Elementary algebra

The elementary algebra is the most basic form of algebra . In contrast to arithmetic , in elementary algebra, in addition to numbers and the basic arithmetic operations , variables also appear. In contrast to abstract algebra , elementary algebra does not consider algebraic structures such as vector spaces .

variables

The addition of variables to the numbers and the basic arithmetic operations has the advantage that general principles can be formulated precisely and, above all, clearly. Fundamental laws of real numbers are for example the commutative law , the associative law or the distributive law .

You can also use variables to set up equations or inequalities and examine them for solvability . An example of an equation with one variable is . If the set of definitions for the set of rational numbers , then this equation has exactly one solution, namely . If one substitutes this number for in the equation, a true statement arises , with all other substitutions wrong statements. If, however, one only allows substitutions with whole numbers , then the equation has no solution at all. ${\ displaystyle 3x + 2 = 10}$ ${\ displaystyle x}$ ${\ displaystyle {\ tfrac {8} {3}}}$ ${\ displaystyle x}$ ${\ displaystyle x}$

The description of functional dependencies can also be represented with the help of variables: For example, if you sell tickets at a unit price of € 3 and have fixed costs of € 10, you make a profit of €. ${\ displaystyle x}$ ${\ displaystyle (3x-10)}$

Terms

A term is clearly a meaningful series of mathematical characters. To put it more precisely, a term in algebra consists of numbers, variables, arithmetic operations (these include the four basic arithmetic operations, exponentiation, taking the root and taking the logarithm) and brackets as auxiliary symbols.

One example is . If a term contains variables, it changes into a number when all variables are replaced by elements of the basic set . When dividing, it should be noted that it is not divided by 0. When extracting the root, only nonnegative numbers may appear as radicands and only positive numbers as arguments when taking the logarithm. ${\ displaystyle x ^ {2} -4}$

As in arithmetic, knowing exactly how mathematical terms are interpreted is important in algebra. This is determined by the priority rules of the operations (for example " point calculation before line calculation ", calculate brackets first).

Term transformations are required to solve equations and inequalities . For example, the phrase can also be written as. These two terms are equivalent. The most important term transformations are obtained by applying the laws and rules of number arithmetic. Such rules for generating equivalent terms are: ${\ displaystyle 4 (2a-3) -a}$ ${\ displaystyle 7a-12}$

The commutative and associative law of addition and multiplication,
the distributive law (rules in brackets),
binomial formulas ,
the power laws as well
the logarithmic laws .

Equations and inequalities

An equation consists of two terms with an equal sign between them . An inequality consists of two terms with an inequality sign between them . If there are no variables in both terms, the (in) equation is a statement , otherwise a statement form . The set of elements that can be used for the variables is called the basic set or definition set . Those elements of the definition set whose substitution for the variables makes the (in) equation a true statement are called solutions of the (in) equation. All solutions are combined to form the solution set . ${\ displaystyle L}$

For example, the equation is only true for the values 2 and −2 of . So the solution set consists of the two elements -2 and 2, that is . ${\ displaystyle x ^ {2} -4 = 0}$ ${\ displaystyle x}$ ${\ displaystyle L = \ left \ {- 2; 2 \ right \}}$

Some equations become true statements with every substitution from the definition set, such as, for example . Such equations are called universal . ${\ displaystyle a + (b + c) = (a + b) + c}$

The main method of solving equations (inequalities) is through equivalence transformations . They do not change the solution set of the equation (inequality). Examples of equivalent transformations are:

Replace a term with an equivalent term.
Addition or subtraction of equal numbers (terms) on both sides of the equation (inequality).
Multiplication or division of both sides of the equation (inequality) by the same term, if this takes the value 0 with no permissible substitution. In the case of inequalities, the "direction" of the inequality sign must be reversed if the number that is being multiplied by or divided by is negative.
Take logarithm, provided that all terms only take positive values for all permissible substitutions. In the case of inequalities, a distinction may have to be made for term values greater than and for term values less than or equal to 1.
If you take the root of the terms on both sides of the equals sign, you get the disjunction of two equations as an equivalent form of statement . The equation is equivalent to the disjunction . ${\ displaystyle x ^ {2} -4 = 0}$ ${\ displaystyle x = 2 \ vee x = -2}$

For quadratic inequalities with :

{\ displaystyle a \ in \ mathbb {R}, a> 0}

{\ displaystyle x ^ {2}> a \ Leftrightarrow x> {\ sqrt {a}} \ quad {\ text {or}} \ quad x <- {\ sqrt {a}},}

{\ displaystyle x ^ {2} <a \ Leftrightarrow - {\ sqrt {a}} <x <{\ sqrt {a}}.}

No equivalence conversion is, for example, squaring when solving root equations .

Equations that are considered in elementary algebra are for example:

linear equations ,
linear systems of equations ,
quadratic equations ,
simple cubic equations ,
biquadratic equations ,
Fractional equations ,
Root equations ,
simple exponential and logarithmic equations
and the associated inequalities.

The use of at least graphical pocket calculators, or even better pocket calculators with a computer algebra system, considerably expands the possibilities for solving equations or inequalities. It becomes possible to visualize sets of solutions and to dispense with complicated term transformations.

Connections

A product costs 140 € net. What does it cost gross at 19% VAT? The relationship between net price, gross price and VAT can be expressed in words like this: The gross price is obtained by adding the VAT (19% of the net price) to the net price. Expressed with word variables, the relationship is: gross price = net price + 19% of the net price. It becomes even clearer if you use letters: B = N + 19% of N. Or converted equivalent: B = 1.19 · N. This equation now describes the relationship with the associated gross prices B for all possible net prices N.

literature

Student Duden: Die Mathematik 1, Dudenverlag Mannheim 1990, ISBN 3-411-04205-2