About Mathematics and Real World Mathematics Applications
Time 2021-10-30 15:04:45Web Name: About Mathematics and Real World Mathematics Applications
WebSite: http://explainingmath.blogspot.com
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keywords: description:Unlocking the secrets of quantitative reasoning. Rewiring your existing math knowledge into a new, powerful web of innovation generating ideas. Axioms Discovery, Axiomatic Frontiers, Directional Thinking, Specific Consequences, Logic, Intuition, Innovation, Invention.
No comments: Thursday, July 2, 2015 One Derivation of Euler's Equation for Complex NumbersYou can download all the important posts asPDF book "Unlocking the Secrets of Quantitative Thinking".
One derivation of Euler's equation using series expansions for e^x, cos(f), sin(f).
Reference http://www.ee.nmt.edu/~elosery/lectures/Quadrature_signals.pdf
[complex numbers, Euler, Euler's identity, Euler's equation]No comments: Wednesday, July 2, 2014 Why does zero factorial (0!) equal one, i.e. 0! = 1?You can download all the important posts asPDF book "Unlocking the Secrets of Quantitative Thinking".
Here are the most useful links with answers why does zero factorial equal one, i.e. 0! = 1:
http://www.quora.com/Mathematics/Why-does-zero-factorial-0-equal-one
http://wiki.answers.com/Q/Why_is_zero_factorial_equal_to_one
http://en.wikipedia.org/wiki/Factorial
http://math.stackexchange.com/questions/25333/why-does-0-1
https://ca.answers.yahoo.com/question/index?qid=20090116132324AACQIGU
http://www.zero-factorial.com/whatis.html
http://statistics.about.com/od/ProbHelpandTutorials/a/Why-Does-Zero-Factorial-Equal-One.htm
http://mathforum.org/library/drmath/view/57128.html
[factorial, factoriel, factorial function, zero factorial]4 comments: Wednesday, June 11, 2014 What is Mathematics?You can download all the important posts asPDF book "Unlocking the Secrets of Quantitative Thinking".
Here is the link for the essay " A Mathematician's Lament" by Paul Lockhart.
"What is Mathematics, Really?" by Reuben Hersh, in Google Books.
"What is Mathematics? An Elementary Approach to Ideas and Methods", Richard Courant, Herbert Robbins, in Google Books.
Mathematical Intuition (Poincaré, Polya, Dewey) by Reuben Hersh, University ofNew Mexico, TMME, vol8, nos.12, p .35
The Number Sense : How the Mind Creates Mathematics, by Stanislas Dehaene, Google Books.
The Psychology of Number and its Applications to Methods of Teaching Arithmetic by James A. McLellan; John Dewey, at www.openlibrary.org.
How Math Can Be Applied To So Many Different Fields, Mathematics and Quantitative Reasoning web site, B. Harford
No comments: Thursday, February 28, 2013 From Basketball, Financial Math to Pure Math and BackYou can download all the important posts asPDF book "Unlocking the Secrets of Quantitative Thinking".
After some initial counting and some thinking put into it, you may haveasked yourself, what is there more to investigate about numbers? A number is anumber, there are a few operations on it, I have just seen that, a clean anddry concept, a quite straightforward count of objects you have been dealingwith. Five apples, seven pears, six pencils. The number five is common to allof them. We have abstracted it, and together with other fellow numbers (three,four, seven,, 128, 349, ...) it is a part of a number system we are familiarand we work with.
From our everyday encounters with mathematics, we may have a feeling thereare only integers present in the world of math, and that it is not really clearwhere and how those mathematicians find so many exotic numerical concepts, somany other kinds of numbers, like rational, irrational, algebraic ... Moreover,you may even think that, without some real objects to count or to measure,there would be no mathematics, and that mathematics is, actually, always linkedto a real world examples, that numbers are intrinsically linked to thequantification of things in the real world, to the objects counted, measured,that they are inseparable. You may think that a number, despite its "mathematicalpurity", somehow shares other, non mathematical properties, of the objectsit represents the count of.
In this article I will discuss these thoughts, assumptions, maybe evenmisconceptions. But, no worries, you are on the right track by very action thatyou want to put a thought about math and numbers.
Before I go to the exciting world of basketball and poker, as anillustration, let me discuss a few statements. A famous mathematician, LeopoldKronecker, once said that there are only positive integers in the mathematicalworld, and that everything else, i.e. definition of other kinds of numbers, isthe work of men. I support that view. Essentially,many mathematicians do as well. Here is the flavour of that perspective. Negativenumbers are positive numbers with a negative sign. Rational numbers are ratiosof two integers, m/n, (where n is not equal 0). Real numbers (rational andirrational) are limiting values of rational numbers sums and sequences (whichare in turn ratios of integers), convergent sums of rational numbers, whererational numbers are smaller and smaller as there are more and more of them. Aswe can see that all these numbers are, fundamentally, constructed from positiveintegers.
As for "purity" of a number here is a comment. Number has only onepersonality! Take number 5, for instance. It's the same number whether we countapples, pears, meters, cars...That's why we need labels below, or beside,numbers, to remind us what is measured, what is counted. For real world mathapplications thats absolutely necessary, because by looking at the number only,we can not conclude where the count comes from. When you write 5 + 3 = 8, youcan apply this result to any number of objects with these matching counts. So,numbers do not hold or hide properties of the objects they are counts of. As a matterof fact, you can just declare a number you will be working with, say number 5,and start using it with other numbers, adding it, subtracting it etc, withoutany reference to a real life object. No need to explain if it is a count ofanything. Pure math doesn't care aboutwho or what generated numbers, it doesn't care where the numbers are comingfrom. Math works with clear, pure numbers, and numbers only. It is a veryimportant conclusion. You may think, that properties of numbers depend on theobjects that have generated them, and there are no other intrinsic propertiesof numbers other than describing them as a part of real world objects. But, itis not so. While you can have a rich description of objects and millions of colourfulreasons why you have counted five objects, the number five, once abstracted,has properties of its own. That's why it is abstracted at the first place, as acommon property! When you read any textbook about pure math you will see thatapples, pears, coins are not part of theorem proofs.
Now, you may ask, if we have eliminated any trace of objects that a numbercan represent a count of, that might have generated the number, what are theproperties left to this abstracted number? What are the numbers'properties?
That's the focus of pure math research. Pure means that a concept of anumber is not anymore linked to any object whose count it may represent. Inpure math we do not discuss logic or reasoning why we have counted apples, orwhy we have turned left on the road and then drive 10 km, and not turned right. Pure mathis only interested in numbers provided to it. Among those properties of numbersare divisibility, which number is greater or smaller, what are the differentsets of numbers that satisfy different equations or other puzzles, differentsets of pairs of numbers and their relationships in terms of their relative differences,what are the prime numbers, how many of them are there, etc. That's what puremath is about, and these are the properties a number has.
In applied math, of course, we do care what is counted! Otherwise, wewouldn't be in situation to "apply" our results. Applied math meansthat we keep track what we have counted or measured. Don't forget though, westill deal with pure numbers when doing actual calculations, numbers are justmarked with labels, because we keep track by adding small letters beside numbers,which number represent which object. When you say 5 apples plus 3 apples is 8apples, you really do two steps. First step is you abstract number 5 from 5apples, then, abstract number 3 from 3 apples, then use pure math to add 5 and3 (5 + 3) and the result 8 you return back to the apples world! You say thereare 8 apples. You do this almost unconsciously! You see the two way streethere? When developing pure math we are interested in pure numbers only. Then,while applying math back to real world scenarios, that same number isassociated with a specific object now, while we kept in mind that the numberhas been abstracted from that or many other objects at the first place. This isalso the major advantage of mathematics as a discipline, when considering itsapplications. The advantage of math is that the results obtained by dealingwith pure numbers only, can be applied to any kind of objects that have thesame count! For instance, 5 + 3 = 8 as a pure math result is valid for any 5objects and for any 3 objects we have decided to add together, be it apples,cars, pears, rockets, membranes, stars, kisses.
While, as we have seen, pure math doesn't care where the numbers come from,when applying math we do care very much how the counts are generated, where thecounts are coming from and where the calculation results will go. We even haveinvented mechanical, electrical, electronic devices to keep track of thesecounted objects. We have all kinds of dials that keep track of fuel consumption,temperature, time, distance, speed. Imagine that! We have devices which keeptrack of counted objects so when we look at them and see number five, or seven,or nine, we will know what that number represents the count of! Say, you have severaldials in front of you, and they show all number 5. It is the same number 5,with the same numerical, mathematical, properties, but represents counts ofdifferent objects or measurements. We can say that the power of mathematics isderived from noticing that number 5 is the same for many objects andabstracting that number 5 from them, then investigating number 5 properties. Aftermathematical investigation we can go back, from pure number 5 to the realworld!
There are dials in cars, for instance, for fuel consumption, speed, time,engine temperature, ambient temperature, fan speed, engine shaft speed. If itwas not up to us, those numbers would float around, enjoying their own puritylike, 5, 23, 120, 35, 2.78 without knowing what they represent until weassigned them a proper dial units. This example shows the essential differencebetween applied and pure math, and how much is up to our thinking andinitiative, what are we going to do with the numbers and objects counted ormeasured. Pure math deals with numbers only, while in applied math we drag thenames of objects, associated them with numbers. In other words, we keep trackwhat is counted.
Now, when dealing with pure numbers, we may go to a great extent toinvestigate all kinds of numerical, mathematical properties of all kinds ofnumbers and sets of numbers. Hence a spectrum of mathematical areas like linearalgebra, calculus, real analysis, etc. These mathematical disciplines are alluseful and there is, frequently, a beauty and elegance in their results. But, often,we do not need to apply or use all those mathematical properties, and pure mathresults, in everyday situations. Excelling in some business endeavourfrequently depends on actually knowing what and why something is counted,while, at the same time, mathematics involved, can be quite simple. When I saybusiness, I mean business in usual sense, like finance, trading, engineering,but also, I mean, for instance, as we will see soon, basketball, and evenpoker.
Lets go now into a basketball game. When playing basketballwealso need to know some math, at least working with positive integers and zero.However, in the domain of basketball game, knowledge of basketball rules areway more important than math,
Those basketball rules are mostly non mathematical. Most of basketball rulesdo not deal with any kind of quantification, which doesn't make them at allless significant. Moreover, they are way more important ingredient, and representmore complex part, for that matter, of a basketball game, than adding thenumbers.
You can posses knowledge of adding integers, but without knowing basketballrules, and without knowing how to play basketball, you will not move anywherein a basketball team or in a game.Moreover, basketball rules are actual axioms of a basketball game. And,every move in the basketball court, any 30 seconds strategy development by oneteam or the other, corresponds to theorems of a basketball game! Anyuninterrupted part of the game, without fouls or penalties, is an actualtheorem proof, with basketball rules as axioms. We can say that basketballrules are those statements that define what belongs to a set "number ofscored points"! You see here how we have whole book of basketball gamerules that serve the purpose just to define what belongs to a set (of scored points). Compare that to thoseboring, and sometimes, ridiculous examples, in many math texts, with apples,pears, watermelons (although they may illustrate the point at hand well). Withridiculing the importance of rules of what belongs to a set, belittling theirsignificance and logic associated to obtain them, those authors,unintentionally, pull you away from an essential point of "applied"math. In order to define what belongs to a set, and then, count its elements(like points in basketball) you need to know areas other than math, and todevelop logic, creativity, even intuition in those non mathematical areas, inorder to decide what really belongs to a set and what needs to be counted.Because, accuracy of rules and logic to determine what belongs to a setdictates the set's cardinality, the size of the set, the number of itselements. And this is the number you will enter in all your calculations later!That number has to be accurate!
Note, also, that only knowing rules of basketball game doesn't make you afirst class player, nor your team can be a winner just knowing the rules. Youhave to develop strategies using those rules. You have to play within thoserules a winning game. The same is in math. Knowing the fundamental axioms ofmath will not make you a great mathematician per se. You have to play the "winninggame" inside math too, as you would in basketball game. You have to showcreativity in math as well, mostly in specifying theorems, and constructingtheir proofs!
In business, it is often more important to know where the numbers are comingfrom than to know in detail the numbers properties. For instance, in poker.again, only integers and rational numbers (in calculating probabilities) areinvolved ( we will skip stochastic processes and calculus for now). You have toremember that the same number 5 can be any of the card suits, and, in addition,can belong to one or more players. Note how abstracting number 5 here andtrying to develop pure math doesn't help us at all in the game. We have to goback to the real world rules, in this case world of poker,, we have to use thatabstracted number 5 and put it back to the objects it may have been abstractedfrom, in this case cards and players. You have to somehow distinguish that purenumber 5, and associate it with different suit, different player. And strategyyou develop, you do with many numbers 5, so to speak, but belonging todifferent sets, suits, players, game scenarios. Hence, being a successful pokerplayer, among other things, you need to memorize, not exotic properties ofintegers and functions, but how the same number 5 (or other number) can belongto so many different places, can be associated, linked to different players,suits, strategies, scenarios.
Lets consider another example, in finance.Any contract you have signed, for instance contract for a credit card, isactual detailed list of definitions what belongs to a certain set. For example,whether $23,789.32 belongs to your account under the conditions outlined in thecontract. Note how even your signature is a part of the definition what belongsto a set, i.e. are those $23,789.32 really belong to your account. You see,math here is quite simple, it is just a matter of declaring a rational number23.789,32, but what sets it belongs to is extraneous to mathematics, it'sin the domain of financial definitions, even in the domain of required signatures.Are you, or someone else, is going to pay the bill of $23,789.32, is a nonmathematical question (its even a legal matter), while mathematics involved isquite simple. It's a number 23,789.32.
Note, when you are paid for your basketball game, suddenly you have mathfrom two domains fused together! It may be that the number of points you scoredare directly linked to a number of dollars you will be paid. Two domains, of sport and finance, are linked togethervia monetary compensation rules,which can have quite a bit of legal background too, and all these (nonmathematical in nature!) rules dictate what number, of dollars, may be picked andassigned to you, as a basketball player, after the set of games.
No comments: Wednesday, February 27, 2013 Mathematical Proof for Enthusiasts - What It Is And What It is NotYou can download all the important posts asPDF book "Unlocking the Secrets of Quantitative Thinking".
Important things you can learn from mathematics are not about counting only,but also about mathematics methods of discovering new truths about numbers.
With the term mathematical proof wewant to indicate a logical proof, i.e. proof using logical inference rules, inthe field of mathematics, as oppose to other disciplines or area of humanactivity. Hence, it should really be a proof in the field of mathematics.Also, we have to assume, and be fully aware, that proof must be logicalanyway. There are really no illogical proofs. Proof that appears to be obtained(whatever that means) by any other way, other than using rules of logic, is nota proof at all.
Assumptions and axioms need no proof. They are starting points and theirtruth values are assumed right at the start. You have to start from somewhere.If they are wrong assumptions, axioms, the results will show to be wrong. Hence,you will have to go back and fix your fundamental axioms.
Often when you have first encountered a need or a task for a mathematicalproof, you may have asked yourself "Why do I need to prove that, it's so obvious!?".
We used to think that we need to prove something if it is not clear enoughor when there are opposite views on the subject we are debating. Sometimes,things are not so obvious, and again, we need to prove it to some party.
In order to prove something we have to have an agreement which things weconsider to be true at the first place, i.e. what are our initial, startingassumptions. Thats where the debate most likely will kick in. In most cases,debate is related to an effort to establish some axioms, i.e. initial truths, andonly after that some new logical conclusions, or proofs will and can be obtained.
The major component of a mathematical proof is the domain of mathematicalanalysis. This domain has to be well established field of mathematics, andmathematics only. The proof is still a demonstration that something is true,but it has to be true within the system of assumptions established in mathematics. The true statement, the proof, has to(logically) follow from already established truths. In other words, when usingthe phrase "Prove something in math..." it means "Show that itfollows from the set of axioms and other theorems (already proved!) in thedomain of math..".
Which axioms, premises, and theorems you will start the proof with is a matter of art, intuition, trial and error, or even true genius. You can not useapples, meters, pears, feelings, emotions, experimental setup, physicalmeasurements, to say that something is true in math, to prove a mathematicaltheorem, no matter how important or central role those real world objects orprocesses had in motivating the development of that part of mathematics. Inother words, you can not use real world examples, concepts, things, objects, realworld scenarios that, possibly, motivated theorems development, inmathematical proofs. Of course, you can use them as some sort of intuitive guidelinesto which axioms, premises, or theorems you will use to start the construction of a proof. You can use your intuition,feeling, experience, even emotions, to select starting points of a proof, tochose initial axioms, premises, or theorems in the proof steps, which, whencombined later, will make a proof. But, you can not say that, intuitively, youknow the theorem is true, and use that statement about your intuition, as anargument in a proof. You have to use mathematical axioms, already proved mathematicaltheorems (and of course logic) to prove the new theorems.
The initial, starting assumptions in mathematics are called fundamentalaxioms. Then, theorems are proved using these axioms. More theorems are provedby using the axioms and already proven theorems. Usually, it is emphasized thatyou use logical thinking, logic, to prove theorems. But, that's not sufficient.You have to use logic to prove anything, but what is important in math is thatyou use logic on mathematical axioms,and not on some assumptions and facts outside mathematics. The focus of yourlogical steps and logic constructs in mathematical proofs is constrained (butnot in any negative way) to mathematical (and not to the other fields) axiomsand theorems.
Feeling that something is "obvious" in mathematics can still be auseful feeling. It can guide you towards new theorems. But, those new theoremsstill have to be proved using mathematical concepts only, and that has to bedone by avoiding the words "obvious" and "intuition"!Stating that something is obvious in a theorem is not a proof.
Again, proving means to show that the statement is true by demonstrating itfollows, by logical rules, from established truths in mathematics, as oppose toestablished truths and facts in other domains to which mathematics may beapplied to.
As another example, we may say, in mathematical analysis, that something is"visually" obvious. Here "visual" is not part ofmathematics, and can not be used as a part of the proof, but it can play importantrole in guiding us what may be true, and how to construct the proof.
Each and every proof in math is a new, uncharted territory. If you like tobe artistic, original, to explore unknown, to be creative, then try toconstruct math proofs.
No one can teach you, i.e. there is no ready to use formula to follow, howto do proofs in mathematics. Math proof is the place where you can show yourtrue, original thoughts. No comments: More About the Concept of a Set and the Concept of a NumberYou can download all the important posts asPDF book "Unlocking the Secrets of Quantitative Thinking".
For instance, let's take a look at the cars on a highway, apples on a table,coffee cups in a coffee shop, apples in the basket. Without our intellectinitiative, our thought action, will, our specific direction of thinking, objectswill sit on the table or in their space, physically undisturbed and conceptuallyunanalyzed. They are and will be apples, cars, coffee cups, pears. But then, onthe other hand, we can think of them in any way we wish. We can think how wefeel about them, are they edible, we can think about theory of color, theirsocial value, utility value, psychological impressions they make. We can thinkof them in any way we want or find interesting or useful, or we can think ofthem for amusement too. They are objects in the way they are and they need notto be members of any set, i.e. we dont need to count them.
Now, imagine that our discourse of thought is to start thinking of them interms of groups or collections, what whatever reason. Remember, it's just cameto our mind that we can think of objects in that way. The fact that the applesare on the table and it looks like they are in a group is just a coincidence. Wewant to form a collection of objects in our mind. Hence, apples on a table arenot in a group, in a set yet. They are just spatially close to each other.Objects are still objects, with infinite number of conceptual contexts we canput them in.
Again, one of the ways to think about them is to put them in a group, forwhatever reason we find! We do not need to collect into group only similarobjects, like, only apples or only cars. Set membership is not always dictatedby common properties of objects. Set membership is defined in the way we wantto define it! For example, we can form set of all objects that has no commonproperty! We can form a group of any kind of objects, if our criterion says so.We can even be just amused to group objects together in our mind. Hence, theset can be specified as all objects we are amused to put together. Like, onegroup of a few apples, a car, and several coffee cups. Or, a collection ofapples only. Or, another collection of cars and coffee cups only. All in ourmind, because, from many directions of thinking we have chosen the one in whichwe put objects together into a collection.
Without our initiative, our thought action, objects will float around bythemselves, classified or not, and without being member of any set! Objects areonly objects. It is us who grouped them into sets, in our minds.In reality, they are still objects,sitting on the table, driven around on highways, doing other function that areintrinsic to them or they are designed for, or they are analyzed in any otherway or within another scientific field.
Since, as we have seen, we invented, discovered a direction of thinkingwhich did not exist just a minute ago, to think of objects in a group, we maywant to proceed further with our analysis.
Roughly speaking, with the group, collection of objects we have introduced aconcept of a set. Note how arbitrary we even gave name to our new thought thatresulted in grouping objects into collections. We had to label it somehow.Let's use the word set!
Now, if we give a bit more thought into set, we can see that set can haveproperties even independent of objects that make it. Of course, for us, in realworld scenarios, and set applications, it is of high importance whether wecounted apples or cars. We have to keep tracks what we have counted. However,there are properties of sets that can be used for any kind of counted objects.Number of elements in a set is such one property. If we play more with countsand number of elements in a set we can discover quite interesting things. Threeobjects plus six objects is always nine objects, no matter what we have counted!The result 3 + 6 = 9 we can use in any set of objects imaginable, and itwill always be true. Now, we can see that we can deal with numbers only,discover rules about them, in this case related to addition that can be usedfor any objects we may count.
Every real world example for mathematics can generate mathematical concepts,mainly sets, numbers, sets of numbers, pair of numbers. Once obtained, allthese pure math concepts can be, and are, analyzed independently from realworld and situations. They can be analyzed in their own world, withoutreferencing any real world object or scenario they have been motivated with orthat might have generate them, or any real world example they are abstractedfrom. How, then, conception of the math problems come into realization, if thereal world scenarios are eliminated, filtered out? Roughly speaking, you willuse word IF to construct starting points. Note that this word IF replacesreal world scenarios by stipulating what count or math concept is given as thestarting point.
But, it is to expect. Since a number 5 is an abstracted count thatrepresents a number of any objects as long as there are 5 of them, we can not,by looking at number 5, tell which objects they represent. And we do not needto that since we investigate properties of sets and numbers between themselves,like their divisibility, which number is bigger, etc. All these pure numberproperties are valid for any objects we count and obtain that number! Quiteamazing!
Moreover, even while you read a book in pure math like "TopologyFundamentals" or "Real Variable Analysis" or "LinearAlgebra" you can be sure that every set, every number, every set ofnumbers mentioned in their axioms and theorems can represent abstractedquantity, common count, and abstracted number of millions different objectsthat can be counted, measured, quantified, and that have the same count denotedby the number you are dealing with. Hence you can learn math in the way of thinkingonly of pure numbers or sets, as a separate concepts from real world objects,knowing they are abstraction of so many different real world, countable objectsor quantifiable processes (with the same, common count), or, you can use,reference, some real world examples as helper framework, so to speak, toillustrate some of pure mathematical relationships, numbers, and sets, whileyou will still be dealing, really, with pure numbers and sets.
There may be, also, a question, why it is important to discover propertiesof complements, unions, intersections, of sets, at all? These concepts look sosimple, so obvious, how such a simple concepts can be applied to so manycomplex fields?
Lets find out! Looking at sets, there is really only a few things you cando with them. You can create their unions, intersections, complements, and thenfind out their cardinalities, i.e. sizes of sets, how many elements are therein a set. There is nothing else there. Note how, in math, it is sufficient todeclare sets that are different from each other, separate from each other. Youdont have to elaborate what are the sets of, in mathematics. You do not evenneed to use labels for sets, A, B, C, Its sufficient to imagine two (or more)different sets. In mathematics, there are no apples, meters, pears, cars,seconds, kilograms, etc. So, if we remove all the properties of these objects,what properties are left to work with sets then? Now, note one essential thinghere! By working with sets only, by creating unions, complements, intersectionsof sets, you obtain their different cardinalities.And, in most cases, we are after thesecardinalities in set theory, as one of the major properties of sets, andhence in mathematics. Roughly speaking, cardinality is the size of a set, butalso, after some definition polishing, it represents a definition of a numbertoo. Hence, if we get a good hold on union, complement, intersectionconstructions and identity when working with sets, we have a good hold on theircardinalities and hence counts and numbers. And, again, that's what we areafter, in general, in mathematics!
As for real world examples, you may ask, how distant is set theory or puremathematical, number theory from real world applications? Not distant at all.Remember the fact how we obtained a number? A number is an abstraction of allcounted objects with the same count, of all sets of objects with the samenumber of elements (apples, cars, rockets, tables, coffee cups, etc). Hence,the result we have obtained by dealing with each pure, abstracted number can beimmediately applied to real world by deciding what that count represents orwhat objects we will count that many times. Or, the other way is, even if wedealt with pure math, pure numbers all the time, we would've kept track what iscounted, with which objects we have started with. There is only one number 5 in mathematics, but in real worldapplications we can assign number 5 to as many objects as we want. Hence, 5apples, 5 cars, 5 rockets, 5 thoughts, 5 pencils, 5 engines. In real world mathapplications scenarios it matter what you have counted. But that fact andinformation, what you have counted (cars, rockets, engines, ..) is not part ofmath, as we have just seen. Math needs to know only about a specific numberobtained. Number 5 obtain as a number of cars is the same as number 5 obtainedfrom counting apples, from the mathematical point of view. But, it can and doesrepresent sizes of two sets, cars and apples. For math, it is sufficient towrite 5, 5 to tell there are two counts, but for us, it is practical to drag adescription from the real world, cars, apples, to keep track what number 5represents. 1 comment: Older PostsHomeSubscribe to:Posts (Atom)Think multi-disciplinary!
Dynamic, creative thinking method with examples in software, engineering, math, art, architecture, physics and more. June, 2020.Unlocking the Secrets of Quantitative Reasoning - 88 pages
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