Проще говоря: если разбить все цифры числа на 2 группы – через одну цифру (в одну группу попадут все цифры с нечетными позициями, в другую – с четными), сложить все цифры в каждой группе и вычесть две полученные суммы друг из друга, то остаток от деления на 11 результата будет такой же, что и у первоначального числа.
Theory of Divisibility
basics
trivial divisor
A divisor of n is called a trivial divisor of n if it is either 1 or n itself.
nontrivial divisor
A divisor of n is called a nontrivial divisor if it is a divisor of n, but is neither 1, nor n.
Инверсия по модулю – такое натуральное число, которое при умножении по модулю на данное число дает в результате единицу. Инверсию по модулю m можно вычислять с помощью так называемого обобщенного алгоритма Евклида.
4. Выбираем число e: простое, меньше k, оно должно быть взаимно простое с k. Выберем e = 43 (открытая экспонента)
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
5. Вычислить d, обратно по модулю k:
(d * e) % k = 1
(d * 43) % 60 = 1
d = 7 или 67
public key
Algorithms
DSA
Digital Signature Algorithm
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.
DSA algorithm
Key Generation:
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.
Select an integer g, where g is a generator of the multiplicative group of integers modulo p. This group is denoted as Zp*.
Choose a private key x, a randomly selected integer from the range [1, q-1].
Compute the corresponding public key y, where y = g^x mod p.
Signing Process:
Hash the message to be signed using a secure hash function, such as SHA-256, to obtain a fixed-size message digest.
Generate a random number k from the range [1, q-1].
Compute r, where r = (g^k mod p) mod q.
Compute s, where s = ((hash + x*r) * k^(-1) mod q), and hash represents the message digest.
The signature for the message is the pair (r, s).
Signature Verification:
Obtain the public key (y) of the signer from a trusted source.
Hash the received message to obtain the message digest.
Compute w, where w = s^(-1) mod q.
Compute u1, where u1 = (hash * w) mod q.
Compute u2, where u2 = (r * w) mod q.
Compute v, where v = ((g^u1 * y^u2) mod p) mod q.
The signature is valid if and only if v is equal to r.
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.
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.
ECDSA
Elliptic Curve Digital Signature Algorithm
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.
ECDSA algorithm
Key Generation:
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.
Choose a private key d, a random integer from a specific range defined by the curve's order.
Compute the corresponding public key Q, where Q = d * G, and G is the base point on the elliptic curve.
Signing Process:
Hash the message to be signed using a secure hash function, such as SHA-256, to obtain a fixed-size message digest.
Generate a random number k within the specified range.
Compute the point R = k * G on the elliptic curve, where G is the base point.
Derive the x-coordinate of R as r, where r = R.x mod n, and n is the order of the base point.
Compute the value s, where s = (k^(-1) * (hash + r * d)) mod n.
The signature for the message is the pair (r, s).
Signature Verification:
Obtain the public key Q of the signer from a trusted source.
Hash the received message to obtain the message digest.
Compute the value w, where w = s^(-1) mod n.
Compute the value u1, where u1 = (hash * w) mod n.
Compute the value u2, where u2 = (r * w) mod n.
Compute the point R' = u1 * G + u2 * Q on the elliptic curve.
The signature is valid if and only if the x-coordinate of R' is equal to r.
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.
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.
RSA
Rivest-Shamir-Adleman
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.
RSA algorithm
Key Generation:
Select two large prime numbers, p and q.
Compute the modulus N as N = p * q.
Compute Euler's totient function ϕ(N) as ϕ(N) = (p-1) * (q-1).
Choose a public exponent e, which is typically a small prime number, coprime to ϕ(N) (i.e., gcd(e, ϕ(N)) = 1).
Compute the private exponent d as the modular multiplicative inverse of e modulo ϕ(N) (i.e., d ≡ e^(-1) (mod ϕ(N))).
The public key is (N, e), while the private key is (N, d).
Encryption:
Convert the plaintext message into a numerical representation (e.g., using ASCII or Unicode encoding).
Split the message into blocks if necessary.
For each block m, compute the ciphertext c as c ≡ m^e (mod N).
The resulting ciphertext represents the encrypted message.
Decryption:
Obtain the private key (N, d).
For each ciphertext block c, compute the plaintext block m as m ≡ c^d (mod N).
If necessary, combine the decrypted blocks to obtain the original plaintext message.
Digital Signatures:
To create a digital signature for a message, the signer uses their private key (N, d).
The signer computes the hash of the message using a secure hash function.
The hash is then encrypted using the private key: s ≡ hash^d (mod N).
The resulting value s represents the digital signature.
Signature Verification:
To verify the signature, the recipient uses the public key (N, e) and the received signature s.
The recipient decrypts the signature using the public key: hash ≡ s^e (mod N).
The recipient computes the hash of the original message using the same secure hash function.
If the computed hash matches the decrypted hash, the signature is valid.
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.
Standarts
DES
Data Encryption Standard
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.
DES algorithm
Key Generation
The DES algorithm uses a 56-bit key, which is generated by the user or system.
The key undergoes a process called key schedule, involving permutations and transformations, to generate 16 subkeys of 48 bits each.
Encryption Process
The data to be encrypted is divided into blocks of 64 bits.
Each block goes through an initial permutation (IP) stage.
The permutation rearranges the bits according to a fixed table.
The permuted block is then divided into two halves, left and right, each consisting of 32 bits.
Rounds
The DES encryption process consists of 16 rounds of similar operations.
In each round, the right half of the data block is expanded from 32 bits to 48 bits using an expansion permutation.
The expanded right half is then XORed (bitwise exclusive OR) with a subkey generated during the key schedule.
The XOR result is passed through a series of S-boxes (substitution boxes).
The S-boxes perform a non-linear substitution, replacing each 6-bit input with a 4-bit output.
The outputs of the S-boxes are combined and passed through a permutation called a P-box.
The result of the P-box permutation is XORed with the left half of the data block.
The updated left half becomes the new right half, and the previous right half becomes the new left half.
This process is repeated for 16 rounds, with the subkeys used in a predetermined order.
Final Permutation and Output
After 16 rounds, the left and right halves of the data block are swapped.
The combined block goes through a final permutation (IP inverse).
The output of the final permutation is the encrypted data block.
Decryption
Decryption in DES is essentially the reverse of the encryption process.
The subkeys are used in the reverse order during decryption.
The same operations of expansion, XOR, S-boxes, and permutation are applied, but in the reverse order.
After 16 rounds, the final permutation is applied to obtain the decrypted data block.
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.
AES
Advanced Encryption Standard
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.
AES algorithm
Key Generation:
AES supports key sizes of 128, 192, or 256 bits.
The key is generated by the user or system and must match the chosen key size.
Encryption Process:
The data to be encrypted is divided into blocks of 128 bits.
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.
Each round consists of several transformation stages, including SubBytes, ShiftRows, MixColumns, and AddRoundKey.
SubBytes: Each byte of the data block is substituted using a predefined substitution box (S-box), which provides non-linearity and confusion.
ShiftRows: The bytes in each row of the data block are shifted cyclically to the left.
MixColumns: The columns of the data block are mixed using a matrix multiplication operation to achieve diffusion.
AddRoundKey: The data block is XORed with a round key derived from the main encryption key.
Key Expansion:
The original encryption key is expanded to generate a set of round keys for each round of encryption.
The key expansion algorithm involves a series of transformations, including SubBytes, RotWord, and XOR with a round constant.
The round keys are derived from the main encryption key and used in the AddRoundKey operation during each round.
Decryption:
Decryption in AES is essentially the reverse of the encryption process.
The round keys are used in reverse order during decryption.
The inverse of each transformation stage (InvSubBytes, InvShiftRows, InvMixColumns) is applied in the reverse order.
After the final round, the data block is XORed with the last round key to obtain the decrypted data.
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.
DSS
Digital Signature Standard
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.
Components
Digital Signature Algorithm (DSA):
DSA is the core algorithm used for generating and verifying digital signatures in the DSS.
It is based on the mathematical properties of modular exponentiation and the difficulty of solving the discrete logarithm problem in finite fields.
DSA utilizes a specific elliptic curve (FIPS 186-4) or a set of predefined parameters for prime fields (FIPS 186-3).
The algorithm provides a high level of security with relatively short key sizes.
Key Generation:
DSS specifies the key generation process for DSA.
A large prime number p and a smaller prime number q are generated according to specific criteria.
The private key is randomly generated as an integer within a certain range.
The corresponding public key is computed based on the private key and the generated primes.
Signature Generation and Verification:
DSS defines the procedures for generating and verifying digital signatures using DSA.
To generate a signature, the private key holder computes specific mathematical operations based on the message and private key.
The resulting signature consists of two values: r and s.
To verify the signature, the public key holder performs calculations using the received signature, message, and public key.
The verification process ensures that the signature is valid and has not been tampered with.
Key Management and Certification:
DSS provides guidelines for key management and certification, including the storage, backup, and revocation of keys.
It outlines practices for generating, storing, and protecting keys to maintain the security and integrity of digital signatures.
The standard also addresses the use of certificates to establish trust and verify the authenticity of public keys.
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.
methods
CFRAC
Continued FRACtion method (for factoring)
Choose a number to factorize: Let's say we have a composite number N that we want to factorize.
Choose a quadratic function: The method uses a quadratic function, typically f(x) = (x^2) mod N, to generate a sequence of numbers.
Generate a sequence: We generate a sequence of numbers using the quadratic function and a chosen starting value.
Create fractions: Each number in the sequence is used to create a fraction. The fraction is then converted into a continued fraction.
Find convergents: The convergents (best rational approximations) of the continued fraction are computed.
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.
Repeat: If no factor is found, the process is repeated with a different quadratic function or a different starting value.
ECM
Elliptic Curve Method (for factoring)
Choose a number to factorize: Let's say we have a composite number N that we want to factorize.
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.
Perform point multiplication: We compute kP where k is a product of small primes, and P is the point chosen on the elliptic curve.
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.
Repeat: If no factor is found, the process is repeated with a different elliptic curve or a different point.
NFS
Number Field Sieve (for factoring)
Choose a number to factorize: Let's say we have a composite number N that we want to factorize.
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.
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.
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.
Square root computation: We compute these square roots and use them to find a factor of N.
Select a smoothness bound B.
Choose a quadratic polynomial f(x) = (ax + b)^2 - N.
Sieve using a set of primes p1, p2, ..., pk, such that their squares are smaller than or equal to B:
Compute f(x) modulo pi for x = 0 to B.
If f(x) is divisible by pi, divide f(x) by pi until it is not divisible anymore.
Record the exponent of pi.
difference
QS
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.
MPQS
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.
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.
Construct a matrix "relations" to store the relationships between smooth values.
For each smooth value (x, exponent):
For each prime pi:
Compute pi^exponent modulo N and store it in the "relations" matrix.
Use Gaussian elimination or other linear algebra techniques to find equations summing up to zero modulo 2.
Solve the system of equations to find a set of x values.
Compute the product of the corresponding f(x) values.
If the product is a perfect square modulo N, you have found non-trivial factors of N.
If no factors are found, repeat with different quadratic polynomials until successful or a limit is reached.
Multiple Polynomial Quadratic Sieve (MPQS)
ECPP
Elliptic Curve Primality Proving
Select a random elliptic curve E defined over a finite field of size N.
Choose a random point P on the elliptic curve E.
Generate a random prime q such that q is small enough for efficient computation and q does not divide N.
Compute the point Q = [q]P, where [q] denotes the scalar multiplication of the point P by the integer q.
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.
Compute the order r of the point Q. The order r is the smallest positive integer such that [r]Q = O.
If r does not divide N, return to step 2 and choose a different random point P.
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.
If the primality test for r determines that r is composite, return to step 2 and choose a different random point P.
Repeat steps 2 to 9 until a prime r is found.
Perform a set of consistency checks to ensure that the computed values are correct and that N is a prime number.
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
Перевод из десятичной
целая часть
последовательно делим на основание, записываем остаток от деления как новую цифру числа, результат снова делим на основание. Цифры записываем в обратном порядке (первый остаток - последняя цифра числа)
дробь
Последовательно умножаем на основание, записываем целую часть как цифру, а нецелую часть снова умножаем на основание. Цифры записываем в прямом порядке
0.515625 = 0.41 (8)
0.515625 * 8 = 4.125
> 4
0.125 * 8 = 1
> 1
Двоичная
Дроби
0.1 = 0.5
1/2
0.01 = 0.25
1/4
0.001 = 0.125
1/8
0.0001 = 0.0625
1/16
в десятичную
0.75 = 0 + 7 * (1/8) + 5 * (1/64)
Восьмеричная
Перевод из двоичной по 3 разряда
10 101.101110 = 25.56
дроби
0.1 = 0.125
1/8
0.01 = 1/64
Шестнадцатиричная
Перевод из двоичной по 4 разряда
1.01 = 1.4
10 0111 0010 = 272
дроби
0.1 = 0.0625
1/16
умножение
FAF9 * 6FFD = 16AF6
Числа
Числа
1
01
1
1
2
10
2
2
3
11
3
3
4
100
4
4
5
101
5
5
6
110
6
6
7
111
7
7
8
1000
10
8
9
1001
11
9
10
1010
12
A
11
1011
13
B
12
1100
14
C
13
1101
15
D
14
1110
16
E
15
1111
17
F
16
10 000
20
10
Последовательности (sequences)
арифметическая
геометрическая
module (random)
Series
Это сумма членов последовательности
пример арифметической
Сумма всех членов арифметической прогрессии равна половине произведения суммы её крайних членов на количество всех её членов.
сумма всех положительных нечетных чисел = n**2
геометрической
Примеры
Сумма n членов геометрической прогрессии со знаменателем q != 1 равна частному от деления разности между произведением последнего члена на знаменатель прогрессии и первым членом на разность между знаменателем прогрессии и единицей.
Сумма квадратов
Сумма кубов
примеры
Треугольные числа
decomposition
sum decomposition
moving multiplier
don't do this for n
limit
стремление последовательности - ее предел
сходимость (convergent)
geometric series
The limit of geometric series
harmonic series
Критерий Коши
Root test
Ratio test
дополнительно
сходимость последовательности между двумя другими
условие
Binomial Theorem
limits
Popular
Bernoulli
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.
Fermat
Mersenne primes
Functions and graphs
Graphs
Координаты
сложение
Q(1,2)
P(-1, 3)
Q+P(0, 5)
Q-P(2,-1)
плоскость
Cartesian Coordinates
System of two perpendicular axes, x,y to map and label points on the plane
origin
(0, 0)
Formulas
distance between points
midpoint
description
Tangent to curve
The gradient of a curve at any point is equal to the gradient of the tangent at that point
Asymptote
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.
intersections with axes
x = 0
y = 0
symmetry
types
Straight line
Any straight line has an equation of the form y = mx + c where m and c are constants
m
In the equation y = mx + c the value m is known as the gradient and is a measure of the steepness of the line
or slope
m
if the point (a, b) lies on the line y = mx + c then equation is satisfied by letting x = a and y = b
parallel
perpendicular
two-point form
base changing
a - vertical dilation
b - horizontal dilation
c - horizontal translation
d - vertical translation
Quadratic function
y = 0
Cubic function
fractional
higher order polynomials
circle
center: (h, k)
Function
Definitions
Domain of a function
elements of set X on which f is defined
Codomain of a function
elements of Y linked by f to X
range or image
Intervals
closed
An interval that includes its end-points is called a closed interval
[1, 3]
open
Any interval that does not include its end-points is called an open interval
strictly greater
strictly less
(1, 3)
semi-open/semi-closed
[1, 3)
types
Surjective function
to each y of set Y is associated at least one element x of set X
onto
vertical line test
Injective function
to each x of set X is associated only one distinct y of set Y
one-to-one
horizontal line test
variants
A non-injective surjective
An injective surjective
An injective non-surjective
A non-injective non-surjective
inverse function
graph
Kinematics
variables
u - initial velocity
v - final velocity
S - distance
formulas
graphs
Useful
Exponential
for all a, y intercept of 1, that is the graph passes through (0,1)
a > 1, the function is increasing
a < 1, the function id decreasing
for all a, the function is positive
the x-axis is an asymptote
e
2.71828
natural exponential function
Logarithms
Types
Natural logarithm
the inverse of the natural exponential function
Logarithm function with base a
the domain
the range
Logarithm with base 10
Properties
Product Rule
Quotient Rule
Reciprocal Rule
Power Rule
Inverse properties
Base a
Base e
Common
Change of base
Every logarithmic function is a constant multiple of the natural logarithm
Graphs
for all a, x intercept of 1, that is the graph passes through (1,0)
for all a, the graph passes through (a, 1)
a > 1, the function is increasing
a < 1, the function id decreasing
for all a, the function is positive
the y-axis is an asymptote
the function defined for a > 0 and x > 0
for a > 1 the bigger a is more slowly the function increases
for a < 1 the smaller a is the more slowly the function decreases
Limits and differentiation
Limit of sequence
If limit exists finite, the sequence is convergent
If limit doesn't exists the sequence is said to be divergent
Laws
Sum Law
The limit of a sum is the sum of the limits
Difference Law
The limit of the difference is the difference of the limits
Constant Multiple Law
The limit if a constant times a function is the constant times the limit of the function
Product Law
The limit if a product is the product of the limits
Quotient Law
The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0)
Limit and continuity of a function
Discontinious
Derivative of a function
Производная функции - это предел отношения приращения функции к приращению ее аргумента, при условии, что приращение аргумента стремится к нулю
Slope
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.
Gradient
Rules
Trigonometric
L'Hôpital's rule
Derivate and study of a function
Max and Min
Second test
Concavity test
Introduction
Symbols
The Greek alphabet
Latex
Vocabulary
Prime numbers
It is a positive integer, larger than 1, which cannot be expressed as the product of two smaller positive integers
2, 3, 5, 7, 11, 13, 13, 17, 19, 23
factor
3 * 4 = 12
3 and 4 are factors of 12
when a number is written as a product of prime numbers we say the number has been factorised
Highest common factor
h.c.f
greatest common divisor
g.c.d
Lowest common factor
fraction
fraction = numerator/denominator = p /q
proper fraction
p < q
improper fraction
p > q
inverted
q / p
reciprocal
reciprocal = inverted fraction
The reciprocal of a number is found by inverting it, so, for example, the reciprocal of 4/5 is 5/4
equivalent fractions
simplest form
when there are no factors common to both numerator and denominator
common denominator
q / a and p / a. a - common denominator
mixed fraction
whole number and fraction part
least common multiply
l.c.m
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
l.c.d
decimal
decimal point
first decimal place
number of significant figures
number of decimal places
rounded
rounded up
rounded down
percentage
percentage change
ratio
Ratios are simply an alternative way of expressing fratcions
Divide 170 in the ratio 3 : 2
3/5 of 170
102
2/5 of 170
68
Divide 250 cm in the ratio 1 : 3 : 4
1/8
BODMAS
Brackets
()
Of
x
Division
/
numerator
denumerator
quotient
Multiplication
*
Addition
+
Substruction
-
Algebra
superscript
power
index
y
indices
plural
laws of indices
the first law
base
x
negative powers
fractional powers
scientific notation
quadratic expressions
a and b - coefficients
constant term
subscript
root
substitution
Substitution means replacing letters by actual numerical values
formula
A formula is used to relate two or more quantities
subject
transpose
If we asked to transpose formula for r, then we must rearrange the formula so that r becomes the subject
like terms
Like terms are multiples of the same quantity
Like terms can be collected together and added or subtracted in order to simplify them
fraction
partial fractions
it is a part of the original fraction
linear factor
ax + b
repeated linear factor
quadratic factor
equations
unknown quantity
solve
solution
root of the equation
satisfy the equation
linear equations
ax + b = 0
b - constant term
simultaneous equations
eliminating
quadratic equations
discriminant
> 0
distinct real roots
= 0
repeated root
equal roots
verbs
evaluate
simplify
express
factorise
determine
obtain
inequalities
x > y
y < 5
sequence
term
finite sequence
infinite sequence
limit
converge
When a sequence possesses a limit it is said converge
diverge
arithmetic progressions
common difference
geometric progressions
common ratio
series
sigma notation
arithmetic series
geometric series
set
A set is a collection of clearly defined objects, things or states
{...}
finite set
infinite set
equal sets
subset
union
Venn diagrams
number sets
Number bases
decimal system
binary system
octal system
hexadecimal system
elementary logic
symbolic logic
negation
The negation of a proposition is the proposition that is true whenever the original proposition is false and false when the original is true
not
conjunction
Given any two propositions we can form their conjunction
and
disjunction
or
implication
if then
compound proposition
Trigonometry
angle
measure
degree
360
minutes
60
seconds
60
radian
types
right
90
flat
180
complete
360
triangles
properties
types
similar triangles
Isosceles
two sides and two angles are equal
equilateral
all sides and angles are equal
right triangles
sides
h - hypotenuse
opposite
adjacent
properties
scalene
all three sides are different
General
cosine
sine
tangent
cotangent
trigonometrical ratios
formulas
the cosine rule
the sine rules
secant
cosecant
cotangent
more
Cofunction identities
Double angles
Even/odd
Half angles
Reciprocal functions
Power reducing formulas
Product to sum
Pythagorean identities
Sum and difference of angles
Sum to product
Circle view
projections
Functions
The sine finction
The cosine function
the tangent function
Vectors and Matrices
Common
Vector space
A vector space or a linear space is a group of objects called vectors, added collectively and multiplied (“scaled”) by numbers, called scalars.
Properties
Associatibity
Commutativity
Identity
Inverse
Compatibility
Distributivity
Examples
Euclidean vector
line
plane
Operations
scalar product of vectors
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
The Dot Product
Cross Product
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.
parallelogram area
length
unit vector
A unit vector is a vector whose length is 1
Linear independence
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.
Basis
A basis for a vector space is a sequence of vectors that form a set that is linearly independent and that spans the space.
Linear Transformations and Matrices
Vector Rotations
Clockwise rotation
Linear
Matrix
Multiplication
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.
sum i row A * j column B
Determinant of a matrix
The determinant of a matrix is the scalar value or number calculated using a square matrix.
Inverse Transformation
Inverse matrix is obtained by dividing the adjugate of the given matrix by the determinant of the given matrix.
Systems of equations and matrices
Example
A system of equations can be represented by an augmented matrix.
In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
In this way, we can see that augmented matrices are a shorthand way of writing systems of equations.
Gauss Jordan elimination
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.
Latex
Sets
\mathbb{letter}
Real number
Probability
Probability
Total outcomes
x
variants
n
times
or
and
Independent events
The outcome of one event does not affect the other
Dependent events
The outcome of one event affects the other
conditional probability
probability that, given the occurrence of B, A occurs as well
example
A = clubs
B = king
A and B = king of clubs
Combinatorics
Permutations
the permutation of r elements
in a set of total n elements
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!
Binomial distribution
Statistics
frequency
mean
The mean is the average of all the numbers in a dataset.
You sum up all the numbers and then divide by the total count.
value
amount of value
median
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.
Median=Middle value in sorted list
variance
Variance measures how spread out the numbers are from the mean.
standard deviation
The standard deviation is the square root of the variance. It's useful because it's in the same units as the data.
mode
Mode=Most frequently occurring value(s)
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."
Normal distribution (Gaussian distribution)
Key Points
Symmetry
It's symmetrical around the mean, which means the left and right sides are mirror images of each other.
Mean, Median, Mode
In a perfect normal distribution, the mean, median, and mode are all the same and located at the center of the curve.
Standard Deviation
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.
68-95-99.7 Rule
About 68% of the data falls within one standard deviation of the mean.
About 95% falls within two standard deviations.
About 99.7% falls within three standard deviations.
Mathematical Expression
μ is the mean
σ is the standard deviation
Chebyshev's Theorem
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).
Key Points
Applicability
Unlike the 68-95-99.7 rule for normal distributions, Chebyshev's Theorem applies to any distribution shape.
k Value
k must be greater than 1. The larger the k, the higher the percentage of data within that range.