Translations of Encyclopedia about Mathematics

Algebra is a study of numbers, their fields, and calculation with them. To write numbers, we use the mathematical symbols created for them (numerical figures). A number can be expressed using various symbols, where each symbol identifies a certain number.

If some mathematical relation is valid for some number (or amount of numbers), such as a law, a rule of computation, a formula and so on, it makes sense to write this down in a statement beforehand – to express that the relation is generally binding for a certain amount of numbers in a given set.

The used mathematical symbols are variable in a defined variable number set. This must always be stated, unless it is absolutely obvious.

When writing our statements, we use similar symbols as if we were writing down music. However, in this case, we do not use letters but rather numbers, which means we can use certain numbers to represent certain quantities while keeping the statement true. The variable represents a certain number. Different variables can represent either different numbers or the same number.

Arithmetical Operations from the First to Third Order

Certain arithmetical operations are performed in defined numerical sets. We can divide basic arithmetical operations into three categories: addition and subtraction; multiplication and division; and exponentiation, extraction of root and calculation of logarithms.

Overview of Numerical Sets

Natural numbers historically make up the first numerical set, have been used to count objects in our surroundings and are used above all as:

1. a certain amount (cardinal numbers: one, two, three…) or
2. ordinal numbers (first, second, third, fourth,…).

The first number is the number one. The number zero indicates that none of a certain object exists. This number was considered a number only until later in history. If we consider a and b as two natural numbers, then the result of their arithmetical operation, such as adding, multiplying or exponentiation, will always generate another natural number. Therefore, these operations take place only within the set of natural numbers without extending beyond it. With other mathematical operations, such as subtraction and division, this does not need to apply. In this case, the natural numbers would have to fulfill certain conditions. If not, then the result would not be a natural number, which means that the arithmetical function has transcended the set of natural numbers.

If a mathematical operation based on the set of natural numbers is to be executable without restriction, the set of numbers used must be increased. This leads to an expansion of the number set, or the creation of a new and broader set. This would require the definition of new numbers so as to give the various mathematical operations a definable result. It would also be necessary to base the same mathematical operation on the new numbers which was possible with the natural numbers. Regarding this, there are two points of view which must be adhered to:

1. older sets of numbers must be part of the new ones, which means that the new ones should cover a broader set than the older ones and that the set of the older numbers should be a subset of the new number set;
2. the rules for calculating with the old set of numbers should, if possible, also apply to the new set (Principle of Permanence by Herman Hankel, German mathematician, 1839-1873).

1. Subtraction is possible without restriction thanks to the expansion of the number set by including negative numbers. This new set of numbers is called whole numbers, which are such numbers as –16, -3, -1, 0, +1, +2, +3
2. Division is possible without restriction with the expansion of the set of whole numbers to include absolute numbers, this expanded set called rational numbers. Rational numbers are necessary when dividing with negative numbers and includes the set of absolute numbers and natural numbers.
3. To be able to extract the root and calculate with logarithms without restriction, the set of rational numbers needed to be broadened to included the set of irrational numbers, whose inclusion created the set of real numbers, which contains all the above sets.
4. Extraction of root and computing with logarithms is now unrestricted with rational numbers if real numbers are expanded to include imaginary number, which creates the new set of complex numbers. This set of numbers is the maximum expandable set, meaning that all arithmetical operations are possible within in without restriction and no further expansion is necessary.

In mathematics and other areas, interesting relations can be found with quantities: relations between the number of employees and a company’s turnover; between the speed of a car and consumption of gasoline; between a day’s progress and the temperature of air; between the diameter of a sphere and its volume, etc.

Under certain conditions (which are often fulfilled) in mathematics, the relation between different quantities is called a function. Thanks to their theoretical and practical significance, this concept is possibly the most important in all of mathematics. Now we would like to ask ourselves under what conditions can we call relations between quantities functions (which requires us to define what a function is) while learning of some typical examples of functional relations and how they are expressed.

The function f: X® Y is a formula with which the set of numbers X is converted to the set of numbers Y, where each value of X is allocated a value of Y. The set of numbers X is called the defining set, and Y is called the set value of function f.

Note:

1. We speak of functions only when we have some clear picture or allocation.
2. X is called the argument or the independent variable. The definition of "independent" refers to the fact that x can be freely and independently chosen from the defined set of X without any restriction.
3. Contrary to this, the values of y are always determined by function for each x, they is dependent on the value of x because of the variable, for which reason we call them function values.
4. For the defined set of X we also use a set of arguments and for the set of values Y we use also a picture set. The defined set is therefore the set of independent variables and the set of values is the set of dependent variables.
5. We can also express the function using other symbols, not only f but such symbols as F, g, h, sin, tan, lg, etc.

Expression of Functions

Functions can be described and written in the most varied of methods, such as:

1. verbal explanations, how the individual values are assigned;
2. using a table of values;
3. using graphs;
4. by analytical expressions.

It applies that (A) is possible in the entire defined set, (B) and (C) do not contain the entire set of assigned pairs f, and (D) fails with functions explaining the actual content or which were attained using abstraction.

Note:

The table of values should display all observed values.
The units on the coordinate axes are chosen according to the defined and function sets.

Basic Functions / Inverse Functions

Functions can be inversible if each parallel of axis X intersects at most once with the function graph.

From the function y = f(x) = 2x – 4, their inversion to the inverse function generates:

y = 2x – 4
y + 4 = 2x
1/2 . (y + 4) = x

the inverse function

However, because an argument is usually written like x, we get:

If some function in its defined set is strictly monotonic, then it is also inversible. An inversible function is one which has an unequivocal inverse assignment, which means that not only some definite y value can be assigned to each value of x but that some definite value of x can be assigned to each value of y.

The defined set of function f is the set of values of the inverse function and the contrary. The graph of inverse f-1 functions can be attained by depicting the function f graph on a y=x coordinate.

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