Chapter 02

REVIEWING MATHEMATICS WITH AI

 

Mathematics encompasses a wide variety of concepts that provide a solid foundation for addressing related problems, in a variety of disciplines; its understanding allows students and professionals to apply rigorous principles in practical and theoretical situations; the most useful and fundamental are:

Essential concepts in mathematics used in a variety of contexts, from simple arithmetic problems to more advanced concepts in algebra and calculus.

Natural Numbers (N): used to count elements, starting with 1 and increasing one by one indefinitely (1, 2, 3, ...).

Integer Numbers (Z): include natural numbers, their negative opposites and zero (..., -3, -2, -1, 0, 1, 2, 3, ...).

Rational Numbers (Q): those that can be expressed as the quotient (fraction) of two whole numbers, where the denominator is not zero (for example, 1/2, -3/4, 7).

Irrational Numbers: cannot be expressed as exact fractions; for example, the square root of 2.

Real Numbers (R): include rational and irrational numbers.

Complex Numbers (C): are made up of a real part and an imaginary part (for example, 3 + 2i, where 3 is the real part and 2i is the imaginary part).

Basic operations:

Addition (+): Add two or more numbers to obtain a total amount.

Subtraction (-): another number is taken from one number to find out what remains.

Multiplication (×): Combine two or more numbers to obtain a product.

Division (÷ or /): divide a quantity into equal parts or find how many times one number fits into another.

Potentiation (^o): Raises a number to a power that indicates how many times that number is multiplied by itself (for example, 2^3 = 2 * 2 * 2 = 8).

Radical (√): inverse operation of potentiation. The symbol √ indicates the square root, and in general, you can have the nth root (√n).

Module (%): Returns the remainder of the division of one number by another.

Elementary algebra: solving equations and operating algebraic expressions. Functions: relationships between variables and their graphs, concepts that provide a solid basis for addressing more advanced topics in mathematics and applied sciences; Constant practice and problem solving are key to understanding and mastering them.

Polynomials:

A polynomial is an algebraic expression that involves addition, subtraction, and multiplication, but not division by variables. For example, 32−27+53x2−2x+5. The degree of a polynomial is the greatest exponent of its variable. In the example above, the degree is 2.
Factoring: is the process of formulating an algebraic expression as a product of factors. For example, 2−4x2−4 can be factored as (x+2) (x−2). Factoring is useful for simplifying expressions and solving equations.
Quadratic equations: Quadratic equations are polynomial equations of the second degree, such as ax2+bx+c=0.
The quadratic formula √ 2a−2b ± b2−4ac) is used to find the roots of a quadratic equation.

Inequalities:
Inequalities are mathematical expressions that indicate that one quantity is greater or less than another. For example, 2 >82x>8 is an inequality. Solving it is finding the values of the variable that satisfy the inequality.

Equation systems:
A system of equations is a set of two or more equations that share the same variables. The common solution to all equations is called the solution of the system. Methods for solving systems of equations include substitution, elimination, and matrices.

Trigonometric functions: such as sine, cosine, and tangent, they are extensions of algebra that are used to model relationships in triangles and circles and are adjustable in geometry, physics, and other areas of applied mathematics.•

Linear algebra: studies vectors, matrices and systems of linear equations. It is applied in fields such as physics, statistics, engineering and computer science.•

Number theory: deals with the properties and relationships of numbers, especially integers. Includes topics such as prime numbers, congruencies, and modular arithmetic.

Prime numbers: They are like the superstars of integers; they are like the famous people in the world of numbers. A prime number is simply greater than 1; He is special because he only has two close friends: number 1 and himself. No other number can divide it without leaving a remainder. For example, 2, 3, 5, 7, 11, and so on, are all prime numbers.

Congruencies: It is like a way of measuring the "closeness" between two numbers after dividing them by another number. If you have two numbers, say a and b, and you subtract them, and the result is divisible by another number (say m), then we say that a and b are "congruent modulo m." It's like saying they have the same "fingerprint" after being divided by that number m. For example, 17 and 5 are congruent modulo 6, because when you subtract 5 from 17, you get 12, which is divisible by 6.

Modular arithmetic: It is like a special version of arithmetic that works in a world where numbers "wrap around" upon reaching a certain value called a modulus; It's like they're going around a clock: when they reach a certain number, they go back to the beginning. This is useful in math, computer security, and other places. If two numbers are congruent modulo m, it means that they have the same remainder when divided by m. You can do basic operations such as addition, subtraction and multiplication, but with a special touch. For example, if you work modulo 5, then 3 + 4 is congruent with 2, because when you add 3 and 4, you get 7, which has a remainder of 2 when divided by 5. It's like doing math with a clock that only goes up to number 5.

Geometric figures: triangles, squares, circles, etc. Pythagorean Theorem and Thales theorem: fundamental concepts in geometry.

Geometry is a powerful tool for understanding and describing shapes and spatial relationships in the world around us; a branch of mathematics concerned with the study of the properties and relationships of figures in space, as well as the properties of space itself; It also covers more advanced areas such as differential geometry, topology, and algebraic geometry, each of which has specific applications in various fields of mathematics and physics. It is a discipline that has been fundamental in the development of mathematics and has wide application in various fields, from physics to architecture.

Geometric figures: These are specific shapes that can be described by properties such as length, area, volume and angles. Some of the most basic geometric figures include triangles, squares, circles, rectangles, and polygons in general. Each of these forms has unique properties that can be studied and applied in different contexts. For example, triangles are fundamental in trigonometry, while circles are essential in the study of analytical geometry.

Pythagorean Theorem: It is one of the fundamental concepts in geometry and establishes a crucial relationship between the sides of a right triangle, in which the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem has practical applications in solving problems related to distances and measurements, and is fundamental in trigonometry.

Thales' Theorem: focuses on proportions in similar triangles. According to this theorem, if we draw two parallel lines cut by a series of secant lines, the segments formed by the secant lines are proportional. This theorem is fundamental in solving geometric problems and also has applications in trigonometry and analytical geometry.

 

Trigonometric functions: cosine, tangent, etc. Trigonometric identities: important relationships between trigonometric functions. Calculation:

Sine (sinθ): in a right triangle, it is the ratio between the opposite leg and the hypotenuse.

Cosine (cosθ): is the ratio between the adjacent leg and the hypotenuse in a right triangle.

Tangent (tanθ): relationship between the opposite leg and the adjacent leg in a right triangle.

Cotangent (cotθ), secant (secθ), cosecant (cscθ): they are reciprocal trigonometric functions of the tangent, cosine and sine, respectively.

Trigonometric identities:
Fundamentals: such as the Pythagorean identity (sin2θ + cos2θ = 1).•
Double and half angle relationships: example: sin (2θ) = 2sinθcosθ. •
Reciprocal: they relate trigonometric functions.
Addition and subtraction identities: help express trigonometric functions of addition and subtraction of angles.•

Calculus: is a branch of mathematics that is divided into two main areas: differential calculus and integral calculus; They are fundamental in mathematics and have extensive applications in physics, engineering, science.

Trigonometry

Trigonometric functions: cosine, tangent, etc. Trigonometric identities: important relationships between trigonometric functions. Calculation:

Sine (sinθ): in a right triangle, it is the ratio between the opposite leg and the hypotenuse.

Cosine (cosθ): is the ratio between the adjacent leg and the hypotenuse in a right triangle.

Tangent (tanθ): relationship between the opposite leg and the adjacent leg in a right triangle.

Cotangent (cotθ), secant (secθ), cosecant (cscθ): they are reciprocal trigonometric functions of the tangent, cosine and sine, respectively.

Trigonometric identities:
Fundamentals: such as the Pythagorean identity (sin2θ + cos2θ = 1).•
Double and half angle relationships: example: sin (2θ) = 2sinθcosθ.•
Reciprocal: they relate trigonometric functions.
Addition and subtraction identities: help express trigonometric functions of addition and subtraction of angles.

Calculus: is a branch of mathematics that is divided into two main areas: differential calculus and integral calculus; they are fundamental in mathematics and have extensive applications in physics, engineering, computer science, and various scientific disciplines.

Differential calculus:
Derivatives: represent the instantaneous rate of change of a function with respect to a variable.
Differentiation rules: allow you to calculate derivatives of more complex functions using specific rules.
Maximum and minimum: derivatives are used to find critical points and determine relative extremes of a function.
Integral calculus:
Definite and indefinite integrals: the indefinite integral represents an anti-derivative, while the definite integral calculates the area under a curve.
Integration rules: help find the integral of more complex functions.
Fundamental theorem of calculus: relates the calculation of areas under curves with the evaluation of functions at specific points.

Probability and Statistics: Exploring the World of Uncertainty and Data; Probability and statistics form a powerful duo that allows researchers, scientists, and practitioners to address uncertainty, make informed decisions, and reveal meaningful patterns in a sea of data. These disciplines are not only fundamental to academic research, but also play a crucial role in everyday decision-making and problem-solving in various fields, providing a solid foundation for understanding the complexity inherent in our environment.
Probability: it stands as an essential tool to quantify the certainty or possibility of events occurring. In the context of probability theory, events are evaluated in terms of their probability of happening, represented by numbers between 0 and 1, where 0 indicates absolute certainty that an event will not occur, while 1 implies absolute certainty. That it will happen. This mathematical approach provides a solid foundation for addressing the uncertainty inherent in many phenomena, from games of chance to complex processes in science and technology.
Statistics: beyond numbers, data science is a comprehensive discipline that encompasses the collection, analysis and interpretation of data. In an information-saturated world, statistics provide the tools needed to extract meaning from data sets, identify underlying patterns, and make informed decisions. From basic data description to advanced modeling techniques, statistics provides an analytical framework for understanding complex phenomena and making informed decisions in fields as diverse as scientific research, financial analysis, and strategic planning. 

Mathematical logic is a branch that deals with the study of the structure and laws of deductive reasoning; provides the formal tools to analyze and evaluate arguments, as well as to establish the validity of propositions.

Probability and Statistics: Exploring the World of Uncertainty and Data; Probability and statistics form a powerful duo that allows researchers, scientists, and practitioners to address uncertainty, make informed decisions, and reveal meaningful patterns in a sea of data. These disciplines are not only fundamental to academic research, but also play a crucial role in everyday decision-making and problem-solving in various fields, providing a solid foundation for understanding the complexity inherent in our environment.
Probability: it stands as an essential tool to quantify the certainty or possibility of events occurring. In the context of probability theory, events are evaluated in terms of their probability of happening, represented by numbers between 0 and 1, where 0 indicates absolute certainty that an event will not occur, while 1 implies absolute certainty. That it will happen. This mathematical approach provides a solid foundation for addressing the uncertainty inherent in many phenomena, from games of chance to complex processes in science and technology.
Statistics: beyond numbers, data science is a comprehensive discipline that encompasses the collection, analysis and interpretation of data. In an information-saturated world, statistics provide the tools needed to extract meaning from data sets, identify underlying patterns, and make informed decisions. From basic data description to advanced modeling techniques, statistics provides an analytical framework for understanding complex phenomena and making informed decisions in fields as diverse as scientific research, financial analysis, and strategic planning.

Mathematical logic is a branch that deals with the study of the structure and laws of deductive reasoning; provides the formal tools to analyze and evaluate arguments, as well as to establish the validity of propositions.

Propositions: is a statement that is true or false, but not both at the same time. For example, "2 + 2 = 4" is a true statement.
Types of Proposals:
Atomic: they cannot be divided into simpler propositions.
Composite: they are formed by combining atomic propositions using logical connectives.

Propositional variables:
Letters (p, q, r, etc.) are used to represent simple or atomic propositions.

Logical connectives: conjunction (∧): the compound proposition is true only when both p and q are true.

Disjunction (∨): the compound proposition is true if at least one of p or q is true.

Negation (¬): the negated proposition p is true if p is false, and vice versa.

Implication (→): the compound proposition p→q is false only when p is true and q is false. In all other cases, it is true.

Bicondicional (↔): The compound proposition p↔q is true if p and q have the same truth value (both true and both false).

Truth Tables: Truth tables show the possible truth values for a compound proposition based on the values of its components.

Laws of Logic: These concepts and tools form the basis for logical reasoning and are essential in various areas, including computer science, computer science, and mathematics in general.
Law of Identity: it is like saying that if something is true, then it is true. It's like saying that if it rains (p), then yes, it is raining.
Law of Domain: If something is true and you mix it with "false", you always get false. If you say "it is day (p) and it is night (F)", that is false. Also, if you say "it is not day (¬p) or it is true (T)", that will always be true.
Law of Contradiction: if something is said to be true and at the same time it is not true, that is always false. For example, saying "it is day (p) and it is not day (¬p)" does not make sense and is false. But saying "it's daytime (p) or it's not daytime (¬p)" is always true.
De Morgan's Laws: two rules, the first is that if you deny that two things are true at the same time that is the same as saying that one of them is not true. For example, "it is not daytime and it is not raining" is the same as "it is not daytime or it is not raining." The second is if you deny that at least one of two things is true, that is the same as saying that neither of them is true. For example, "it is not daytime or it is not raining" is the same as "it is not daytime and it is not raining." They are like rules to play with words and truths.

A look at the importance of vectors and matrices in Linear Algebra and their applications in various disciplines.

Vectors: A vector is like an arrow in mathematics. If you are in a three-dimensional space (like the world around us), and you want to say where something is, you need three numbers for that. In a city, you can say "I am 12 kilometers north, 3 kilometers east and 4 kilometers up." You can also put it in a vector: [12, 3, 4]. If you want to add two vectors or multiply a vector by a number, you simply do those operations with each number in the vector.

Arrays: Now, a matrix is like a table of numbers. Instead of having just one row (like a vector) or one column, it has rows and columns, like a spreadsheet. Each number in the matrix has a specific location, such as being in row 2, column 3. Arrays are very useful for doing more complex calculations.

Matrix multiplication: to multiply two matrices, A and B, a row of the first matrix is multiplied by a column of the second and these products are added to calculate the resulting matrix.

Analytical Geometry is a branch of mathematics that combines geometric concepts with algebraic tools. Within this field, Cartesian coordinates play a fundamental role in providing an algebraic representation of points and geometric figures in the plane or in three-dimensional space. Analytical Geometry provides a powerful framework for the algebraic representation and study of geometric objects, playing an essential role in mathematical analysis and numerous practical applications.

Cartesian coordinate system: The representation of points on a plane is achieved using a Cartesian coordinate system. This system consists of two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point in the plane is identified by an ordered pair (x, y), where "x" is the coordinate on the horizontal axis and "y" is the coordinate on the vertical axis.

Geometric transformations: algebraic operations on Cartesian coordinates allow geometric transformations to be performed. This includes translations, rotations, reflections and scales, which affect the position and shape of geometric figures.

Geometry in space: Analytical Geometry extends to three-dimensional space, where each point is represented by a set of three coordinates (x, y, z). Algebraic equations are used to describe three-dimensional objects, such as planes, spheres, and cylinders. 

Applications in Science and Engineering: Analytical Geometry has extensive applications in various disciplines, including physics, engineering, and computer science. It allows modeling and solving practical problems using algebraic and geometric methods.

Vectors and Linear Algebra: vector representation is integrated with Analytical Geometry. Vectors are used to describe directions and magnitudes, making it easier to analyze displacements and forces in space.

Modern Developments: Analytical Geometry has evolved with mathematical and technological developments to more advanced methods, such as differential geometry and algebraic geometry, expanding the possibilities of modeling and analysis. Differential geometry focuses on understanding geometric properties through the use of derivative and integral concepts. This helps analyze how shapes change and relate at different points. Algebra uses concepts to study the solutions of polynomial equations, which has direct applications in the description of curves and surfaces. New advances have significantly expanded the capabilities to model and analyze geometric shapes in more complex and detailed ways. Furthermore, with the support of technology, such as specialized software, these shapes can now be visualized and manipulated more efficiently and accurately, which has revolutionized the way we approach geometric problems today.

Set Theory is a fundamental branch of mathematics that provides the basis for logical reasoning and extends to various areas of study. In this discipline, concepts related to sets and the operations that can be performed with them are explored, playing an essential role in the structure and understanding of mathematical logic.
  Definition of Sets: a well-defined collection of elements that can be numbers, letters, or any other mathematical object. Set Theory is not only essential for mathematical logic, but also plays a central role in the structure and reasoning of various branches of mathematics, providing a solid conceptual framework for the formulation and solution of mathematical problems.

Basic Operations with Sets:
Union (⋃): a new set is generated that contains all the elements present in both original sets, eliminating duplicates.
Intersection (⋂) produces a new set containing only the elements common to both sets.
Difference (−): The difference between two sets creates a new one that includes all the elements of the first set that are not in the second set.
Complement ('): the complement of a set with respect to a universal set is made up of all the elements that are not in the original set.
Axioms of set theory: it is based on propositions that establish the fundamental rules that govern the manipulation of sets. These axioms provide the logical basis for the development of the theory. 

Relationship with other branches of mathematics: Set theory is intertwined with various branches of mathematics, such as mathematical logic, algebra, topology and number theory. Its application is crucial in the formulation and resolution of problems in these disciplines.
Cantor's Theorem and the Cardinality of Sets: This theorem states that there are sets of different infinite sizes, and the notion of cardinality is used to compare and classify the "number" of elements in infinite sets; There are infinite sets, some being larger than others, like two boxes, one with all the even numbers and the other with all the numbers. Although both sets are infinite, the set of all numbers is larger.
Zermelo-Fraenkel (ZF) theorem: main Idea: precise rules for playing with sets and avoiding strange problems. It's like having clear rules in a game. The ZF ensures that when we play with sets, we will not encounter confusing situations or strange problems. Together with the Choice Proposition, the ZF system is a set of axioms that provides a more rigorous and complete basis for Set Theory, addressing possible paradoxes and establishing the rules for the construction of sets.

 

 

A branch that focuses on the study of financial operations and how time affects the value of money. Understanding these principles is essential to making informed financial decisions and evaluating investments, loans and other transactions. Practice with problems and examples is essential to strengthen these concepts and calculations are greatly facilitated using Excell.
 

Simple interest:
It is calculated on the original capital during a specific period.
The basic formula is: interest=capital×rate×time.
Placing $1000 at an interest rate of 5% per year. Simple interest is calculated by multiplying the initial capital by the interest rate and the time in years.
i=p.r.t
P: initial capital ($1000)
r: Interest rate (5% or 0.05)
t: Time in years (say, 3 years)

Calculate simple interest:

=1000⋅0.05⋅3=150I=1000⋅0.05⋅3=150

After 3 years, the simple interest is $150. The total amount would be =1000+150=1150P+I=1000+150=1150.

Compound interest:
Unlike simple interest, the returns generated are added to the original principal, and future interest is calculated on this new amount. The formula is: amount=capital (1+rate) time.

Compound interest formula: (1+A=P (1+r/n) nt
P: initial capital (in this case, $1000)
r: interest rate per period (in this case, 5% or 0.05)
n: number of compounding periods per year
t: time in years (in this case, 3 years)

If the interest rate is capitalized annually (1n=1).

=1000(1+0.05/1)1⋅3A=1000⋅(1+0.05/1)1⋅3

=1000(1.05)3A=100⋅(1.05)3

We calculate A and get the total amount after 3 years.

It is clearly noticed that the total amount after 3 years with compound interest is more than the total amount with simple interest. Compound interest takes into account the interest accumulated in each period, which leads to exponential capital growth.

Time value of money:

Principle holds that the value of an amount of money varies over time due to factors such as inflation or interest. The present and future value formula is used to adjust monetary amounts to different points in time.
Annuities:
An annuity is a series of equal payments made or received at regular intervals. Annuities can be ordinary (payments at the end of the period) or anticipated (payments at the beginning of the period).
Net present value (NPV) and internal rate of return (IRR):
The NPV is the difference between the present value of the incoming and outgoing cash flows of an investment.
The IRR is the discount rate that makes the NPV equal to zero.

Calculation of Present Value (PV):

If you have the opportunity to invest money in a project that will give a return of 8% per year for 5 years and you want to calculate how much you would have to invest today to obtain a return of $10,000 at the end of those 5 years.

The present value formula is:   (1+VP= (1+r) nVF, where:

VP is the present value,
VF is the future value,
r is the interest rate per period, and
n is the number of periods.

In this case, VF is $10,000, r is 0.08 (8% expressed as a decimal), and n is 5.

=10,000(1+0.08)5VP= (1+0.08)510,000

If $5,000 is invested in a deposit that pays an interest rate of 6% per year, how much will it accumulate at the end of 8 years.

The Future Value formula is:

Where:
VF is the future value,
VP is the present value,

r is the interest rate per period, and
n is the number of periods.

In this case, PV is $5,000, r is 0.06 (6% expressed as a decimal), and n is 8.

=5,000(1+0.06)8VF=5,000 (1+0.06)8

This will give you the Future Value after 8 years.
Amortization:
It is the process of paying a debt over time in regular payments. Amortization formulas are used to calculate loan installments that cover both principal and interest.
Nominal and effective interest rate:
The nominal interest rate is the interest rate before adjusting for compounding frequency and time. The effective interest rate is the actual interest value after considering these factors. 

The effective rate is a measure that reflects the actual cost or return of a loan or investment, taking into account factors such as the frequency of compounding. The general formula to calculate the effective rate is:

Assuming that there is an annual nominal rate of 5%, and interest is capitalized quarterly (4n=4).

(1+0.054)4−1 (1+40.05)⁴⁻¹ 

Step by step solution:
Calculate the quotient ni: 0.054=0.012540.05=0.0125
Add 1 to the quotient: 1+0.0125=1.01251+0.0125=1.0125
Raise this result to the power n: (1.0125)4≈1.0506015625 (1.0125)4≈1.0506015625
Subtract 1 from the result: 1.0506015625−1≈0.05060156251.0506015625−1≈0.0506015625

Therefore, the effective rate in this case would be 0.0506 or 5.06%. This example assumes that interest is capitalized quarterly. If the capitalization is different, simply adjust the value of n in the formula. The effective rate is useful for comparing different loan or investment offerings, as it reflects the impact of compounding frequency on actual performance or cost.

Concept of AI: "Reviewing Mathematics with AI" by José Saúl Velásquez Restrepo offers an exhaustive tour of fundamental mathematical concepts, from natural numbers to advanced areas such as calculus, analytical geometry, and set theory. The work highlights the importance of these concepts in various disciplines and highlights their practical application in theoretical and practical situations.
The author begins by introducing natural, integer, rational, irrational, real, and complex numbers, providing a solid foundation for understanding basic operations such as addition, subtraction, multiplication, and division. Additionally, he explores concepts such as potentiation, radicals, and modulus. This section lays the foundation necessary to move into more complex areas of mathematics.
Elementary algebra and solving equations are clearly covered, along with concepts such as polynomials and factoring. He delves into quadratic equations, inequalities and systems of equations, highlighting their importance in solving problems in various fields.
The inclusion of trigonometric functions expands the application of mathematics to problems in triangles and circles, fundamental in geometry and applied sciences. Number theory, with its emphasis on prime numbers and congruences, adds an intriguing dimension to mathematical study.
The work highlights the relevance of geometry, introducing fundamental concepts such as the Pythagorean theorem and Thales theorem. Analytical geometry, with its connection to vectors and matrices, is presented as a powerful tool in physics and computer science.
Calculus, divided into differential and integral calculus, is addressed exhaustively, highlighting its importance in areas such as physics, engineering, and computer science. Practical applications are explored such as the calculation of areas and volumes, the resolution of optimization problems, and the modeling of phenomena that change over time using differential equations.
The section dedicated to probability and statistics highlights their role in decision making and revealing patterns in data sets. Mathematical logic is presented as a fundamental tool to analyze and evaluate arguments, establishing the validity of propositions through logical connectives and truth tables.
The work concludes with a review of vectors and matrices in linear algebra, highlighting their applications in physics and computer science. Additionally, modern developments in analytical geometry, including differential and algebraic geometry, and the influence of technology on the visualization and manipulation of complex geometric shapes, are addressed.
In summary, "Revisiting Mathematics with AI" provides a comprehensive and accessible overview of essential mathematical concepts, highlighting their applicability in various disciplines and highlighting their importance in solving theoretical and practical problems.

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