Introduction to Mathematics

This is a great playlist from a fantastic teacher, I hope to put more and more here about him. So here it is, introduction to math.

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) includes the study of such topics as quantity, structure, space, and change. It has no generally accepted definition.

Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from countingcalculationmeasurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid‘s Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciencesApplied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

Physicists like math, usually. People who hate math just don’t understand it. But like that weird ethnic food you’re scared of trying, or that odd kid in class that doesn’t say much, if you come to understand something it’s not so scary. In fact, if we can learn enough about math, we will come to love it, because there is so much we can do with it. Watch this series to supplement your math classes, or just to learn enough math that you can become friends. It starts out super easy so don’t be afraid! Okay, let’s go! “

Arithmetic

Addition (often signified by the plus symbol “+”) is one of the four basic operations of arithmetic; the others are subtractionmultiplicationand division. The addition of two whole numbers is the total amount of those values combined. For example, in the adjacent picture, there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression “3 + 2 = 5” i.e., “3 add 2 is equal to 5″.

Besides counting items, addition can also be defined on other types of numbers, such as integersreal numbers and complex numbers. This is part of arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can be performed on abstract objects such as vectors and matrices.

Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.

Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some members of other animal species. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. The result of a subtraction is called a difference. Subtraction is signified by the minus sign (−). For example, in the adjacent picture, there are 5 − 2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. Therefore, the difference of 5 and 2 is 3, that is, 5 − 2 = 3. Subtraction represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbersfractionsirrational numbersvectors, decimals, functions, and matrices.

Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is also not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because 0 is the additive identity, subtraction of it does not change a number. Subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that continue these patterns are studied in abstract algebra.

Performing subtraction is one of the simplest numerical tasks. Subtraction of very small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems.

In advanced algebra and in computer algebra, an expression involving subtraction like A − B is generally treated as a shorthand notation for the addition A + (−B). Thus, A − B contains two terms, namely A and −B. This allows an easier use of associativity and commutativity.

Ring Theory

In the same way “integers” are a set of objects, “numbers” have rules about how they interact. For example “addition” or “multiplication” of two “numbers”. These rules on how they interact are called “axioms” and they tell us about the nature and how these groups of numbers interact. Rings are similiar in terms of having elements or sybols that we defined and then add or subtract based on specific rules.

“Integers have lots of properties that continue to work if we scale back on the number of “rules” we are willing to take for granted? What can we still prove? For every set of “rules” we take, we can develop an entire theory of results which require nothing more than the set of rules we started out with. For one specific set of rules, any object which is found satisfying all of those properties is called a ring (though people differ slightly on exactly what they call a ring).

There is a different set of rules which define what people call a group (the definition of a group is more set than that of a ring), a different one still for a vector space, one for a field, … The list goes on. People have dreamed up lots of different sets of rules which, for some reason or another, are more relevant to what they find interesting. More generally, this whole idea is the idea of abstraction; a ring is just one of many abstract objects.

On the other hand, when you strip many of the extra details out and just focus on these smaller sets of rules, some facts become more obviously true, because it is clear which properties imply the result you are looking for.
Rings are objects with sufficiently general rules as to gain insight on objects from the set of integers or the set of complex numbers, to the set of 3×3 matrices, or the set of all functions from the real numbers to the real numbers.

These are at least the thoughts that went into the formation of the theory (though they don’t much distinguish between, say, ring theory and group theory). A bit more specifically, the notion of a ring is a generalization of standard number systems, in that they retain two distinct operations with “multiplication” distributing over “addition” (though a general ring does not require that multiplication commutes as is the case for matrices). An exact definition of a ring is easy enough to put down, but not particularly enlightening without more context. Hopefully this helps provide some of the context. ” – Source: mymathforum.com/abstract-algebra/20100-simplified-explanation-algebraic-ring-theory.html

Group theory

Math and Algebra study Algebraic groups. These are concepts in abstract algebra like ringsfields, and vector spaces, are all forms or algebraic groups.

We can see algrebraic groups in nature, like crystals and the hydrogen atom, in other words, we can translate them to math through socmeting called symmetry groups. This is why group theory is important to create mathematical models of phisical things like chemistry, and materials science. Group theory is also central to public key cryptography. But I´ll let the experts explain this they way experts know how.

Computational Economics? Quick Summary.

Computational economics is a discipline that is related to computer science, economics and management. Computational models are developed to predict and understand economic dynamics.

” Computational economics uses computer-based economic modeling for the solution of analytically and statistically formulated economic problems. A research program, to that end, is agent-based computational economics (ACE), the computational study of economic processes, including whole economies, as dynamic systems of interacting agents. As such, it is an economic adaptation of the complex adaptive systems paradigm. Here the “agent” refers to “computational objects modeled as interacting according to rules,” not real people. Agents can represent social, biological, and/or physical entities. The theoretical assumption of mathematical optimization by agents in equilibrium is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces, including game-theoretical contexts. Starting from initial conditions determined by the modeler, an ACE model develops forward through time has driven solely by agent interactions. The ultimate scientific objective of the method is “to … test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before.”

Computational solution tools include for example software for carrying out various matrix operations (e.g. matrix inversion) and for solving systems of linear and nonlinear equations. For a repository of public-domain computational solution tools.

The following journals specialize in computational economics: ACM Transactions on Economics and ComputationComputational EconomicsJournal of Applied EconometricsJournal of Economic Dynamics and Control, and the Journal of Economic Interaction and Coordination.”  Source: https://en.wikipedia.org/wiki/Computational_economics

What is Cybersecurity?

” Computer security, also known as cybersecurity or IT security, is the protection of computer systems from the theft and damage to their hardware, software or information, as well as from disruption or misdirection of the services they provide.

Cybersecurity includes controlling physical access to the hardware, as well as protecting against the harm that may come via network access, data, and code injection.  Also, due to malpractice by operators, whether intentional, accidental, IT security is susceptible to being tricked into deviating from secure procedures through various methods.

The field is of growing importance due to the increasing reliance on computer systems and the Internet,  wireless networks such as Bluetooth and Wi-Fi, and the growth of “smart” devices, including smartphones, televisions and tiny devices as part of the Internet of Things.

A vulnerability is a weakness in design, implementation, operation or internal control. As they are discovered many vulnerabilities are documented in the Common Vulnerabilities and Exposures (CVE) database. An exploitable vulnerability is one for which at least one working attack or “exploit” exists. Vulnerabilities are often hunted or exploited with the aid of automated tools. To secure a computer system, it is important to understand the attacks that can be made against it.” https://en.wikipedia.org/wiki/Computer_security

What is Algebra?

“Algebra (from Arabic “al-jabr” meaning “reunion of broken parts”) is one of the broad parts of mathematics, together with number theory, geometry, and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in {\displaystyle x+2=5} the letter {\displaystyle x} is unknown, but the law of inverses can be used to discover its value: {\displaystyle x=3} . In E = mc2, the letters {\displaystyle E} and {\displaystyle m} are variables, and the letter {\displaystyle c} is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words. The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an “algebra”, and the word is used, for example, in the phrases linear algebra and algebraic topology.” https://en.wikipedia.org/wiki/Algebra

What Are Computer Networks?

A computer network is a group of devices all linked between each other that are able to share information or data. Data is shared using a cable wire or can even be wireless These networks contain a large number of computers and other electronic devices capable of composing a node.

“A computer network or data network is a digital telecommunications network which allows nodes to share resources. In computer networks, networked computing devices exchange data with each other using a data link. The connections between nodes are established using either cable media or wireless media.

Network computer devices that originate, route and terminate the data are called network nodes. Nodes can include hosts such as personal computers, phones, servers as well as networking hardware. Two such devices can be said to be networked together when one device is able to exchange information with the other device, whether or not they have a direct connection to each other. In most cases, application-specific communications protocols are layered (i.e. carried as payload) over other more general communications protocols. This formidable collection of information technology requires skilled network management to keep it all running reliably.

Computer networks support an enormous number of applications and services such as access to the World Wide Web, digital video, digital audio, shared use of application and storage servers, printers, and fax machines, and use of email and instant messaging-applications as well as many others. Computer networks differ in the transmission medium used to carry their signals, communications protocols to organize network traffic, the network’s size, topology and organizational intent. The best-known computer network is the Internet.” https://en.wikipedia.org/wiki/Computer_network

What is The Central Processing Unit (CPU)?

Computers have their brains in an electronic system called Central Processing Unit, this contains all the commands and memory to save what we make into computer. It bases in arithmetic and logical input and outputs. In this article, we will learn and introduce you some details on how CPU work and what does it really do.

“A central processing unit (CPU) is the electronic circuitry within a computer that carries out the instructions of a computer program by performing the basic arithmetic, logical, control and input/output (I/O) operations specified by the instructions. The computer industry has used the term “central processing unit” at least since the early 1960s. Traditionally, the term “CPU” refers to a processor, more specifically to its processing unit and control unit (CU), distinguishing these core elements of a computer from external components such as main memory and I/O circuitry.

The form, design, and implementation of CPUs have changed over the course of their history, but their fundamental operation remains almost unchanged. Principal components of a CPU include the arithmetic logic unit (ALU) that performs arithmetic and logic operations, processor registers that supply operands to the ALU and store the results of ALU operations, and a control unit that orchestrates the fetching (from memory) and execution of instructions by directing the coordinated operations of the ALU, registers and other components.

Most modern CPUs are microprocessors, meaning they are contained on a single integrated circuit (IC) chip. An IC that contains a CPU may also contain memory, peripheral interfaces, and other components of a computer; such integrated devices are variously called microcontrollers or systems on a chip (SoC). Some computers employ a multi-core processor, which is a single chip containing two or more CPUs called “cores”; in that context, one can speak of such single chips as “sockets”. Array processors or vector processors have multiple processors that operate in parallel, with no unit considered central. There also exists the concept of virtual CPUs which are an abstraction of dynamical aggregated computational resources.” https://en.wikipedia.org/wiki/Central_processing_unit

What is Computer Science?

Computers fill a crucial role in the function of our society, and it’s our hope that over the course of this series you will gain a better understanding of how far computers have taken us and how far they may carry us into the future. Computer science consists studies of the engineering, theory, experimentation of the computer. It studies the diverse applications of its system.

“Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. It is the scientific and practical approach to computation and its applications and the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to information. An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems.

Its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory(which explores the fundamental properties of computational and intractable problems), are highly abstract, while fields such as computer graphics emphasize real-world visual applications. Other fields still focus on challenges in implementing computation. For example, programming language theory considers various approaches to the description of computation, while the study of computer programming itself investigates various aspects of the use of programming language and complex systems. Human-computer interaction considers the challenges in making computers and computations useful, usable, and universally accessible to humans.” https://en.wikipedia.org/wiki/Computer_science

Who is Alan Turing?

Today we’re going to take a step back from programming and discuss the person who formulated many of the theoretical concepts that underlie modern computation – the father of computer science himself: Alan Turing. From his theoretical Turing Machine and work on the effort to break Nazi Enigma codes during World War II, to his contributions in the field of Artificial Intelligence (before it was even called that), Alan Turing helped inspire the first generation of computer scientists – despite a life tragically cut short.

Alan Turing the man who approached to solve the decision problem through a hypothetical computing machine, actually known as a Turing machine. Watch this interesting video and learn where does your smartphone, computer, laptop and iPad basic system come from.