in each row of the data block are shifted cyclically to the left.
      13. MixColumns: The columns of the data block are mixed using a matrix multiplication operation to achieve diffusion.
      14. AddRoundKey: The data block is XORed with a round key derived from the main encryption key.
      15. Key Expansion:
      16. The original encryption key is expanded to generate a set of round keys for each round of encryption.
      17. The key expansion algorithm involves a series of transformations, including SubBytes, RotWord, and XOR with a round constant.
      18. The round keys are derived from the main encryption key and used in the AddRoundKey operation during each round.
      19. Decryption:
      20. Decryption in AES is essentially the reverse of the encryption process.
      21. The round keys are used in reverse order during decryption.
      22. The inverse of each transformation stage (InvSubBytes, InvShiftRows, InvMixColumns) is applied in the reverse order.
      23. After the final round, the data block is XORed with the last round key to obtain the decrypted data.
      24. AES provides a high level of security, even against sophisticated attacks, and is widely adopted in various applications, including secure communication, data storage, and digital systems. Its flexibility in supporting different key sizes allows for a balance between security and performance, making it a versatile encryption algorithm.
    3. DSS
      1. Digital Signature Standard
      2. The Digital Signature Standard (DSS) is a standard for digital signatures that was established by the National Institute of Standards and Technology (NIST) in the United States. It specifies the algorithms and protocols to be used for generating and verifying digital signatures.
      3. Components
      4. Digital Signature Algorithm (DSA):
      5. DSA is the core algorithm used for generating and verifying digital signatures in the DSS.
      6. It is based on the mathematical properties of modular exponentiation and the difficulty of solving the discrete logarithm problem in finite fields.
      7. DSA utilizes a specific elliptic curve (FIPS 186-4) or a set of predefined parameters for prime fields (FIPS 186-3).
      8. The algorithm provides a high level of security with relatively short key sizes.
      9. Key Generation:
      10. DSS specifies the key generation process for DSA.
      11. A large prime number p and a smaller prime number q are generated according to specific criteria.
      12. The private key is randomly generated as an integer within a certain range.
      13. The corresponding public key is computed based on the private key and the generated primes.
      14. Signature Generation and Verification:
      15. DSS defines the procedures for generating and verifying digital signatures using DSA.
      16. To generate a signature, the private key holder computes specific mathematical operations based on the message and private key.
      17. The resulting signature consists of two values: r and s.
      18. To verify the signature, the public key holder performs calculations using the received signature, message, and public key.
      19. The verification process ensures that the signature is valid and has not been tampered with.
      20. Key Management and Certification:
      21. DSS provides guidelines for key management and certification, including the storage, backup, and revocation of keys.
      22. It outlines practices for generating, storing, and protecting keys to maintain the security and integrity of digital signatures.
      23. The standard also addresses the use of certificates to establish trust and verify the authenticity of public keys.
      24. The Digital Signature Standard (DSS) is widely adopted and used in various applications where digital signatures are required for authentication, integrity, and non-repudiation of digital data. It ensures the security and interoperability of digital signature systems and plays a crucial role in secure communication, electronic transactions, and data integrity verification.
10. methods
  1. CFRAC
    1. Continued FRACtion method (for factoring)
      1. Choose a number to factorize: Let's say we have a composite number N that we want to factorize.
      2. Choose a quadratic function: The method uses a quadratic function, typically f(x) = (x^2) mod N, to generate a sequence of numbers.
      3. Generate a sequence: We generate a sequence of numbers using the quadratic function and a chosen starting value.
      4. Create fractions: Each number in the sequence is used to create a fraction. The fraction is then converted into a continued fraction.
      5. Find convergents: The convergents (best rational approximations) of the continued fraction are computed.
      6. Check for factorization: For each convergent, we check if the denominator divides N. If it does, we've found a non-trivial factor of N.
      7. Repeat: If no factor is found, the process is repeated with a different quadratic function or a different starting value.
  2. ECM
    1. Elliptic Curve Method (for factoring)
      1. Choose a number to factorize: Let's say we have a composite number N that we want to factorize.
      2. Choose an elliptic curve and a point: We randomly choose an elliptic curve E over the field of integers modulo N, and a point P on E.
      3. Perform point multiplication: We compute kP where k is a product of small primes, and P is the point chosen on the elliptic curve.
      4. Check for factorization: If during the computation of kP, we find a non-trivial divisor of N, then we've found a factor of N.
      5. Repeat: If no factor is found, the process is repeated with a different elliptic curve or a different point.
  3. NFS
    1. Number Field Sieve (for factoring)
      1. Choose a number to factorize: Let's say we have a composite number N that we want to factorize.
      2. Polynomial selection: We choose two polynomials f(x) and g(x) such that f(x) is irreducible over the integers and g(x) is a simpler polynomial, typically linear. The root of f(x) modulo N should be a root of g(x) modulo N.
      3. Sieving: We search for values of x such that both f(x) and g(x) are B-smooth, meaning all their factors are less than some bound B. These values of x are used to form a matrix over the field with two elements.
      4. Matrix reduction: We use linear algebra techniques to find a nontrivial kernel of this matrix. This gives us a set of x values such that the product of f(x) is a square modulo N and the product of g(x) is a square in the integers.
  4. QS/MPQS
    1. Quadratic Sieve/Multiple Polynomial Quadratic Sieve (for factoring)
      1. Square root computation: We compute these square roots and use them to find a factor of N.
      2. Select a smoothness bound B.
      3. Choose a quadratic polynomial f(x) = (ax + b)^2 - N.
      4. Sieve using a set of primes p1, p2, ..., pk, such that their squares are smaller than or equal to B:
      5. Compute f(x) modulo pi for x = 0 to B.
      6. If f(x) is divisible by pi, divide f(x) by pi until it is not divisible anymore.
      7. Record the exponent of pi.
      8. difference
      9. QS
      10. In the QS algorithm, a single quadratic polynomial is selected, and sieving is performed using this polynomial. The goal is to find x values for which the polynomial evaluates to a perfect square modulo N.
      11. MPQS
      12. On the other hand, MPQS extends the QS algorithm by using multiple quadratic polynomials. A set of quadratic polynomials is chosen, and sieving is performed using each of these polynomials. The aim is still to find x values for which each polynomial evaluates to a perfect square modulo N.
      13. In summary, the key difference between QS and MPQS is that QS uses a single quadratic polynomial for sieving, while MPQS employs multiple quadratic polynomials to increase the probability of finding suitable x values for factorization.
      14. Construct a matrix "relations" to store the relationships between smooth values.
      15. For each smooth value (x, exponent):
      16. For each prime pi:
      17. Compute pi^exponent modulo N and store it in the "relations" matrix.
      18. Use Gaussian elimination or other linear algebra techniques to find equations summing up to zero modulo 2.
      19. Solve the system of equations to find a set of x values.
      20. Compute the product of the corresponding f(x) values.
      21. If the product is a perfect square modulo N, you have found non-trivial factors of N.
      22. If no factors are found, repeat with different quadratic polynomials until successful or a limit is reached.
      23. Multiple Polynomial Quadratic Sieve (MPQS)
  5. ECPP
    1. Elliptic Curve Primality Proving
      1. Select a random elliptic curve E defined over a finite field of size N.
      2. Choose a random point P on the elliptic curve E.
      3. Generate a random prime q such that q is small enough for efficient computation and q does not divide N.
      4. Compute the point Q = [q]P, where [q] denotes the scalar multiplication of the point P by the integer q.
      5. If the point Q is the identity element on the elliptic curve (i.e., Q = O, where O denotes the point at infinity), return to step 2 and choose a different random point P.
      6. Compute the order r of the point Q. The order r is the smallest positive integer such that [r]Q = O.
      7. If r does not divide N, return to step 2 and choose a different random point P.
      8. Verify the primality of N using the primality test for r. This step involves checking if r is a prime number using a separate primality-testing algorithm.
      9. If the primality test for r determines that r is composite, return to step 2 and choose a different random point P.
      10. Repeat steps 2 to 9 until a prime r is found.
      11. Perform a set of consistency checks to ensure that the computed values are correct and that N is a prime number.
      12. If all the checks pass, conclude that N is a prime number. Otherwise, repeat the process with a different elliptic curve or different random point P.
Number systems
1. Conversion from Decimal
  1. Whole Part
    1. Sequentially divide by the base, record the remainder as the new digit of the number, and then divide the result again by the base. Write the digits in reverse order (the first remainder is the last digit of the number).
  2. Fraction
    1. Sequentially multiply by the base, record the whole part as a digit, and multiply the fractional part again by the base. Write the digits in the order obtained.
    2. 0.515625 = 0.41 (8)
    3. 0.515625 * 8 = 4.125
      1. > 4
    4. 0.125 * 8 = 1
      1. > 1
2. Binary
  1. Fractions
    1. 0.1 = 0.5
      1. 1/2
    2. 0.01 = 0.25
      1. 1/4
    3. 0.001 = 0.125
      1. 1/8
    4. 0.0001 = 0.0625
      1. 1/16
  2. To Decimal
    1. 0.75 = 0 + 7 * (1/8) + 5 * (1/64)
3. Octal
  1. Conversion from Binary using 3 bits
    1. 10 101.101110 = 25.56
  2. Fractions
    1. 0.1 = 0.125
      1. 1/8
    2. 0.01 = 1/64
4. Hexadecimal
  1. Conversion from Binary using 4 bits
    1. 1.01 = 1.4
    2. 10 0111 0010 = 272
  2. Fractions
    1. 0.1 = 0.0625
      1. 1/16
  3. Multiplication
    1. FAF9 * 6FFD = 16AF6
5. Numbers
  1. Numbers
    1. 1
      1. 01
      2. 1
      3. 1
    2. 2
      1. 10
      2. 2
      3. 2
    3. 3
      1. 11
      2. 3
      3. 3
    4. 4
      1. 100
      2. 4
      3. 4
    5. 5
      1. 101
      2. 5
      3. 5
    6. 6
      1. 110
      2. 6
      3. 6
    7. 7
      1. 111
      2. 7
      3. 7
    8. 8
      1. 1000
      2. 10
      3. 8
    9. 9
      1. 1001
      2. 11
      3. 9
    10. 10
      1. 1010
      2. 12
      3. A
    11. 11
      1. 1011
      2. 13
      3. B
    12. 12
      1. 1100
      2. 14
      3. C
    13. 13
      1. 1101
      2. 15
      3. D
    14. 14
      1. 1110
      2. 16
      3. E
    15. 15
      1. 1111
      2. 17
      3. F
    16. 16
      1. 10 000
      2. 20
      3. 10
Sequences and series
1. Arithmetic progression
2. Geometric progression
3. module (random)
4. Series
  1. It is the sum of the terms of a sequence.
    1. Arithmetic progression
      1. The sum of all terms of an arithmetic progression is equal to half the product of the sum of its extreme terms and the number of all its terms.
      2. sum of all positive odd numbers
    2. Geometric progression
      1. Examples
      2. The sum of n terms of a geometric progression with denominator q != 1 is equal to the quotient of dividing the difference between the product of the last term by the denominator of the progression and the first term by the difference between the denominator of the progression and one.
    3. Sum of squares
    4. Sum of cubes
    5. Examples
      1. Triangular numbers
    6. decomposition
      1. sum decomposition
      2. moving multiplier
      3. don't do this for n
5. limit
  1. sequences limit
  2. convergents
    1. geometric series
    2. The limit of geometric series
    3. harmonic series
    4. Cauchy Criterion
      1. English
    5. Root test
    6. Ratio test
    7. additional
      1. convergence of a sequence between two others
      2. conditions
    8. Binomial Theorem
    9. limits
6. Popular
  1. Bernoulli
    1. Bernoulli numbers are a sequence of rational numbers which are deeply connected to number theory. They appear in the series expansions of trigonometric functions, in formulas for the sum of powers of the first n positive integers, in the Euler-Maclaurin formula, and many other areas in mathematics.
  2. Fermat
  3. Mersenne primes
7. compound percent
  1. FV is the future value of the annuity.
  2. P is the monthly deposit (annuity payment).
  3. r is the monthly interest rate (as a decimal).
  4. n is the total number of payments (or months, in this case).
Functions and graphs
1. Graphs
  1. Coordinates
    1. Addition
      1. Q(1,2) P(-1, 3) Q+P(0, 5) Q-P(2,-1)
  2. Plane
  3. Cartesian Coordinates
    1. System of two perpendicular axes, x,y to map and label points on the plane
    2. origin
      1. (0, 0)
  4. Formulas
    1. distance between points
    2. midpoint
  5. description
    1. Tangent to curve
      1. The gradient of a curve at any point is equal to the gradient of the tangent at that point
    2. Asymptote
      1. an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
    3. intersections with axes
      1. x = 0
      2. y = 0
    4. symmetry
  6. types
    1. Straight line
      1. Any straight line has an equation of the form y = mx + c where m and c are constants
      2. m
      3. In the equation y = mx + c the value m is known as the gradient and is a measure of the steepness of the line
      4. or slope
      5. m
      6. if the point (a, b) lies on the line y = mx + c then equation is satisfied by letting x = a and y = b
      7. parallel
      8. perpendicular
      9. two-point form
    2. base changing
      1. a - vertical dilation
      2. b - horizontal dilation
      3. c - horizontal translation
      4. d - vertical translation
    3. Quadratic function
      1. y = 0
    4. Cubic function
    5. fractional
    6. higher order polynomials
    7. circle
      1. center: (h, k)
2. Function
  1. Definitions
    1. Domain of a function
      1. elements of set X on which f is defined
    2. Codomain of a function
      1. elements of Y linked by f to X
      2. range or image
    3. Intervals
      1. closed
      2. An interval that includes its end-points is called a closed interval
      3. [1, 3]
      4. open
      5. Any interval that does not include its end-points is called an open interval
      6. strictly greater
      7. strictly less
      8. (1, 3)
      9. semi-open/semi-closed
      10. [1, 3)
    4. types
      1. Surjective function
      2. to each y of set Y is associated at least one element x of set X
      3. onto
      4. vertical line test
      5. Injective function
      6. to each x of set X is associated only one distinct y of set Y
      7. one-to-one
      8. horizontal line test
    5. variants
      1. A non-injective surjective
      2. An injective surjective
      3. An injective non-surjective
      4. A non-injective non-surjective
  2. inverse function
    1. graph
3. Kinematics
  1. variables
    1. u - initial velocity
    2. v - final velocity
    3. S - distance
  2. formulas
  3. graphs
4. Useful
5. Exponential
  1. for all a, y intercept of 1, that is the graph passes through (0,1)
  2. a > 1, the function is increasing
  3. a < 1, the function id decreasing
  4. for all a, the function is positive
  5. the x-axis is an asymptote
  6. e
    1. 2.71828
    2. natural exponential function
6. Logarithms
  1. Types
    1. Natural logarithm
      1. the inverse of the natural exponential function
    2. Logarithm function with base a
      1. the domain
      2. the range
    3. Logarithm with base 10
  2. Properties
    1. Product Rule
    2. Quotient Rule
    3. Reciprocal Rule
    4. Power Rule
    5. Inverse properties
      1. Base a
      2. Base e
      3. Common
    6. Change of base
      1. Every logarithmic function is a constant multiple of the natural logarithm
  3. Graphs
    1. for all a, x intercept of 1, that is the graph passes through (1,0)
    2. for all a, the graph passes through (a, 1)
    3. a > 1, the function is increasing
    4. a < 1, the function id decreasing
    5. for all a, the function is positive
    6. the y-axis is an asymptote
    7. the function defined for a > 0 and x > 0
    8. for a > 1 the bigger a is more slowly the function increases
    9. for a < 1 the smaller a is the more slowly the function decreases
7. Limits and differentiation
  1. Limit of sequence
    1. If limit exists finite, the sequence is convergent
    2. If limit doesn't exists the sequence is said to be divergent
  2. Laws
    1. Sum Law
      1. The limit of a sum is the sum of the limits
    2. Difference Law
      1. The limit of the difference is the difference of the limits
    3. Constant Multiple Law
      1. The limit if a constant times a function is the constant times the limit of the function
    4. Product Law
      1. The limit if a product is the product of the limits
    5. Quotient Law
      1. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0)
  3. Limit and continuity of a function
    1. Discontinious
  4. Derivative of a function
    1. The derivative of a function is the limit of the ratio of the change in the function to the change in its argument, provided that the change in the argument approaches zero.
    2. Slope
      1. Slope shows the change in y or the change on the vertical axis versus the change in x or the change on the horizontal axis.
    3. Gradient
    4. Rules
      1. Trigonometric
    5. L'Hôpital's rule
    6. Derivate and study of a function
      1. Max and Min
      2. Second test
      3. Concavity test
Introduction
1. Symbols
  1. The Greek alphabet
  2. Latex
    1. Sets
      1. \mathbb{letter}
      2. Real number
2. Vocabulary
  1. Prime numbers
    1. It is a positive integer, larger than 1, which cannot be expressed as the product of two smaller positive integers
    2. 2, 3, 5, 7, 11, 13, 13, 17, 19, 23
  2. factor
    1. 3 * 4 = 12
      1. 3 and 4 are factors of 12
      2. when a number is written as a product of prime numbers we say the number has been factorised
    2. Highest common factor
      1. h.c.f
      2. greatest common divisor
      3. g.c.d
    3. Lowest common factor
  3. fraction
    1. fraction = numerator/denominator = p /q
    2. proper fraction
      1. p < q
    3. improper fraction
      1. p > q
    4. inverted
      1. q / p
    5. reciprocal
      1. reciprocal = inverted fraction
      2. The reciprocal of a number is found by inverting it, so, for example, the reciprocal of 4/5 is 5/4
    6. equivalent fractions
    7. simplest form
      1. when there are no factors common to both numerator and denominator
    8. common denominator
      1. q / a and p / a. a - common denominator
    9. mixed fraction
      1. whole number and fraction part
    10. least common multiply
      1. l.c.m
      2. In Mathematics, the LCM of any two is the value that is evenly divisible by the two given numbers. The full form of LCM is Least Common Multiple. It is also called the Least Common Divisor
      3. l.c.d
    11. decimal
      1. decimal point
      2. first decimal place
      3. number of significant figures
      4. number of decimal places
      5. rounded
      6. rounded up
      7. rounded down
    12. percentage
      1. percentage change
    13. ratio
      1. Ratios are simply an alternative way of expressing fratcions
      2. Divide 170 in the ratio 3 : 2
      3. 3/5 of 170
      4. 102
      5. 2/5 of 170
      6. 68
      7. Divide 250 cm in the ratio 1 : 3 : 4
      8. 1/8
  4. BODMAS
    1. Brackets
      1. ()
    2. Of
      1. x
    3. Division
      1. /
      2. numerator
      3. denumerator
      4. quotient
    4. Multiplication
      1. *
    5. Addition
      1. +
    6. Substruction
      1. -
  5. Algebra
    1. superscript
      1. power
      2. index
      3. y
      4. indices
      5. plural
      6. laws of indices
      7. the first law
      8. base
      9. x
      10. negative powers
      11. fractional powers
      12. scientific notation
      13. quadratic expressions
      14. a and b - coefficients
      15. constant term
    2. subscript
      1. root
    3. substitution
      1. Substitution means replacing letters by actual numerical values
    4. formula
      1. A formula is used to relate two or more quantities
      2. subject
      3. transpose
      4. If we asked to transpose formula for r, then we must rearrange the formula so that r becomes the subject
    5. like terms
      1. Like terms are multiples of the same quantity
      2. Like terms can be collected together and added or subtracted in order to simplify them
    6. fraction
      1. partial fractions
      2. it is a part of the original fraction
      3. linear factor
      4. ax + b
      5. repeated linear factor
      6. quadratic factor
    7. equations
      1. unknown quantity
      2. solve
      3. solution
      4. root of the equation
      5. satisfy the equation
      6. linear equations
      7. ax + b = 0
      8. b - constant term
      9. simultaneous equations
      10. eliminating
      11. quadratic equations
      12. discriminant
      13. > 0
      14. distinct real roots
      15. = 0
      16. repeated root
      17. equal roots
    8. verbs
      1. evaluate
      2. simplify
      3. express
      4. factorise
      5. determine
      6. obtain
    9. inequalities
      1. x > y
      2. y < 5
  6. sequence
    1. term
    2. finite sequence
    3. infinite sequence
      1. limit
      2. converge
      3. When a sequence possesses a limit it is said converge
      4. diverge
    4. arithmetic progressions
      1. common difference
    5. geometric progressions
      1. common ratio
    6. series
      1. sigma notation
      2. arithmetic series
      3. geometric series
  7. set
    1. A set is a collection of clearly defined objects, things or states
    2. {...}
    3. finite set
    4. infinite set
    5. equal sets
    6. subset
    7. union
      1. Venn diagrams
    8. number sets
  8. Number bases
    1. decimal system
    2. binary system
    3. octal system
    4. hexadecimal system
  9. elementary logic
    1. symbolic logic
    2. negation
      1. The negation of a proposition is the proposition that is true whenever the original proposition is false and false when the original is true
      2. not
    3. conjunction
      1. Given any two propositions we can form their conjunction
      2. and
    4. disjunction
      1. or
    5. implication
      1. if then
    6. compound proposition
Trigonometry
1. angle
  1. measure
    1. degree
      1. 360
    2. minutes
      1. 60
    3. seconds
      1. 60
    4. radian
  2. types
    1. right
      1. 90
    2. flat
      1. 180
    3. complete
      1. 360
2. triangles
  1. properties
  2. types
    1. similar triangles
    2. Isosceles
      1. two sides and two angles are equal
    3. equilateral
      1. all sides and angles are equal
    4. right triangles
      1. sides
      2. h - hypotenuse
      3. opposite
      4. adjacent
      5. properties
    5. scalene
      1. all three sides are different
  3. General
    1. cosine
    2. sine
    3. tangent
    4. cotangent
    5. trigonometrical ratios
    6. formulas
      1. the cosine rule
      2. the sine rules
      3. secant
      4. cosecant
      5. cotangent
      6. more
      7. Cofunction identities
      8. Double angles
      9. Even/odd
      10. Half angles
      11. Reciprocal functions
      12. Power reducing formulas
      13. Product to sum
      14. Pythagorean identities
      15. Sum and difference of angles
      16. Sum to product
    7. Circle view
  4. projections
3. Functions
  1. The sine finction
  2. The cosine function
  3. the tangent function
Vectors and Matrices
1. Common
  1. Vector space
    1. A vector space or a linear space is a group of objects called vectors, added collectively and multiplied (“scaled”) by numbers, called scalars.
  2. Properties
    1. Associatibity
    2. Commutativity
    3. Identity
    4. Inverse
    5. Compatibility
    6. Distributivity
  3. Examples
    1. Euclidean vector
    2. line
    3. plane
  4. Operations
    1. scalar product of vectors
      1. The scalar product of two vectors is defined as the product of the magnitudes of the two vectors and the cosine of the angles between them
      2. The Dot Product
    2. Cross Product
      1. The scalar product is the product of vectors which gives a scalar quantity whereas the vector product is the product of vectors which gives a vector quantity as the product.
      2. parallelogram area
  5. length
  6. unit vector
    1. A unit vector is a vector whose length is 1
2. Linear independence
  1. In the theory of vector spaces, a set of vectors is said to be linearly independent if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be linearly dependent. These concepts are central to the definition of dimension.
3. Basis
  1. A basis for a vector space is a sequence of vectors that form a set that is linearly independent and that spans the space.
4. Linear Transformations and Matrices
  1. Vector Rotations
    1. Clockwise rotation
  2. Linear
  3. Matrix
    1. Multiplication
      1. To perform multiplication of two matrices, we should make sure that the number of columns in the 1st matrix is equal to the rows in the 2nd matrix. Therefore, the resulting matrix product will have a number of rows of the 1st matrix and a number of columns of the 2nd matrix.
      2. sum i row A * j column B
    2. Determinant of a matrix
      1. The determinant of a matrix is the scalar value or number calculated using a square matrix.
    3. Inverse Transformation
      1. Inverse matrix is obtained by dividing the adjugate of the given matrix by the determinant of the given matrix.
    4. Systems of equations and matrices
      1. Example
      2. A system of equations can be represented by an augmented matrix.
      3. In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
      4. In this way, we can see that augmented matrices are a shorthand way of writing systems of equations.
      5. Gauss Jordan elimination
      6. Gauss-Jordan Elimination is an algorithm that can be used to solve systems of linear equations and to find the inverse of any invertible matrix.
Probability
1. Probability
  1. Total outcomes
    1. x
      1. variants
    2. n
      1. times
  2. or
  3. and
  4. Independent events
    1. The outcome of one event does not affect the other
  5. Dependent events
    1. The outcome of one event affects the other
    2. conditional probability
    3. probability that, given the occurrence of B, A occurs as well
    4. example
    5. A = clubs
    6. B = king
    7. A and B = king of clubs
2. Combinatorics
  1. Permutations
    1. the permutation of r elements
    2. in a set of total n elements
    3. if we don't want to count more than once the groups (couples) that are different only for by the order we must divide by r!
    4. Binomial distribution
    5. Permutations of multisets
3. Statistics
  1. frequency
  2. mean
    1. The mean is the average of all the numbers in a dataset.
    2. You sum up all the numbers and then divide by the total count.
      1. value
      2. amount of value
  3. median
    1. The median is the middle value when you arrange the numbers in ascending order. If there's an even number of values, you take the mean of the two middle numbers.
    2. Median=Middle value in sorted list
  4. variance
    1. Variance measures how spread out the numbers are from the mean.
  5. standard deviation
    1. The standard deviation is the square root of the variance. It's useful because it's in the same units as the data.
  6. mode
    1. Mode=Most frequently occurring value(s)
    2. The mode is the number that appears most frequently in a dataset. A dataset can have zero or more modes. If no number repeats, it's called "no mode." If there are multiple numbers that appear most frequently, it's "multimodal."
  7. Normal distribution (Gaussian distribution)
    1. Key Points
      1. Symmetry
      2. It's symmetrical around the mean, which means the left and right sides are mirror images of each other.
      3. Mean, Median, Mode
      4. In a perfect normal distribution, the mean, median, and mode are all the same and located at the center of the curve.
      5. Standard Deviation
      6. The "width" of the bell curve is determined by the standard deviation (σ). A larger σ means a wider curve, and a smaller σ means a narrower curve.
      7. 68-95-99.7 Rule
      8. About 68% of the data falls within one standard deviation of the mean.
      9. About 95% falls within two standard deviations.
      10. About 99.7% falls within three standard deviations.
    2. Mathematical Expression
      1. μ is the mean
      2. σ is the standard deviation
  8. Chebyshev's Theorem
    1. Chebyshev's Theorem states that no matter what the shape of the distribution, at least a certain percentage of the data must lie within k standard deviations (σ) of the mean (μ).
    2. Key Points
      1. Applicability
      2. Unlike the 68-95-99.7 rule for normal distributions, Chebyshev's Theorem applies to any distribution shape.
      3. k Value
      4. k must be greater than 1. The larger the k, the higher the percentage of data within that range.