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Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include: Simplification of algebraic expressions, in computer algebra; Simplification of boolean expressions i.e. logic optimization
Expressions can be evaluated or simplified by replacing operations that appear in them with their result, or by combining like-terms. [21] For example, take the expression +; it can be evaluated at x = 3 in the following steps: +, (replace x with 3)
The expression is considered simplified when all like terms have been combined, and all terms present are unlike. In this case, all terms now have different unknown factors, and are thus unlike, and so the expression is completely simplified.
Simplified formal grammar for arithmetical expressions in a programming language (left), [45] and derivation of the example expression (a+b)^2/2 (right). The latter corresponds to a hierarchical structure (" syntax tree ") which is unique for the given expression.
Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example, Added terms are simplified using coefficients.
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.
Simplifying radical expressions involving nested radicals can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced Galois theory . Moreover, when complete denesting is impossible, there is no general canonical form such that the equality of two numbers can be tested by simply looking at ...
Sometimes it is useful to simplify complex expressions made up of bitwise operations, for example when writing compilers. The goal of a compiler is to translate a high-level programming language into the most efficient machine code possible. Boolean algebra is used to simplify complex bitwise expressions.