The main Classification of real numbers Is divided into natural numbers, whole numbers, rational numbers and irrational numbers. The real numbers are represented by the letter R.
Real numbers refer to the combination of rational and irrational sets of numbers. To form these groups you need natural numbers and integers.
There are many ways in which different real numbers can be constructed or described, ranging from simpler to more complex forms, depending on the mathematical work that is to be done.
How are real numbers classified?
Natural numbers
They are the numbers that are used to count, as for example"there are four flowers in the glass".
Some definitions begin with natural numbers at 0, while other definitions start at 1. Natural numbers are those that use to count: 1,2,3,4,5,6,7,8,9,10... etc; Are used as ordinal or cardinal numbers.
Natural numbers are the basis on which many other sets of numbers can be constructed by extension: integers, rational numbers, real numbers, and complex numbers among others.
These chains of extensions make up the natural numbers canonically identified in the other number systems.
The properties of natural numbers, such as the divisibility and distribution of primary numbers, are studied in number theory.
Problems related to counting and sorting, such as enumerations and partition, are studied in combinatorics.
In common parlance, as in primary schools, natural numbers can be called accounting numbers to exclude negative integers and zero.
They have several properties, such as addition, multiplication, subtraction, division, etc.
Integer numbers
Whole numbers are those numbers that can be written without a fractional component. For example: 21, 4, 0, -76, etc. On the other hand, numbers like 8.58 or √2 are not integers.
It can be said that the integers are the complete numbers together with the negatives of the natural numbers. They are used to express money that is owed, depths with respect to sea level or temperature below zero, to name a few uses.
A set of integers consists of zero (0), positive natural numbers (1,2,3...), and negative integers (-1, -2, -3...). Generally this is called with a ZZ or with a Z in bold (Z).
Z is a sub set of the group of rational numbers Q, which in turn form the group of real numbers R. Like natural numbers, Z is an infinite accounting group.
Whole numbers form the smallest group and the smallest set of natural numbers. In the theory of algebraic numbers, integers are sometimes called irrational integers to distinguish them from algebraic integers.
Rational numbers
A rational number is any number that can be expressed as the component or fraction of two integers p / q, a numerator p and a denominator q. Since q can be equal to 1, each integer is a rational number.
The set of rational numbers, often referred to as"the rational,"is denoted by a Q.
The decimal expansion of a rational number always ends after a finite number of digits or when it begins to repeat the same finite sequence of digits again and again.
Additionally, any repeated decimal or terminal represents a rational number. These statements are true not only for base 10, but also for any other base of integer.
A real number that is not rational is called irrational. Irrational numbers include √ 2, a π and e, for example. Since the whole set of rationable numbers is countable, and the group of real numbers is not countable, it can be said that almost all real numbers are irrational.
Rational numbers can be formally defined as classes of equivalences of pairs of integers (p, q) such that q ≠ 0 or the equivalent relation defined by (p1, q1) (p2, q2) only if p1, q2 = p2q1.
Rational numbers, together with addition and multiplication, form fields that make up whole numbers and are contained by any branch containing integers.
Irrational Numbers
Irrational numbers are all real numbers that are not rational numbers; Irrational numbers can not be expressed as fractions. Rational numbers are numbers composed of fractions of integers.
As a consequence of Cantor's proof that all real numbers are non-countable and that the rational numbers are countable, we can conclude that almost all real numbers are irrational.
When the line radius of two line segments is an irrational number, it can be said that these line segments are incommensurable; Meaning that there is not a sufficient length so that each of them could be"measured"with a particular multiple integer of it.
Among the irrational numbers are the radius π of a circumference of circle to its diameter, the number of Euler (e), the golden number (φ) and the square root of two; Yet all the square roots of natural numbers are irrational. The only exception to this rule are the perfect squares.
It can be seen that when irrational numbers are posi- tionally expressed in a numeral system, (for example in decimal numbers) they do not end or repeat.
This means that they do not contain a sequence of digits, the repetition by which a line of the representation is made.
For example, the decimal representation of the number π starts with 3.14159265358979, but there is no finite number of digits that can represent π exactly, nor can they be repeated.
The proof that the decimal expansion of a rational number must end or repeat is different from the proof that a decimal extension must be a rational number; Although basic and somewhat long, these tests take some work.
Usually mathematicians do not generally take the notion of"ending or repeating"to define the concept of a rational number.
Irrational numbers can also be treated via non-continuous fractions.
References
- Classifying real numbers. Retrieved from chilimath.com.
- Natural number. Retrieved from wikipedia.org.
- Classification of numbers. Retrieved from ditutor.com.
- Retrieved from wikipedia.org.
- Irrational number. Retrieved from wikipedia.org.