## 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.

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**Mathematics** (from Greek μάθημα *máthēma*, “knowledge, study, learning”) includes the study of such topics as quantity,^{[1]} structure,^{[2]} space,^{[1]} and change.^{[3]}^{[4]}^{[5]} It has no generally accepted definition.^{[6]}^{[7]}

Mathematicians seek and use patterns^{[8]}^{[9]} 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 counting, calculation, measurement, 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.^{[10]}

Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied 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.^{[11]}^{[12]}

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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 subtraction, multiplicationand 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 integers, real 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 numbers, fractions, irrational numbers, vectors, 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 rings, fields, 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.

## 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* = *mc*^{2}, 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