Algebraic expression

An algebraic expression is a mathematical way to represent relationships between different values. These expressions are constructed using three main building blocks: constants, variables, and algebraic operations. Constants are values that stay the same, such as algebraic numbers. Variables are symbols, often letters, that represent values that can change. Unlike an algebraic equation, an expression does not include relational signs like an equals sign (=) or a less-than sign (<).

To build an expression, we use specific operators to perform actions on our building blocks. These operators include addition, subtraction, multiplication, and division. We can also use whole number powers, which are known as exponents, and roots, which are fractional powers. For example, taking a square root is the same as raising a number to the power of one-half. When we use these operations on integer constants, the result is an algebraic number. If we restrict the constants to just numbers, the expression is called an arithmetic expression.

Algebraic expressions have a specific set of parts that mathematicians name. A term is a single part of an expression, such as a number or a variable. A coefficient is a number placed in front of a variable. For example, in the term 3x, the number 3 is the coefficient. An exponent, or power, is a small number written high up that tells us how many times to multiply a value by itself. We also use operators to connect these different parts.

There are many conventions used to make writing these expressions clear and easy to read. By convention, letters from the beginning of the alphabet, like a or b, usually represent constants. Letters from the end of the alphabet, like x or y, are typically used as variables. These letters are usually written in italics. When writing exponents, terms with the highest power are placed on the left. If a coefficient is one, it is usually left out. If an exponent is one, it is also usually omitted. If the exponent is zero, the result is always 1.

We can group expressions into different types based on their structure. A rational expression is a quotient, or a division, of two polynomials. We call a rational expression "proper" if the degree of the top polynomial is smaller than the bottom. If the top degree is larger, the expression is "improper." An improper fraction can be written as the sum of a polynomial and a proper rational fraction. We can also perform a process called resolving into partial fractions. This involves breaking a proper rational fraction into a sum of two or more smaller fractions.

Some expressions are more complex and are categorized as irrational. An irrational fraction contains a variable under a fractional exponent, meaning the variable is inside a root. There is a mathematical process called rationalization used to transform an irrational fraction into a rational one. To do this, one might find the least common multiple of the indices of the roots. One can then substitute a new variable with that least common multiple as the exponent.

Algebraic expressions are essential for solving polynomial equations. A polynomial equation is a type of equation involving polynomials, and algebraic expressions can serve as its solutions. If the degree of a polynomial is less than five, the roots can always be written as algebraic expressions. This includes well-known tools like the quadratic formula. However, the Abel–Ruffini theorem provides a significant limit to this. It states that algebraic solutions do not exist for all such equations if the degree is five or higher.

These expressions connect to many different areas of mathematics. They can be used in Abstract algebra to work with more abstract objects. They also relate to the concept of transcendental numbers. Transcendental numbers, such as pi or e, are not algebraic because they are not derived from integer constants and algebraic operations. While some expressions like pi are constructed as geometric relationships, others, like e, require an infinite number of algebraic operations. This shows how algebraic expressions help us map the boundaries of mathematical logic.