Equations | Research & Encyclopedia Articles

Equations

Equations are statements that use numbers and symbols to demonstrate that two groups of mathematical data are equal. The roots or solutions to an equation are numbers that replace the variables (unknown quantities) to make the equation true.

The first records of equations come from early Greek scholars, 540 B.C.-250 B.C. Euclid, Pythagoras, and his Pythagorean followers used algebraic problems with geometric proofs. This method of mixing algebra and geometry led to complex constructions. The Greek system of mathematics used layers of numbers, letters, and punctuation marks piled on top of each other. This approach probably favored interpretations of geometry rather than arithmetic calculations. Other early mathematical records show that Egyptians, Indians, Muslims, and Greeks have been using cubic equations for more than 2000 years.

Some of the best known categories of equations include:

• Algebraic equations: The variables appear with coefficients that can have addition, subtraction, multiplication, or division performed on them. Some algebraic equations have non-algebraic coefficients.
• Root equations: The variable is a radicands (a number under a radical sign).
• Transcendental equations: These equations are not algebraic, but they may have algebraic coefficients. They often deal with relationships between non-algebraic numbers and quantities.
• Exponential equations: These are a form of transcendental equation that is usually solvable by algebra. The variables often appear as exponents but can appear as bases or coefficients.
• Logarithmic equations: These are also a form of transcendental equation and are difficult to distinguish from exponential equations.
• Trigonometric equations: These are also a form of transcendental equation and have trigonometic terms with algebraic coefficients. These equations are often solvable by algebra.
• Identity equations: These equations use a statement of equality that is true for all values in a universal set.
• Conditional equations: These equations are true only for certain values of variables.
• Linear equations: In these equations, the unknown variable is expressed in first-degree terms.
• Polynomial equations: There are two kinds of polynomial equations: linear equations with integer coefficients, and fractional equations.
• Equations with absolute values: In these equations, a first-degree equation with one variable usually has one solution. However, first-degree equations with absolute-value expressions have more than one solution.
• Quadratic equations: These are defined as a polynomial equations of the second degree.
• Cubic equations: These require third-degree equations. Mathematical records show that although cubic equations were written for many centuries, Italian mathematicians of the sixteenth century were the first to actually solve such an equation.
• Quartic equations: These are regarded as very cumbersome and can be solved with rational operations and rational expressions.
• Simultaneous equations: In this form of problem, all equations must be true at the same time.
• Diophantine equations: These are named for the mathematician Diophantus of Alexandria (c. A.D. 220). The definition of Diophantine equations have evolved over time and today this term refers to equations that can be solved with integers.