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Unidad I. Conjuntos numéricos, Unit II. Algebraic expressions, Unit IV.…
Unidad I. Conjuntos numéricos
Modular arithmetic
It is a branch of mathematics that deals with the properties of integers obtained by working with remainders of divisions by a fixed number. Some applications are:
Criptografía: la aritmética modular se utiliza en muchos sistemas criptográficos para cifrar y descifrar mensajes.
Programación informática: La aritmética modular se utiliza en muchos lenguajes de programación para trabajar con índices de matrices, que son números enteros que indican la posición de un elemento en una matriz.
It is a branch of mathematics that deals with the properties of integers obtained by working with remainders of divisions by a fixed number. Some applications are:Calculating the remainder of a division: modular arithmetic is used to find the remainder of a division by a fixed number.
Horario: La aritmética modular puede utilizarse para gestionar el tiempo y los horarios.
Numeros naturales, enteros, racionales, irracionales y reales.
Los números enteros son los que incluyen a los naturales y a sus negativos, así como también el número cero. Ejm: ...-3, -2, -1, 0, 1, 2, 3...
Números irracionales son aquellos que no se pueden expresar como una fracción exacta de dos números enteros y por lo tanto tienen una expansión decimal infinita no periódica. Se denotan con el símbolo "I" ejemplo: π(3.14159265358979...)
Números racionales son aquellos que se pueden expresar como una fracción, donde el numerador y el denominador son números enteros. Ejm: 1/2, 5/3, -2/7
Numeros reales es el conjuntos formado por los numeros racionales e irracionales, es decir, cualquier númeroque pueda representarse en una recta numérica.
Números naturales son los que usamos para contar personas, objetos, animales, etc. Comienzan desde el número 1 y se extienden hasta el infinito ejm: 1,2,3,4,5,6,7...
Exponentes, radicales, y sus propiedades
Radicales: son la inversa de los exponentes y se utilizan para hallar el cuadrado, el cubo u otra raíz de un número. Ejemplo: √25=5
Propiedades de los exponentes: existen varias propiedades de los exponentes que facilitan el cálculo.
-La multiplicación de bases iguales se resuelve sumando los exponentes.
-La división de bases iguales se resuelve restando los exponentes.
-La potencia de una potencia se resuelve multiplicando los exponentes.
Exponentes: son una forma abreviada de escribir la multiplicación repetida de un número por sí mismo. El número que se multiplica se llama base y el exponente indica el número de veces que se multiplica. Ejemplo: 2^3= 2×2×2=8
Propiedades de los radicales: Existen propiedades de los radicales que facilitan el cálculo. -La raíz cuadrada del producto de dos números es igual al producto de las raíces cuadradas de los números. -La raíz cuadrada del cociente de dos números es igual a la raíz cuadrada del numerador dividida por la raíz cuadrada del denominador. -La raíz cúbica de una potencia es igual a la base elevada a la fracción n/3.
Desigualdades, intervalos reales y valor absoluto
Intervalos reales: son conjuntos de números reales qué se encuentran entre dos valores dados. Los intervalos pueden ser abiertos cerrados o semiabiertos.
Ejm: [3,7] es cerrado, (0,4] es abierto.
Valor absoluto: es la distancia entre un número y el cero en la recta numérica, siempre dando positivo o cero. Ejm: |5|=5
|-3|=3
|0|=0
Desigualdades: son expresiones matemáticas que comparan dos cantidades indicando si una es mayor, menor o igual que las otras . Ejm: 5<10 x+3>7
Operaciones y propiedades
Operación de multiplicación: consiste en repetir una suma varias veces. Ejm: 2×3=6
5×4×2=20
Operación de la división: consiste en separar una cantidad en partes iguales. Ejm:8÷2=4
24÷4÷2=3
Operación de resta: consiste en encontrar la diferencia entre dos números. Ejm: 8-3=6
10-6-2=2
Propiedad conmutativa: en las operaciones de suma y multiplicación, el resultado es el mismo sin importar como se agrupan los números. Ejm:3+4=4+3 2×5=5×2
Propiedad asociativa: en las operaciones de suma y multiplicación, el resultado es el mismo sin importar como se agrupan los números. Ejm: (2+3)+4=2+(3+4)
(3×4)×2=3×(4×2)
Operación de suma: consiste en combinar dos o más números para obtener una suma. Ejm: 2+3=5
7+4+1=12
Distributive property: multiplication can be distributed over addition or subtraction. Example: 2×(3+4)=(2×3)+(2×4)
4×(10-6)=(4×10)-(4×6)
Racionalización: existen diferentes técnicas que consisten en eliminar radicales del denominador de una fracción.
-Racionalización de una raíz cuadrada en el denominador: para racionalizar una raíz cuadrada en el denominador, multiplique el numerador y el denominador por el conjugado de la expresión en el denominador.
-Racionalización de una raíz cúbica en el denominador: para racionalizar una raíz cúbica en el denominador, multiplica el numerador y el denominador por la expresión que produce la misma raíz en el denominador.
-Racionalización de una expresión con dos términos en el denominador: para racionalizar una expresión con dos términos en el denominador, multiplicar el numerador y el denominador por el conjugado de la suma o resta de los términos.
Unit II. Algebraic expressions
Polynomials and operations with polynomials
These are algebraic expressions that are constructed from additions and products of terms that have a variable raised to a non-negative power.
Remarkable products
They are algebraic expressions that appear frequently in mathematics and can be factored using common patterns. Some notable products are:
-Square of an addition:(a+B)^2=a^2+2ab+b^2
-Square of a subtraction:(a-b)^2=a^2-2ab+b^2
-Product of a sum by a difference:(a+B)(a-b)=a^2-b^2
-cube of a sum:(a+B)^3=a^3+3a^2b+3ab^2+b^3
-cube of a subtraction:(a-b)^3=a^3-3a^2b+3ab^2-b^3
-Product of two binomials with opposite terms:(a+B)(a-b)=a^2-b^2
-Product of the sum by a constant: k(a+B)=Ka+kb
-Product of a constant by a binomial: k(a+b)=Ka+kb
Rational algebraic expressions
They are expressions that include both polynomials and algebraic fractions. An algebraic fraction is an expression of the form p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not the null polynomial. Rational algebraic expressions are used in many areas of mathematics, including calculus, physics and engineering.
Quotient of polynomials
The quotient of polynomials is an expression of the form P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) is not the null polynomial. The quotient of polynomials can be calculated using the polynomial division algorithm.
Factorization
It is the process of writing an algebraic expression as a product of simpler factors. Factorization is used in many fields of mathematics and is important for simplifying expressions and solving equations. Some techniques are:
Grouping: the similar terms of the expression are grouped and factored by groups.
Perfect square trinomial: a perfect square trinomial is an algebraic expression of the form a^2+2ab+b^2 or a^2-2ab+b^2.
Common factor: the common factor of the terms of the terms of the expression is found and the common factor is factored out of the parentheses.
Difference of squares: a difference of squares is an algebraic expression of the form a^2-b^2.
Algebraic expressions
Algebraic expressions are mathematical expressions that contain one or more letters representing numbers or variables. They can be constructed from numbers, variables, mathematical operations and parentheses.
Root of a polynomial
A root of a polynomial is a value of the independent variable that makes the polynomial equal to zero. There are different techniques for finding the roots of a polynomial:
Hint method: this method is used for polynomials of degree two or three and consists of finding values that satisfy the equations formed from the coefficients of the polynomial.
Newton-Raphson method: this method is used to find roots of any polynomial and is based on the use of an iterative formula to approximate the roots.
Factorization: if the polynomial can be factored, its roots are the values that make the factors equal to zero.
Bairstow's method: this method is used for polynomials of degree three or higher and consists of finding the coefficients of a polynomial of lower degree that approximates the solution.
Remainder Theorem
It is a fundamental theorem in polynomial algebra that allows one to calculate the value of a polynomial at a specific point. The theorem states that if a polynomial P(x) is divided by the polynomial x-a, the remainder of the division is equal to P(a).
Factor Theorem
The factor theorem states that if a polynomial P(x) has a root a, then (x-a) is a factor of polynomial. In other words, if P(a)=0, then (x-a) is a factor of P(x).
Unit IV. Vectors
Geometric vectors
A vector is a quantity that has magnitude and direction. In geometry vectors are used to represent displacements, velocities, forces and other physical concepts. Operations with vectors include vector addition, vector subtraction, vector multiplication by scalars, and dot product.
Graphical representation
The graphical representation of vectors in the R2 plane and in R3 space is a useful tool to visualize the geometric properties of vectors and the operations performed with them.
In R2, vectors are represented as arrows extending from a starting point to an end point.
In R3, vectors are represented similarly to vectors in R2, but they extend from an initial point in space to an end point in space.
Operations in analytical and graphical form
They can be performed analytically using the components of the vectors or graphically using the arrows representing the vectors.
Norm of a vector
The norm of a vector is a measure of its length or magnitude and is denoted by ||v||. It is defined as the square root of the sum of the squares of its components. It is used in many fields of mathematics and physics to calculate distances, forces, and other important values.
Trigonometry
The branch of mathematics that studies the relationships between the sides and angles of triangles. The three main trigonometric functions are sine, cosine and tangent.
Components in R2 and R3
In analytic geometry, vectors in the R2 and R3 plane can be represented by their components in Cartesian coordinates. Vectors in R2 and R3 can be added and subtracted by adding and subtracting their components, they can also be multiplied by scalars.
Scalar product
It is an operation between two vectors that results in a scalar number. The scalar product is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them.
Unit III. Matrices and determinants
Matrix
A matrix is a rectangular table of numbers or elements that are arranged in rows and columns. Matrices are used in mathematics and other disciplines such as physics, engineering and computer science to represent and manipulate these of different types.
Special dies
There are many special matrices that are commonly used in mathematics and applications. Some examples are:
Triangular matrix: is a square matrix in which all elements above or below the main diagonal are equal to zero. If the elements above the main diagonal are equal to zero, the matrix is called a lower triangular matrix. If the elements below the main matrix are equal to zero, the matrix is called an upper triangular matrix.
Symmetric matrix: is a square matrix in which the corresponding elements of the transpose matrix. That is, if A is a symmetric matrix, then A=AT.
Identity matrix: a square matrix in which the elements of the main diagonal are equal to 1 and all other elements are equal to zero.
Orthogonal matrix: is a square matrix in which the columns form a set of orthonormal vectors, that is, each column has a unit length and the columns are orthogonal to each other.
Elementary operations with rows and columns
These are transformations that are applied to the rows and columns of a matrix to produce a new matrix.
We have three elementary operations with rows
Multiplication of a row by a non-zero constant
Sum of a row multiplied by a constant to another row
Interchange of two rows
We have three elementary operations with columns
Multiplication of a column by a non-zero constant
Sum of a column multiplied by a constant to another column
Interchange of two columns
Order of a matrix
The order of a matrix refers to the number of rows and columns the matrix has. If a matrix has m rows and n columns, it is said to have m×n order. It is important to note that the order of a matrix affects the operations that can be performed on it.
Inverse matrix
The inverse matrix is a special matrix used to solve systems of linear equations and perform other matrix operations.
CERPA
Name: Julieth Nicole Lorenzo
Mathematics work
Date: Saturday, August 03, 2024