Algebra it is the branch of mathematics that generalizes arithmetic. This means that concepts and operations from arithmetic (addition, subtraction, multiplication, division etc.) will be tested and their effectiveness will be proven for all numbers belonging to certain sets numeric.
Does the “addition” operation, for example, really work on all numbers belonging to the set of natural numbers? Or is there some very large natural number, close to infinity, that behaves differently from others when added? The answer to this question is given by algebra: First, the set of natural numbers is defined and the operation adds; then it is proven that the addition operation works for any natural number.
US algebra studies, letters are used to represent numbers. These letters can represent either unknown numbers or any number belonging to a numerical set. If x is an even number, for example, then x can be 2, 4, 6, 8, 10,... In this way, x is any number belonging to the set of even numbers and it is clear what kind of number x is: a multiple of 2.
Properties of Mathematical Operations
Knowing that any number belonging to a set can be represented by a letter, consider the numbers x, y and z as belonging to the set of numbers. real and the operations addition and multiplication represented by “+” and “·”, respectively. So, the following properties are valid for x, y and z:
1 - Associativity
(x + y) + z = x + (y + z)
(x·y)·z = x·(y·z)
2 – Commutativity
x + y = y + x
x·y = y·x
3 – Existence of a neutral element (1 for multiplication and 0 for addition)
x + 0 = x
x·1 = x
4 – Existenceof opposite (or symmetric) element.
x + (–x) = 0
x· 1 = 1
x
5 – Distribution (also called the distributive property of multiplication over addition)
x·(y + z) = x·y + x·z
These five properties are valid for all real numbers x, y, and z, as these letters were used to represent any real number. They are also valid for addition and multiplication operations.
algebraic expressions
In Mathematics, expression is a sequence of mathematical operations performed with some numbers. For example: 2 + 3 – 7 is a numeric expression. When this expression involves unknown numbers (unknowns), it is called algebraic expression. An algebraic expression that has only one term is called a monomium. Any algebraic expression that is the result of addition or subtraction between two monomials is called a polynomial.
algebraic expressions, monomials and polynomials are examples of elements belonging to algebra, as they are constituted from operations performed with unknown numbers. Remember that an unknown number can represent any number in a numerical set.
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Equations
Equations they are algebraic expressions who have an equality. Thus, equation it is a content of Mathematics that relates numbers to unknowns through an equality.
The presence of the unknown is what classifies the equation as algebraic expression. The presence of equality allows finding the solution of an equation, that is, the numerical value of the unknown.
Examples
1) 2x + 4 = 0
2) 4x - 4 = 19 - 8x
3) 2x2 + 8x – 9 = 0
Roles
The formal definition of function is as follows: occupation it is a rule that relates each element of a set to a single element of a second set.
This rule is mathematically represented by an algebraic expression that has an equality, but that relates the unknown to the unknown. This is the difference between function and equation: the equation relates an unknown to a fixed number; at occupation, the unknown represents an entire numerical set. For this reason, within functions, unknowns are called variables, as they can take any value within the set they represent.
As it involves algebraic expressions, occupation it is also a content belonging to Algebra, since the letters represent any number belonging to any set of numbers.
Examples:
1) Consider the function y = x2, where x is any real number.
In this occupation, the variable x can take any value within the set of real numbers. Since the rule connecting the numbers represented by x to the numbers represented by y is a basic mathematical operation, so y also represents real numbers. The only detail about this is that y cannot represent a negative real number in this function, since y is the result of an exponent power of 2, which will always have a positive result.
2) Consider the function y = 2x, where x is a natural number.
In this occupation, the variable x can take any value within the set of natural numbers. These numbers are the positive integers, so the values that y can take are natural numbers multiples of 2. In this way, y is a representative of the set of even numbers.
From classical algebra to abstract algebra
The concepts listed so far make up the classical algebra. This part of algebra is more linked to sets of natural, integer, rational, irrational, real and complex numbers and is studied in both elementary and higher education. The other part of algebra, known as abstract, studies these same structures, but for any sets.
Thus, given any set, with any elements (numbers or not), it is possible to define an operation "addition", an operation "multiplication" and verify the existence or not of the properties of these operations, as well as the validity of "equations", "functions", "polynomials" etc.
By Luiz Paulo Moreira
Graduated in Mathematics
