Numerical sets are the categories in which numbers are classified, based on their different characteristics. For example, whether or not they have a decimal part, or if they have a negative sign in front.
Number sets are, in other words, the types of numbers that people have at our disposal to perform operations, both everyday and at a more sophisticated level (by engineers or scientists, for example).
These sets are the creation of the human mind, and are part of an abstraction. That is, they do not exist materially speaking.
Next, we will explain the main examples of numerical sets, which can be seen represented in the image above.
Natural numbers are those that take discrete intervals of one unit, and begin with the number 1, extending to infinity. One way to distinguish these numbers is as those used for counting.
In formal terms, the set of natural numbers is expressed with the letter N and as follows:
The integers include the natural numbers, plus those that also take discrete intervals, but have a negative sign before them, and zero is included. We can express it as follows:
Within this set, each number has its corresponding opposite with another sign. For example, the opposite of 10 is -10.
Rational numbers include not only those integers, but also those that can be expressed as the quotient of two whole numbers, so they can have a decimal part.
The set of rational numbers can be expressed as follows:
It should be noted that the decimal part of a rational number can be repeated indefinitely, in which case it is called periodic. Thus, it can be a pure periodic, when the decimal part contains one or more numbers that repeat to infinity, or a mixed periodic, when after the decimal point there is some number, or some numbers, that do not repeat themselves, while that the rest does extend to infinity.
Irrational numbers cannot be expressed as the quotient of two whole numbers, nor can a repeating periodic part be specified, although they extend to infinity.
Irrational numbers and rational numbers are disjoint sets. That is, they do not have elements in common.
Let’s look at some examples of irrational numbers:
Real numbers are those that include both rational and irrational numbers.
That is, the real numbers go from minus infinity to most infinity.
Imaginary numbers are the product of any real number by the imaginary unit, that is, by the square root of -1.
Imaginary numbers can be expressed as follows:
r = n i
- r is an imaginary number.
- n is a real number.
- i is the imaginary unit.
It should be noted that imaginary numbers are not part of real numbers.
Complex numbers are those that have a real part and an imaginary part. Its structure is as follows:
h + ui
- h is a real number.
- u is the imaginary part.
- i is the imaginary unit.