Whole numbers must be positive

The division of the numbers¶

Natural numbers¶

The payment used to count things is called the set of natural numbers :

The set of natural numbers without the number is shown with the symbol shown.

Order of natural numbers

With the help of natural numbers one can count how many elements are contained in a set of things, for example how many apples are in a box. [1] It is thus also possible to compare the number of elements of two different quantities, for example to check whether there are the same number of apples in two boxes or in which box there are more apples.

The boxes can thus be sorted, i.e. sorted based on the number of apples they contain. A “number line” is often chosen as the graphic form of representation, with the numbers arranged in ascending order from left to right according to their size.

Calculating with natural numbers

Lots of things can be enlarged by adding more elements. Mathematically, this corresponds to adding two natural numbers. The result of an addition is called the sum.

Likewise, a set of things can be reduced by removing individual elements, with the condition that no more elements can be removed from the set than are contained in it. Mathematically, this corresponds to a subtraction of two natural numbers. [2] The result of a subtraction is called a difference.

A multiplication of two natural numbers is also always possible; mathematically, it corresponds to performing an addition multiple times. The result, called the product, can again be represented by a natural number, the size of which corresponds to the respective multiple of the original number.

A division of two natural numbers, i.e. a division of a set of things into several items, gives an integer result if and only if the number of elements in the set is a multiple of the number of items - otherwise there is a remainder that turns out to be The whole cannot be further divided.

Whole numbers¶

In order to represent a lack of things numerically, the natural numbers are not enough. The set of natural numbers however, it can be related to the set of integers expand.

The whole numbers as a superset of the natural numbers

All natural numbers are a subset of the set of whole numbers contain. In addition, there is a corresponding negative “opposite number” for every natural number, which expresses a lack of the corresponding value. The number zero is used to represent the event that no element is present (but also none is missing).

The whole numbers can thus be represented as a number line in the same way as the natural numbers. The individual numbers are again sorted according to their size in ascending order from left to right.

While the minus sign must always be added to negative numbers, the plus sign can be omitted for positive numbers.

Calculating with whole numbers

By expanding the natural numbers to include the negative numbers, not only any addition and multiplication, but also any subtraction can be carried out with the help of the whole numbers.

An illustrative representation of negative numbers is not immediately possible, since the number of elements in a set is always greater than or equal to zero - rather negative numbers can be understood as quantities that are able to compensate for correspondingly large positive quantities - like a pile of earth a correspondingly large one Can fill a hole in the ground.

As a restriction, however, the extended system of integers remains that a division of two numbers is only possible if the first number (the dividend) is an integer multiple of the second number (the divisor) - otherwise, the division does not remain a further divisible remainder.

Rational numbers¶

Rational numbers (sometimes also called “fractions”) are an extension of the number range of integers to allow a general division of two (or more) numbers.

The rational numbers as a superset of the whole numbers

All integers are a subset of the set of rational numbers contain. In addition, all numbers that are a fraction of two whole numbers are added as additional elements and display:

The number above the fraction line is called the numerator, the number below the fraction line as the denominator of the fraction. The only condition is that it is not allowed to divide by zero, so must be. [3]

The rational numbers can also be arranged according to their size as a number line; the whole numbers are embedded as part of the rational numbers at the appropriate places. [4]

The rational numbers are “close” to one another, so in the “gaps” between two whole numbers there are an infinite number of values ​​that can be represented as rational numbers. This can be clearly explained by the fact that, for example, every natural number a reciprocal can be assigned for which the following applies:

In terms of amount, the same also applies to reciprocal values ​​of negative numbers; here only the sign has to be observed. [5]

Expand and simplify fractions

A peculiarity of rational numbers is that one and the same number can be represented by several equivalent fractions. The following applies:

In general, any fraction can be converted into another, equally large fraction by taking both the numerator as well as the denominator multiplied by any integer. This method, which occurs frequently in practice, is called “expanding” a fraction.

Examples:

  • On the occasion of a celebration, you want to distribute pieces of cake. For example, if a guest should get a quarter of a cake, it can be cut in half and thus served two eighth pieces.

In the opposite case, a fraction whose numerator and denominator have (at least) one common factor can be simplified to a rational number of equal value by reducing the common factor (or reducing the common factors).

Example:

Rounding fractions

Every rational number can be represented by an integer part and a residual fraction, the value of which is less than one. If this residual fraction is also to be specified as a decimal number, two different cases can arise:

  • Some fractions can be represented as decimal numbers with a finite number of decimal places.

    Example:

  • Some fractions correspond to a decimal number with a finite period. With numbers like this, one or more decimal places are repeated infinitely often from a certain point onwards.

    Examples:

When calculating with decimal numbers, only a finite number of decimal places can be taken into account; rational numbers are therefore rounded to a desired degree of accuracy. This accuracy is determined by specifying a certain number of "counting" digits, i.e. digits except for zeros at the beginning or at the end.

Examples:

Excessive counting digits will simulate unjustified accuracy in the results of measurements or estimates. To avoid this, the respective numbers are increased or. rounded. To do this, the superfluous digits are first replaced by zeros. Then the last non-superfluous digit is either increased by one ("round up" if the first superfluous digit is) or left unchanged ("round off").

In the case of physical quantities, the number of counting digits implicitly also indicates the measurement accuracy. For example, a length specification of indicates a measurement accuracy in the millimeter range, while an indication of only means a measurement accuracy in the centimeter range. [6]

In order to avoid too large rounding errors, however, early rounding should be avoided for every calculation and the rounding should instead only be carried out at the end (in the result).

Calculating with rational numbers

By using rational numbers, all four basic arithmetic operations - apart from division by zero - can be carried out without restriction and combined with one another as desired:

Further information on calculating with rational numbers is described in the section Calculation of fractions.

Real numbers¶

A large number of mathematical problems cannot be solved with the help of rational numbers. For example, there is no such thing as a rational number showing the equation solves; likewise there is no rational number that represents the ratio from the diagonal of a square and its side length or the ratio from the scope and the diameter of a circle could express. In order to eliminate such “deficiencies”, the range of rational numbers can be expanded to include the range of real numbers.

The newly added numbers, for example or , are referred to as "irrational" numbers. Although they can be classified according to their value in the number line, they cannot be expressed by any rational number and have an infinite number of non-periodic decimal places in the representation as a decimal number.

Special symbols are used for two particularly important real numbers:

  • The number is referred to as the "circle number". It gives the relationship between the diameter and the scope of a circle to:

Calculating with real numbers

With the help of the real numbers, not only can all four basic arithmetic operations - apart from division by zero - be carried out without restriction; The exponentiation of arbitrary real numbers and the extraction of the root of non-negative real numbers always produce unambiguous results.

If square, cube and general roots are calculated with the help of a pocket calculator or computer, the often irrational results are rounded in the same way as when rounding fractions according to the possible display accuracy.

Complex numbers¶

Complex numbers represent an extension of the number range of real numbers. The basis for this was the considerations of Gerolamo Cardano and Rafael Bombelli, in which way roots of negative numbers could be defined.

The number range created in this way of complex numbers has proven extremely useful for a variety of applications in the natural and engineering sciences. However, a complex number can no longer be represented by a single number, but rather forms an ordered pair of a two-dimensional vector space.

Since complex numbers are not considered in the current curricula, interested readers are only referred to the excursus: Complex numbers.


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