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Viewing a number comparatively as an integer, real number, etc.: Any thoughts?

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  • Viewing a number comparatively as an integer, real number, etc.: Any thoughts?

    I find it interesting to view a number comparatively as a counting number, a whole number, an integer, a real number, etc. etc. etc. I feel that going through this process can lead us to a further understanding of math, of engineering, of logic, etc. I will throw some questions out there for your possible consideration. As a weird side note, years ago I did a similar thread on Linkedin within an interest group. It drew quite a few comments. One day the thread was gone with no explanation. Only time that ever happened. I asked for an explanation and never got one. I started the thread as looking at the number "4". Only thing I could come up with is, that once when I was involved in selling some real estate, the real estate agent said there might be some reduced interest, because some people consider the number unlucky.

    And just to be clear, we all understand the symbol "4" has clearly different meanings when considered as an integer, a complex number (4 + 0i) etc. In the same way the word "run" or the word "set" has clearly different meanings.

    So, some questions:

    Is there a natural progression to the way human cultures view numbers?

    Does the way it is presented in math classes differ from the above, and is this interesting or informative?

    Are there some numbers which are particularly inherently interesting?

    Do mathematical operations and concepts develop in a natural order - counting, comparison as to quantity, estimation, negation, etc. etc. etc. ?

    Some simple examples - when we say 1 million divided by one million is one/is equal to one, is that really the same as saying you have one penny?

    When we say we add 4 and 1/3 and get 13/3, are we switching between meanings or is it simultaneously an integer and a real number?

    When we say, 0.999999.... with a line over it is the same as one, isn't that by definition, changing meanings, going from one the counting number to one the limit of a process?
    Last edited by lakechampainer; 17 Sep 16, 15:13.

  • #2
    From Wikipedia on number, magnitude, and form.

    https://en.wikipedia.org/wiki/Number

    https://en.wikipedia.org/wiki/History_of_mathematics

    https://en.wikipedia.org/wiki/Number_sense

    excerpt

    A number is a mathematical object used to count, measure, and label.[citation needed] The original examples are the natural numbers 1, 2, 3, and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, number may refer to a symbol, a word, or a mathematical abstraction.

    In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as {\displaystyle {\frac {1}{2}}} {\frac {1}{2}} and {\displaystyle -{\frac {2}{3}}} -{\frac {2}{3}}, real numbers such as {\displaystyle {\sqrt {2}}} {\sqrt {2}} and {\displaystyle \pi } \pi , complex numbers, which extend the real numbers by including {\displaystyle {\sqrt {-1}}} {\sqrt {-1}}, and sometimes additional objects. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic. The same term may also refer to number theory, the study of the properties of the natural numbers.

    Besides their practical uses, numbers have cultural significance throughout the world.[1][2] For example, in Western society the number 13 is regarded as unlucky, and "a million" may signify "a lot."[1] Though it is now regarded as pseudoscience, numerology, the belief in a mystical significance of numbers permeated ancient and medieval thought.[3] Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.[3]

    During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Today, number systems are considered important special examples of much more general categories such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.[4]

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    • #3
      One of my favorite college courses was The History of Mathematics... fascinating stuff.
      Is there a natural progression to the way human cultures view numbers?

      There are definite parallels and similarities, most of which start with counting, quantifying and accounting. But not a common natural progression from there.

      Mathematics are part of the natural world. The expression of mathematics is a human construction.
      Does the way it is presented in math classes differ from the above, and is this interesting or informative?

      It depends on the math class, yes and yes. My Calculus III professor infused the semester with historical discussions regarding the derivations of concepts like L'Hopital's Rule. She also taught the History of Mathematics course.
      Are there some numbers which are particularly inherently interesting?

      42.
      Do mathematical operations and concepts develop in a natural order - counting, comparison as to quantity, estimation, negation, etc. etc. etc. ?

      They often develop as needed. So they tend to start with quantifying. Then they tend to progress along with a particular culture's needs or observations of nature
      Some simple examples - when we say 1 million divided by one million is one/is equal to one, is that really the same as saying you have one penny?

      Only if you started with 1 million pennies.
      When we say we add 4 and 1/3 and get 13/3, are we switching between meanings or is it simultaneously an integer and a real number?

      All integers are real numbers. All complex numbers are real numbers. Fractions aren't integers, but they are real numbers. Fractions are complex numbers.

      An integer plus a complex number equals a complex number.
      When we say, 0.999999.... with a line over it is the same as one, isn't that by definition, changing meanings, going from one the counting number to one the limit of a process?

      I'd have to look that one up. Off the top of my head, I think the superscript line means an exact figure. Limit notation is different.
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      • #4
        I think it is also important to take into account that symbols can have different meanings. For example, as we all learned in school the + symbol can mean addition which involves two numbers, or it can mean the positive number sign. Likewise the - sign can be a subtraction sign or a negative number sign, but it retains a varying notion of negation in both uses or meanings. X can mean multiplication, the variable symbol X, or the cross-product symbol.

        In terms of language in general, as opposed to just math, we can look at the different types of definitions, to help us think about issues related to math definitions:

        https://en.wikipedia.org/wiki/Definition

        excerpt

        A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).[1] Definitions can be classified into two large categories, intensional definitions (which try to give the essence of a term) and extensional definitions (which proceed by listing the objects that a term describes).[2] Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.[3][a]

        In mathematics, a definition is used to give a precise meaning to a new term, instead of describing a pre-existing term. Definitions and axioms are the basis on which all of mathematics is constructed.[4]

        The Wikipedia article on Polysemy

        https://en.wikipedia.org/wiki/Polysemy

        excerpt

        Polysemy (/pəˈlɪsᵻmi/ or /ˈpɒlᵻsiːmi/;[1][2] from Greek: πολυ-, poly-, "many" and σῆμα, sêma, "sign") is the capacity for a sign (such as a word, phrase, or symbol) to have multiple meanings (that is, multiple semes or sememes and thus multiple senses), usually related by contiguity of meaning within a semantic field. It is thus usually regarded as distinct from homonymy, in which the multiple meanings of a word may be unconnected or unrelated.

        Charles Fillmore and Beryl Atkins' definition stipulates three elements: (i) the various senses of a polysemous word have a central origin, (ii) the links between these senses form a network, and (iii) understanding the 'inner' one contributes to understanding of the 'outer' one.[3]
        Last edited by lakechampainer; 24 Sep 16, 18:39.

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        • #5
          Link to an article called, "Ambiguity in Mathematical Notation".

          https://davidwees.com/content/ambigu...ical-notation/

          excerpt

          "To illustrate this, I often ask teachers to write 4x and 4½. I then ask them what the mathematical operation is between the 4 and the x, which most realize is multiplication. I then ask what the mathematical operation is between the 4 and the ½, which is, of course, addition. I then ask whether any of them had previously noticed this inconsistency in mathematical notation — that when numbers are next to each other, sometimes it means multiply, sometimes it means add, some times it means something completely different, as when we write a two-digit number like 43. Most teachers have never noticed this inconsistency, which presumably is how they were able to be successful at school. The student who worries about this and asks the teacher why mathematical notation is inconsistent in this regard may be told not to ask stupid questions, even though this is a rather intelligent question and displays exactly the kind of curiousity that might be useful for a mathematician — but he has to get through school first!" ~ Dylan Wiliam, Embedded Formative Assessment, 2011, p53

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          • #6
            From Berkeley.edu - Order of arithmetic operations, in particular, the 48/2(9 + 3) problem

            https://math.berkeley.edu/~gbergman/...s/ord_ops.html

            excerpt

            A problem that hit the Internet in early 2011 is, "What is the value of 48/2(9+3) ?"

            Depending on whether one interprets the expression as (48/2)(9+3) or as 48/(2(9+3)) one gets 288 or 2. There is no standard convention as to which of these two ways the expression should be interpreted, so, in fact, 48/2(9+3) is ambiguous. To render it unambiguous, one should write it either as (48/2)(9+3) or 48/(2(9+3)). This applies, in general, to any expression of the form a/bc : one needs to insert parentheses to show whether one means (a/b)c or a/(bc).

            In contrast, under a standard convention, expressions such as ab+c are unambiguous: that expression means only (ab)+c; and similarly, a+bc means only a+(bc). The convention is that when parentheses are not used to show the contrary, multiplication precedes addition (and subtraction); i.e., in ab+c, one first multiplies out ab, then adds c to the result, while in a+bc, one first multiplies out bc, then adds the result to a. For expressions such as a−b+c, or a+b−c, or a−b−c, there is also a fixed convention, but rather than saying that one of addition and subtraction is always done before the other, it says that when one has a sequence of these two operations, one works from left to right: One starts with a, then adds or subtracts b, and finally adds or subtracts c.

            Why is there no fixed convention for interpreting expressions such as a/bc ? I think that one reason is that historically, fractions were written with a horizontal line between the numerator and denominator. When one writes the above expression that way, one either puts bc under the horizontal line, making that whole product the denominator, or one just makes b the denominator and puts c after the fraction. Either way, the meaning is clear from the way the expression is written. The use of the slant in writing fractions is convenient in not creating extra-high lines of text; but for that convenience, we pay the price of losing the distinction that came from how the terms were arranged horizontally and vertically.

            Probably another reason why there is not a fixed convention for order of multiplication and division, as there is for addition and subtraction, is that while people frequently do calculations that involve adding and subtracting lengthy strings of numbers, the numbers of multiplications and divisions that come into everyday calculations tends to be smaller; so there is less need for a convention, and none has evolved.

            Finally, the convention in algebra of denoting multiplication by juxtaposition (putting symbols side by side), without any multiplication symbol between them, has the effect that one sees something like ab as a single unit, so that it is natural to interpret ab+c or a+bc as a sum in which one of the summands is the product ab or bc. Without that typographic convention, the order-of-operations convention might never have evolved. When one has numbers rather than letters, one can't use juxtaposition, since it would give the appearance of a single decimal number, so one must insert a symbol such as ×, and there is less natural reason for interpreting 2 × 3 + 4 as (2 × 3) + 4 rather than 2 × (3 + 4), but I suppose that we do so by extension of the convention that arose in the algebraic context. Likewise, because addition and subtraction constitute one "family" of operations, and multiplication and division another, and perhaps also because the slant "/" doesn't seem to separate two expressions as much as a + or − does, we are ready to read a/b+c etc. as involving division before addition. But when it comes to a/bc, where the operations belong to the same family, the left-to-right order suggests doing the division first, while the "unseparated letters" notation suggests doing the multiplication first; so neither choice is obvious.

            It is interesting that in the 48/2(9+3) problem, the last element was written 9+3 rather than 12. If the latter had been used, it would have been necessary to insert a multiplication sign, 48/2×12, and I would guess that a large majority of people would have then made the interpretation (48/2)×12. Perhaps we will never know where this puzzle originated; perhaps it was cunningly designed so that one interpretation would seem as likely as the other; or perhaps it came up as a real expression that someone happened to write down, not thinking of it as ambiguous, but that other people did have trouble with.

            From correspondence with people on the the 48/2(9+3) problem, I have learned that in many schools today, students are taught a mnemonic "PEMDAS" for order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. If this is taken to mean, say, that addition should be done before subtraction, it will lead to the wrong answer for a−b+c. Presumably, teachers explain that it means "Parentheses — then Exponents — then Multiplication and Division — then Addition and Subtraction", with the proviso that in the "Addition and Subtraction" step, and likewise in the "Multiplication and Division" step, one calculates from left to right. This fits the standard convention for addition and subtraction, and would provide an unambiguous interpretation for a/bc, namely, (a/b)c. But so far as I know, it is a creation of some educator, who has taken conventions in real use, and extended them to cover cases where there is no accepted convention. So it misleads students; and moreover, if students are taught PEMDAS by rote without the proviso mentioned above, they will not even get the standard interpretation of a−b+c.

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            • #7
              History of Math Notation - from Wikipedia

              https://en.wikipedia.org/wiki/Histor...tical_notation

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              • #8
                Several thoughts on related topics:

                1. In the same way that we can think about numbers differently depending on the type of number or the context, we can also think about units differently. For example, we can think of units as being synthesized or built up from basic units (length, time, mass, etc.) or we can think of units being analyzed or broken down into these artificial units. For example, we can think of density as being mass divided by length to the third power, or for that matter as mass divided by volume, or for that matter as weight divided by either, etc. Or we can think of the basic units as coming together in a certain combination, where can apply the formal rules of math. I would argue that the density of density arises first: Way, way, way back in the day people noticed that some things floated on water. Animals probably have a similar concept.

                2. Numbers/units usually have several different concepts embedded in them, in everyday thought. For example, if we say that there is a ratio 685:0, most people would probably not agree that is equivalent to the ratio 1:0. The number 1:0 might be one occurrence of a 50/50 probability, whereas 685:0 represents something that is either always so or almost always so. Also, as measure was mentioned before, numbers generally are used both in everyday life and technically, with some more-or-less defined sense of approximation or accuracy.

                Wikipedia article on analysis.

                https://en.wikipedia.org/wiki/Analysis

                Wikipedia article on synthesis

                https://en.wikipedia.org/wiki/Synthesis

                Wikipedia article on SI base units

                https://en.wikipedia.org/wiki/SI_base_unit

                article on binomial theorem

                https://en.wikipedia.org/wiki/Binomial_theorem
                Last edited by lakechampainer; 31 Dec 16, 13:05.

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                • #9
                  Originally posted by lakechampainer View Post
                  2. Numbers/units usually have several different concepts embedded in them, in everyday thought. For example, if we say that there is a ratio 685:0, most people would probably not agree that is equivalent to the ratio 1:0. The number 1:0 might be one occurrence of a 50/50 probability, whereas 685:0 represents something that is either always so or almost always so. Also, as measure was mentioned before, numbers generally are used both in everyday life and technically, with some more-or-less defined sense of approximation or accuracy.
                  Interesting discussion, but you have to be careful in science and math.

                  In (2) above, the ratio of 1:0 = 1/0 and 685:0 = 685/0. BY DEFINITION, you cannot divide by "0" (a ratio is division). The result is undefined, which makes such a ratio meaningless because it is NOT a ratio, it is a "comparison". Both mean the same, because both results are undefined.

                  The other trap you have fallen into is the "ill posed question". This is very important in the development of scientific experiments, because if you ask an invalid question, you will not get an answer that is meaningful.

                  For example, what flavor is 5? What is the radius of a square? What is the last digit of Pi? How many dogs in a cat? How many rungs on a rope? What color is delicious? How often was "HE" pregnant?

                  These questions are meaningless, even though you can ask them.

                  There are MANY more meaningless questions (ill posed) than meaningful questions.

                  ANY answer to a an ill posed question will lead to a meaningless result that will lead to false conclusions.

                  However, given the right context, they may be meaningful. In non-Euclidean geometry a square will have a radius. However, it is not a fixed value, it is a function of angle f(theta)=R(theta). If you are talking about genomes, there may be some "dogs in a cat". In a rope ladder, there may be rungs in a rope.

                  Your elementary teacher that always said "any question is a good question" lied to you. Most questions are stupid. In fact, it takes a certain level of understanding about a subject to ask a GOOD question.
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                  • #10
                    I was thinking about this topic, and realized another way to think about is to think about the parts of speech, and how they relate to mathematical usage.
                    Below is a link to the Wikipedia article on parts of speech. Of course, other languages have different ways of looking at parts of speech, included those that are more inflected (for example German with 4 cases)

                    https://en.wikipedia.org/wiki/Part_of_speech

                    So, for example, 8 - 5 = 3,we could look at the numbers as nouns and the subtraction sign as an action verb. We could look at the equals sign as a form of the verb to be. Or maybe the verb becomes. It seems reasonable to think in terms of nouns/adjectives and verbs/adverbs.

                    The numbers could be considered as abstract numbers, without any units attached. Or they might be considered as pronouns, where they stand in for something else: 8 apples, 8 inches with a tolerance of 0.01 inch, 8 apples that are not showing sign or rot, 16 half apples, etc. The subtraction sign could be looked at for example as subtract exactly, subtract approximately, etc.

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                    • #11
                      There are two type of numbers. Real ones (such as 1, 2, 100, Pi) and those that mathematics have made up, such as the square root of -1.

                      Real ones are more important, but pretend ones can be far more interesting.
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                      • #12
                        Originally posted by lakechampainer View Post
                        Are there some numbers which are particularly inherently interesting?
                        I do know that when people are asked to pick a number between 1 and 10, 3 and 7 are more likely to be chosen, apparently.

                        A simple "fact" that may come in useful on certain occasions
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                        • #13
                          A few more thoughts on equal signs:

                          1. In the literature a relational view views equal signs as the left side is the same as/equivalent to the right side. An operational view views that the is an operation or transformation taking place: like f (x) = 3x + 5. Like a machine.

                          2. To go back to grammar analogies, a relational view can be considered as a present tense - in the sense that 8 - 3 = 5, both now and always/as a rule.
                          An operational view can be viewed as a present progressive: We are in the process of subtracting 3 from 8, for example. Could also be viewed as a future tense or subjunctive tense: What would happen if we subtracted 3 from 8.

                          3. In the relational view, I think it really has to be looked at as the left side and the right side both equal something else or multiple something elses.

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                          • #14
                            When I made these posts I was still a middle-school math teacher (which was before I was a STEM teacher, and before I entered the wonderful retail world). I found that much talk of types of numbers at the middle school level was not a productive use of time - the good/best students could understand any lesson easily, whereas the average student wouldn't get much out of such a lesson. The best way I found to approach these concepts at all was during the learning of percents/decimals/fractions - the rote learning of these three equivalents could lead to a somewhat higher discussion.



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                            • #15
                              I have talked about the abstract nature of language often in these forums. My assumption had been that people would intuitively understand that mathematics was an abstract language and that it could therefore to illustrate how all language is abstract. Perhaps I was wrong :-).

                              If we start with zero and infinity you can illustrate the point easily. 0 is the absence of existence. It's imaginary nature is fairly obvious. For example imagine that you don't exist. If you can imagine you clearly must exist. Even you imagine that you can imagine that you don't exist you have simply added a layer to a circular argument. Infinity has the same problem as to imagine the infinite you would have to be infinite.

                              Even cardinal numbers have the property of being abstract. For example there is obviously no single indivisible real object. Not only are there molecular and subatomic division but the meaning of an object is relative to it's relationship to other objects. As it turns out reality itself is from a scientific perspective a matter of mathematical functions. You can not know the thing itself only it's relative position to other things.

                              This same principle applies to words. A cat is a cat because it is different from a dog and everything else that is not a cat. There is no cat only various examples of things that are cat like.

                              This principle of abstraction applies to other animals as well. An animal knows what something is by knowing what it is not. This concept of relativity is so profound that it has altered our political views. When a post modernists says there are a near infinite number of ways to define reality they are technically correct.

                              Some ways of interrupting reality are of course more accurate or precise. That doesn't necessarily make them better. Pragmatism plays an important role. If a Springbok is to accurate and precise in identifying what is making the grass move it may find it is just the wind or it may take to much time in evaluation and be eaten by a lion. In this case grass moving is an abstract concept that means danger. It is not a precise or accurate definition but it is practical.

                              We can make a similar argument for money. Everyone knows that money represent something. The more intelligent you are the more abstract your representation of money will likely be. Asking if money is real however is a meaningless question. It may not physically exist in the way things you can buy with it do. It is however a useful representation of things that have "real" value. How much of those things varies constantly based on other relative abstraction such as those tied to supply and demand. Money clearly only exist relative to it's representation and the representation of other things. Even if the medium of exchange was gold, does gold have intrinsic or imaginary value?

                              Some people take away the wrong message from the abstract nature of reality. Instead of being humbled by ignorance of reality they start to believe their reality is paramount. Signs of this mental illnesses are all around us. You could call it the arrogance of luxus.
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