Numerical sets. Types of numbers and application

Introduction

Numer¬≠ic sets are used to group num¬≠bers that have sim¬≠i¬≠lar char¬≠ac¬≠ter¬≠is¬≠tics. It is one of the basic con¬≠cepts of math¬≠e¬≠mat¬≠ics so it is impor¬≠tant to under¬≠stand what they are and what char¬≠ac¬≠ter¬≠is¬≠tics each has.

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What are number sets used for?

Numer¬≠ic sets are used to sep¬≠a¬≠rate num¬≠bers into dif¬≠fer¬≠ent class¬≠es that have sim¬≠i¬≠lar properties. 

We must see this sim¬≠ply as a form of orga¬≠ni¬≠za¬≠tion, in which giv¬≠en any num¬≠ber we say that this num¬≠ber belongs to such a set.

Numerical Sets

The basic numer­i­cal sets are as follows:

  • Nat¬≠ur¬≠al ‚ÄĒ ‚Ąē
  • Inte¬≠ger ‚ÄĒ ‚Ą§
  • Ratio¬≠nal ‚ÄĒ ‚Ąö
  • Real ‚ÄĒ ‚ĄĚ
  • Com¬≠plex ‚ÄĒ ‚Ąā

Each more gen­er­al set encom­pass­es the pre­vi­ous set, i.e. for exam­ple all nat­ur­al num­bers are inte­gers, but not all inte­gers are natural.

Set of Natural Numbers (‚Ąē)

This set is made up of the num¬≠bers {1,2,3,‚Ķ} (the sus¬≠pen¬≠sion points indi¬≠cate that the enu¬≠mer¬≠a¬≠tion con¬≠tin¬≠ues indef¬≠i¬≠nite¬≠ly), these num¬≠bers are all pos¬≠i¬≠tive and rep¬≠re¬≠sent inte¬≠ger mag¬≠ni¬≠tudes, i.e. they have no dec¬≠i¬≠mal part.

Set of integer numbers (‚Ą§)

If to the nat­ur­al num­bers we add the num­ber 0 and the neg­a­tive num­bers with­out dec­i­mal part we obtain the set of inte­gers. {…,-3,-2,-1,0,1,2,3,…}.

With neg¬≠a¬≠tive num¬≠bers we can rep¬≠re¬≠sent sub¬≠trac¬≠tion oper¬≠a¬≠tions, miss¬≠ing mag¬≠ni¬≠tudes, val¬≠ues below the ref¬≠er¬≠ence zero and so on.

Some exam­ples are out­go­ing flows of mon­ey, that is, mon­ey that we pay and sub­tract from what we have; tem­per­a­tures below zero are expressed as neg­a­tive val­ues of degrees Celsius.

Sets of Rational Numbers (‚Ąö)

The set of ratio¬≠nal num¬≠bers aris¬≠es from mak¬≠ing divi¬≠sions of two whole num¬≠bers. For exam¬≠ple 1 divid¬≠ed 2 is an oper¬≠a¬≠tion that results in a num¬≠ber that is small¬≠er than 1 but larg¬≠er than 0.

These num¬≠bers are used to rep¬≠re¬≠sent non inte¬≠ger mag¬≠ni¬≠tudes, for exam¬≠ple vari¬≠ables of con¬≠tin¬≠u¬≠ous nature such as speed, weight, elec¬≠tric cur¬≠rent; to express frac¬≠tion¬≠al quan¬≠ti¬≠ties such as half a kilo of flour are 0.5 kg of flour. 

Set of Real Numbers (‚ĄĚ)

If to the set of ratio­nal num­bers we add the set of irra­tional num­bers we get the set of real numbers.

Irra­tional num­bers arise from per­form­ing cer­tain oper­a­tions and it is not pos­si­ble to express them as the quo­tient between two whole numbers.

An exam­ple of this type of num­bers is the well-known Pi num­ber which is com­posed of infi­nite dec­i­mal digits.

Set of Complex Numbers (‚Ąā)

If we add imag­i­nary num­bers to the set of real num­bers, we get the set of com­plex num­bers. Any num­ber we choose will be, in gen­er­al terms, a com­plex number.

The imag¬≠i¬≠nary num¬≠ber i is the result of the square root of ‚ÄĎ1.

Com­plex num­bers are com­posed of a real part and an imag­i­nary part and are very use­ful for the study of elec­tri­cal cir­cuits involv­ing capac­i­tive or induc­tive ele­ments. They are also used in Fouri­er trans­form calculations.

Numerical Sets in Videogames

Modeling of game elements

To mod¬≠el the behav¬≠ior of dif¬≠fer¬≠ent ele¬≠ments in our game we will prob¬≠a¬≠bly need to use dif¬≠fer¬≠ent types of numbers. 

For exam­ple a char­ac­ter may have a stan­dard of liv­ing rep­re­sent­ed by hearts. When it is dam­aged it los­es a heart and when it takes a heal­ing potion it recov­ers a heart. In this case the health lev­el is a whole magnitude.

We can also think, for exam¬≠ple, that each heart can be divid¬≠ed into four parts and each blow received takes away a quar¬≠ter of our heart. In addi¬≠tion it could be that a strong ene¬≠my takes away three quar¬≠ters of our heart with each blow. In this case we are think¬≠ing of hearts as ratio¬≠nal mag¬≠ni¬≠tudes and we are going to need dec¬≠i¬≠mal num¬≠bers to rep¬≠re¬≠sent them.

It all depends on how we want our mod¬≠el to be.

Numerical sets in programming

To rep­re­sent the dif­fer­ent types of num­bers we need to build mod­els, we have ele­ments called vari­ables, which are spaces in the mem­o­ry where infor­ma­tion is stored.

Some time ago I made a video about what a vari­able in pro­gram­ming is and also wrote an arti­cle for this page.

Types of numerical variables in C#

The pro­gram­ming lan­guage we use in Uni­ty offers us dif­fer­ent types of variables.

Integers

To rep¬≠re¬≠sent inte¬≠ger mag¬≠ni¬≠tudes we have for exam¬≠ple the byte, the short, int and long

Each has its own range of rep¬≠re¬≠sen¬≠ta¬≠tion, for exam¬≠ple the int allows us to rep¬≠re¬≠sent inte¬≠gers that are between ‚ÄĎ2,147,483,648 and 2,147,483,647. The short instead allows us to rep¬≠re¬≠sent num¬≠bers between ‚ÄĎ32,768 and 32,767, which is also a fair¬≠ly large range.

By default we are going to use the int vari­able type to rep­re­sent inte­ger magnitudes.

Reals

The float­ing point sys­tem is used to rep­re­sent dec­i­mal val­ues in computation.

We have 32-bit float type vari­ables, 64-bit dou­ble type vari­ables and 128-bit dec­i­mal type vari­ables. Each one has a cer­tain range of rep­re­sen­ta­tion and precision.

In com¬≠pu¬≠ta¬≠tion, infi¬≠nite dig¬≠its can¬≠not be stored, so irra¬≠tional num¬≠bers will be rep¬≠re¬≠sent¬≠ed as ratio¬≠nal num¬≠bers and a trun¬≠ca¬≠tion error will arise.

Complex

Com­plex num­bers can be rep­re­sent­ed with two real num­bers, one is the real part and the oth­er is the imag­i­nary part, the lat­ter will be mul­ti­plied by the num­ber i. As expressed in the fol­low­ing equation:

Z = X + i . Y

X and Y being real num­bers and Z being a com­plex number.

We can rep­re­sent com­plex num­bers using two floats with sig­nif­i­cant names for the real and imag­i­nary parts, we can use two-com­po­nent vec­tors, or we can use the Com­plex class of C#.

Conclusion

Numer¬≠ic sets are names we use to clas¬≠si¬≠fy the dif¬≠fer¬≠ent types of num¬≠bers that exist.

Know¬≠ing the dif¬≠fer¬≠ent types of num¬≠bers and their prop¬≠er¬≠ties will help us in build¬≠ing mod¬≠els for our games.

Para imple¬≠men¬≠tar los n√ļmeros en pro¬≠gra¬≠maci√≥n con¬≠ta¬≠mos con ele¬≠men¬≠tos lla¬≠ma¬≠dos vari¬≠ables. Exis¬≠ten dis¬≠tin¬≠tos tipos de vari¬≠ables para rep¬≠re¬≠sen¬≠tar n√ļmeros enteros y racionales, cada tipo tiene su pro¬≠pio ran¬≠go de representaci√≥n.

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