An equation is a mathematical expression consisting of two terms connected by the equal sign. The two terms are called left and. right side of the equation.
On the concept of equations
The term Equation goes back to the Italian mathematician LEONARDO FIBONACCI OF PISA (about 1180 to about 1250).
Equations in which no variables occur are (true or false) statements :
- 4 ⋅ 25 = 100 is a true Statement.
- 3 + 17 = 19 is one false Statement.
Equations, in which (at least) one variable occurs, are not statements, but statement forms . They become (true or false) statements if all occurring variables are occupied (d.h. by numbers from the respective basic range replaced) or by additional conditions bound become:
- 4 ⋅ x = 25 ( m i t x ∈ ℝ ) becomes a true statement for x = 6.25, and a false statement for all other values.
However, if the basic range of variables is the set of natural numbers ( x ∈ ℕ ) , there is no number for which the equation becomes a true statement.
- If in the equation x 2 = 17 the variable is bound by the existential statement "There is an x with x 2 = 17 ", then for the basic variable range of the real numbers ( x ∈ ℝ ) a true statement arises, but for the basic variable range of the rational numbers ( x ∈ ℚ ) a false statement.
- If, in the equation a + b = b + a, the variables are replaced by the all-statement "For all real numbers a and b holds . " bound, so a true statement is created.
Solving an equation – solution set
An equation solve means finding all elements of the basic range that produce a true statement when inserted into the equation.
Each such element of the basic domain is called a solution of the equation. It is also said: One solution satisfied the equation.
All solutions together form the solution set L of this equation. The set of solutions depends on the basic range of variables .
- The equation x 2 = 16 has, over the basic variable range of the natural numbers ( x ∈ ℕ ), the solution set L =< 4>and over the variable base range of integers ( x ∈ ℤ ) the solution set L =< – 4 ; 4>.
To the classification of equations
Equations can first be divided according to the number of unknowns (free variables).
Equations with one unknown (free variable) can be distinguished according to the form in which the unknown (free variable) occurs. If one assumes that possible simplifications are carried out, one can divide equations (following a classification of functions) into algebraic and transcendent equations divide.
Among the algebraic equations are the rational equations (especially the integer ones)and the Root equations, to the transcendental equations the goniometric equations As well as Exponential and logarithmic equations.
A whole-rational equation has the following form:
a n x n + a n – 1 x n – 1 + a n – 2 x n – 2 + . + a 2 x 2 + a 1 x + a 0 = 0
Here the highest exponent of the variable x (here thus n) indicates the degree of the equation.
First degree equations are also linear, Second-degree equations quadratic called. For both there are universally valid solution formulas.
Also for third degree equations (cubic equations) and fourth degree, there are general solution formulas, the so-called cardanic formulas, Named after the Italian mathematician GERONIMO CARDANO (1501 to 1576).
The Norwegian mathematician NILS HENRIK ABEL (1802 to 1829) first proved that there is no general formula for solving equations of a higher than fourth degree.