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An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If you restrict your set of constants to be numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in Abstract algebra.
In elementary algebra, a variable in an expression is a letter that represents a number whose value may change. To evaluate an expression with a variable means to find the value of the expression when the variable is assigned a given number. Expressions can be evaluated or simplified by replacing operations that appear in them with their result ...
An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as a + b = b + a {\displaystyle a+b=b+a} ); such equations are called identities .
Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning. (For a formal explanation, see § Axioms ) For example: Given real numbers a and b , if a = b , then a > 0 {\displaystyle a>0} implies b > 0 {\displaystyle b>0} .
An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients (or equivalently — by clearing denominators — with integer coefficients). Numbers such as π that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.
The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, with rational coefficients). Galois theory was introduced by Évariste Galois to specify criteria for deciding if an algebraic equation may be solved in terms of ...
[e] The expression + is an algebraic expression created by multiplying the number 5 with the variable and adding the number 3 to the result. Other examples of algebraic expressions are 32 x y z {\displaystyle 32xyz} and 64 x 1 2 + 7 x 2 − c {\displaystyle 64x_{1}^{2}+7x_{2}-c} .
An example of using Newton–Raphson method to solve numerically the equation f(x) = 0. In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign.