Real numbers

Real numbers

The set of real numbers belongs in mathematics to the number line that includes rational numbers and irrational numbers.
This means that they include all the positive and negative numbers, the zero symbol, and the numbers that can not be
expressed by fractions of two integers that have non-zero numbers as their denominator.

A real number can be expressed in different ways, on the one hand there are real numbers that can be expressed very
easily, since they do not have complex rules to do so. These are the integers and the fractionals, such as number
67
which is an integer, or also the
3
4
, which is a fractional number composed of two integers, whose numerator is
3
and its denominator is
4
. However, there are also other numbers that can be expressed under different, more complex mathematical rules, such as
numbers whose decimal numbers are infinite, such as the number
π
or
√
2
and that are used to perform mathematical calculations but can not be represented as a unique numeric symbol.

The real numbers are represented by the letter
R
, and appear by the need to perform more complex calculations since in times such as between the sixteenth and
seventeenth century, new figures were needed for technological advances that could no longer be represented by
approximate figures or by colloquial expressions for their inaccuracy. The rigor of the advance of humanity from its
tools, made necessary the creation of new mathematical expressions that give greater accuracy to the calculations.

Therefore, the set of real numbers was formed from other subsets of numbers that arose from needs in mathematics, such
as negative numbers and fractional numbers and decimals. In Europe, cradle of science in modernity, negative numbers
were not used until the late seventeenth century, however, and had been thought many centuries ago by cultures such as
Chinese and Hindu. It was even possible to discard the solutions of calculations that had negative results, because they
were considered unreal numbers.

The fractional numbers for their part, were used by the Egyptians to solve different problems. But it is in the Greek
culture that the present use of rationals is extracted, from rations of numbers, since they were used to define the
space between musical notes with harmonic relations that corresponded to divisions in the melodies of sound. Thus began
to see fractions in other things and substances.

From there, the complexity of the calculations begins to deepen and it is up to the theorem of Pythagoras that arise the
irrational numbers that were discussed, where the decimals of the fraction are infinite and therefore are not
expressible in unique numbers. From here comes the, perhaps, first irrational number known. From the theorem posed as
the Pythagorean constant, whose figure arises from the length of the hypotenuse of a right triangle whose length of each
of its legs is
1
, the figure obtained is
√
2
.So, the concept of real numbers is that they are the numbers that can be expressed with decimals, including those that
have decimals in infinite expansion. This is because in the logic of real numbers, there are no exact numbers. That is,
the accuracy of a result is marked by the infinite expansion of the decimals of a number, whose best example is
π
, and paradoxically, this is not an exact number, since it comes from the division of the circumference for the diameter
of a perfect circle. Clarifying better with another example, is the division of
10
÷
3
whose answer is
3
,
333333333333333
...
Real Numbers System
The system of real numbers consists mainly of two large sets, that of rational numbers which are those that can be
expressed as the division of two integers as
3
4
,
1
5
, even an integer can be expressed as a fraction, since the whole number can be divided for
1
without changing its essence, for example the number
8
can be expressed in fraction like this
8
1
; while the other great set of the system of real numbers is that of irrational numbers whose decimal representation is
expansive, infinite and aperiodic.

Irrational numbers are a set in themselves but, in turn, rational numbers have subsets that are: non-whole fractions
with their respective negative notations; the whole numbers; inside the whole numbers are the negative ones and the
positive integers; the latter in turn include natural numbers and zero. To clarify this conjunction, you can graph as in
the diagram above.

Otherwise, a conceptual map of real numbers is shown below:

Real Numbers System

 Representation of real numbers
On the number line, the representation of real numbers can be done with an approximate accuracy, however, techniques can
be used to represent them accurately. As in the following example of
√
7
:

There you can see that the root of 7 can be decomposed to draw a triangle that complies with the Pythagorean theorem.
First, 7 is decomposed into sum of squares

The addends of this addition will be the points on the Cartesian axis that will give us the location of the number in
each of the axes of the plane. The root of three. For this, the root of 2 or
√
2
must first be represented
, which is obtained by drawing a triangle whose legs have a value of one and whose hypotenuse will be equal to
√
2
. The top vertex should then be moved in a circular manner and with a pivot to zero until it reaches the horizontal line
or X axis

With this representation done, we proceed to search
√
3
, since when breaking this number, we obtain that:

Therefore, on the number line you must place a point between these two addends, be
1
and
√
2
so that the graph, on the previous graph would look like this:

Finally, we already have the location of
√
3
on the X axis and
2
on the Y axis. Now proceed to locate
√
7
on the number line, like this:(numbers facts)