# Thread: Mathematics: the fifth operation

1. Mathematics: the fifth operation

It was known that our calculations in the mathematics are four (Combine, subtraction, multiplication, division), but we discovered the fifth and process that combines two two at the same time.
I want to help in this matter.

2.

3. Originally Posted by zouaoui messaoud
It was known that our calculations in the mathematics are four (Combine, subtraction, multiplication, division), but we discovered the fifth and process that combines two two at the same time.
Are you going to tell us what it is? Or is it a secret?

I want to help in this matter.
I think we all agree that you need help.

4. You left three operations off your list: exponentiation and its two inverses, log and root. Then there's the precursor to addition, the successor operation, and the not-very-well-defined next step past exponentiation, tetration. BTW, 2+2 = 2*2 = 2^2 = 2^^2 = 4.

5. The four branches of arithmetic are ambition, distraction, uglification, and derision.

Ambition, Distraction, Uglification, and Derision: Summary of Chapter Nine (2 of 2) | "An-Alice-Is" In Wonderland

6. Addition and subtraction are combination and intersection respectively, multiplication and devision are the same only non linear,as is epotentiation. I am not for sure about inverse.

7. Quickie, one thing I know for sure in aritmetic is that fractions are division problems, they are not numbers. That is what makes fractions so hard. The line should be a divide by sign, and it probably started out that way.

8. It did probably start that way, but that doesn't mean fractions aren't numbers. 1/4, whether you think of that as 1 divided by 4, one quarter, 0.25, 25% or whatever else still represents a specific real value (a rational value actually, but a number either way).

I should also point out that addition and subtraction can't be directly mapped to union and intersection without at least some extra details. After all, {a, b, c} U {b, c, d} is {a, b, c, d} which can't be arrived at by any form of addition. If you assume the sets are disjoint ({a, b} U {c, d} for example) then you can equate union and addition, but then subtraction doesn't work and the intersection of two disjoint sets is always empty. (It's possible to make arithmetic work on special sets, but then addition and subtraction aren't mapped to union or intersection.)

9. Quickie, one thing I know for sure in aritmetic is that fractions are division problems, they are not numbers. That is what makes fractions so hard. The line should be a divide by sign, and it probably started out that way.
Nonsense. Fractions are numbers representing a value just like any other number. It just depends how finely you divide the number line for your year 6 class.

All numbers can be represented as the result of calculations, including fraction representations. One way I present the concept to algebra students is to talk about all the invisible fraction and exponent notations (and times tables and number families) just waiting for you to make them visible when they're needed.

2 may be just a number with a recognised value. It's also 21, but we only write it that way when the occasion arises. (It's also 2*1 and 2+0 and half of 4 and 10% of 20 ... and ... an infinity of other calculations and representations.)

Fraction is also a more general word and it's used in arithmetic in different ways. Though most of them are really about ensuring that you're using the same units rather than anything about fractions themselves as numbers. You can't subtract 1/4 of an hour from half a pineapple. Nor can you add half a potato to half a mile to come up with one whole anything meaningful.

10. Let me clarify with an example. Hold out your two hands. Here are two rulers. Combine them and the numbers add. The opposite intersection also holds true and is subtraction. This system works for a sliderule. And by the way, not only fractions, but also algebra too was also hard.

11. When I was really little (like eight or nine), the idea of a fifth operator fascinated me. I came up with many of my own, but none really kept, and I don't think I remember what any of them were. Nowadays, extra operators are nothing. Any mapping from f(x,y) --> z works. If I had to add one to the list with +, -, *, and /, it would probably be exponentiation.

12. Originally Posted by Hill Billy Holmes
Let me clarify with an example. Hold out your two hands. Here are two rulers. Combine them and the numbers add. The opposite intersection also holds true and is subtraction. This system works for a sliderule. And by the way, not only fractions, but also algebra too was also hard.
That's not union and intersection under the normal mathematical definition of those words, but yes, you can do almost any operation with a properly constructed slide rule (or a nomogram).

13. Originally Posted by zouaoui messaoud
It was known that our calculations in the mathematics are four (Combine, subtraction, multiplication, division)
These are just the basic algebraic operators; there are many more operators to be found in mathematics, perhaps have a look here :

Mathematical symbols list (+,-,x,º,=,<,>,...)

14. Originally Posted by Hill Billy Holmes
Quickie, one thing I know for sure in aritmetic is that fractions are division problems, they are not numbers.
They are numbers.

The line should be a divide by sign, and it probably started out that way.
The other way round: the division symbol is an abbreviation of a fraction (something over something).

15. The line should be a divide by sign, and it probably started out that way.
The line has a name. It's called a vinculum.

It has other uses as well. Vinculum (symbol) - Wikipedia, the free encyclopedia

16. Good read, nice history. As for the vinculum symbol itself, whatever makes the most sense, considering you are going to teach it to a new generation.

17. You guys are trivializing a rather deep issue, namely that, in talking about operations on a space, you really should specify

a) what space you are talking about and

b) whether you require your operations to be closed or not.

Anyway, it seems the space space you are all referring to is , the space of Real numbers. This is a field, which means by definition it admits of 2 closed binary operations and each with its own inverse. That is and . Notice that only 2 operations are required in this case

Step "back" to the integers. This is a ring not a field (there isno multiplicative inverse), although addition and multiplication are still defined, so it makes sense to define another operation called "division" if and only if we don't require this operation to be closed i.e. we don't require the result to be an integer.

Step further back to the natural numbers . This is apparently controversial; most mathematicians, I think, would say it is not a ring or a field, merely a set without any algebraic structure other than a natural order. (The Wiki disagrees, btw)

One can recover some algebraic structure here by defining the natural numbers as the set of all strictly positive integers which is of course a sub-ring of the ring of all integers.

Disagree if you want

18. That was sort of fancy. I'm not as good. I have some disagreement. I think you can define addition in a space, fine. When your addition is along a slope, its called multiplication, but it is still simple addition, only along a slope. Now you can add the set of integers into your space and apply the simple algebra you have so far. Now another difference is that subtraction and division have already been defined as an inverse. As to Wiki, I think it is wrong in this case, and should be traced to its source.

19. If you're talking about fields, rings and such things, then no, multiplication is not defined in terms of addition in any way. Take the field Z2 for example. There are just two elements, 0 and 1. Addition is defined as 0+0 = 1+1 = 0 and 0+1 = 1+0 = 1. Multiplication is the usual multiplication on those two numbers. I'm pretty sure you can't duplicate multiplication with addition at all here.

If you're talking about slide rules, those slopes (curves actually) require you to already understand multiplication (and in particular the equation ) before you can build the slide rule, so those are not fundamental and not the definition of multiplication.

20. Perhaps you should get five operations?

21. I read threads like this and I really wish the "5th" operation was logic.

22. New and innovative concepts? Sagnation? Why does everybody want learning arithmetic to be hard? Maybe it was just meant to be hard to get people to think harder so they can be smarter. Combining, grouping of numbers, representative intigers, displacment in a space, what ever you want to call it can occur linearly OR on a slope, makes no difference.

23. It's only hard if you want it to be.

But it is important that everyone agree on what terms mean.

24. Originally Posted by Hill Billy Holmes
New and innovative concepts? Sagnation? Why does everybody want learning arithmetic to be hard? Maybe it was just meant to be hard to get people to think harder so they can be smarter. Combining, grouping of numbers, representative intigers, displacment in a space, what ever you want to call it can occur linearly OR on a slope, makes no difference.
To me, mathematics is nothing else but a language, the language of the universe itself - and like any language, when you first try to learn it, it is hard. But the more you practice and use it, the more "natural" it becomes.

25. Mathmatics is good to know and is mostly fun. As a language? It can be thought of as that, such as thinking in it and being fluent in it. But, languages were made by man, and in the cases of mathmatics, early man. So its a language like latin . It is hard to learn.

26. Dominos nabiskos.

27. Originally Posted by Hill Billy Holmes
But, languages were made by man, and in the cases of mathmatics, early man. So its a language like latin . It is hard to learn.
testiculis completum

Spoiler Alert, click show to read:
Latin for complete bollocks if my rusty language skills are correct

28. Originally Posted by MagiMaster
If you're talking about slide rules, those slopes (curves actually) require you to already understand multiplication (and in particular the equation ) before you can build the slide rule, so those are not fundamental and not the definition of multiplication.
Slide rules are of course based on logarithms. Logarithms were developed by John Napier.

Napier also invented a calculating device called Napier's bones.

Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to subtractions. More advanced use of the rods can even extract square roots.

Napier's bones - Wikipedia, the free encyclopedia

Interesting, how much history gets lost as we make scientific progress.

29. Originally Posted by Markus Hanke
To me, mathematics is nothing else but a language, the language of the universe itself
I don't fully agree with this view. Sure, the notation of mathematics is a language, but that notation describes a logical notion that is beyond the language and is not arbitrary. I should remark that I adopt the formalist view of mathematics that regards it as symbol manipulation. But nevertheless, even symbol manipulation has an underlying logic. As for regarding mathematics as the language of the universe, I quite strongly disagree. The universe isn't mathematical at all. Scientists (especially physicists) impose mathematical structure onto the universe. The laws of physics are the result of the mathematical properties of this imposed mathematical structure. The constraints imposed by the universe are real enough, but the descriptions of those constraints are in terms of the imposed mathematical structure.

30. Mathematics has syntax, grammar and a lexicon. I'd say that would probably be sufficient to classify it as a language.

I've used this analogy before, but if Physics is the "operating system" of the Universe, then Mathematics is the programming language it's written in.

31. Originally Posted by KJW
I don't fully agree with this view. Sure, the notation of mathematics is a language, but that notation describes a logical notion that is beyond the language and is not arbitrary. I should remark that I adopt the formalist view of mathematics that regards it as symbol manipulation. But nevertheless, even symbol manipulation has an underlying logic. As for regarding mathematics as the language of the universe, I quite strongly disagree. The universe isn't mathematical at all. Scientists (especially physicists) impose mathematical structure onto the universe. The laws of physics are the result of the mathematical properties of this imposed mathematical structure. The constraints imposed by the universe are real enough, but the descriptions of those constraints are in terms of the imposed mathematical structure.
Hm, I admit that you are raising some interesting points. I should not have stated "mathematics is the language of the universe", but rather "mathematics is the language that we use to describe the universe". It would have been closer to the point...

32. Originally Posted by zouaoui messaoud
I want to help in this matter.[/FONT]
If you want help (or you want to help) then it would be a good idea to engage in the conversation. Have you nothing to say?

33. Eureika, got it now , math doesn't have 5 operations. It has either 0, 2 or a dimentionaly definable infinity. If you are pill science stop reading at this point because I made it up myself. Because I make it up myself doesn't make wrong. Math runs a logical slope from its origin. Its limitations keeping it from infinity are physical dimentions. The slope interacting with 3 or if you prefer spacetime 4 are the alowable logic operations. From source 0 everything equalls nothing. When it hits 2 it starts multiplying 2*first dim,2*2nd and so on so there are really 2^3 or fourth operations the rest is geometry boolien.

34. Originally Posted by Hill Billy Holmes
Eureika, got it now , math doesn't have 5 operations. It has either 0, 2 or a dimentionaly definable infinity. If you are pill science stop reading at this point because I made it up myself. Because I make it up myself doesn't make wrong. Math runs a logical slope from its origin. Its limitations keeping it from infinity are physical dimentions. The slope interacting with 3 or if you prefer spacetime 4 are the alowable logic operations. From source 0 everything equalls nothing. When it hits 2 it starts multiplying 2*first dim,2*2nd and so on so there are really 2^3 or fourth operations the rest is geometry boolien.

35. Originally Posted by Hill Billy Holmes
Because I make it up myself doesn't make wrong.
Are you quite sure about that?

36. Originally Posted by Markus Hanke
I should not have stated "mathematics is the language of the universe", but rather "mathematics is the language that we use to describe the universe".
I don't consider this to be a minor distinction. I think it gets to the heart of the way reality should be viewed, in particular, why the laws of physics are the way they are. Do we consider the laws of physics to be imposed from "above" as if by design, or do we consider the laws of physics to be based on logical reasons that would allow us to deduce them? It also gets to the heart of the scientific method. We expect our physical theories to be falsifiable and accept that they are never proven to be true, but mathematics is based on provable theorems. Therefore, physics and mathematics are at odds with each other with regards to the notion of "truth", exacerbated by the heavy reliance on mathematics by physics. Because of this, we need to closely examine the relationship between mathematics and physics. For example, if physics provide a mathematical description of physical reality obtained by measurement, and mathematics provides proof about the mathematical properties of mathematical descriptions, then where can our physical theory actually fail? Of course, I'm not suggesting that physical theories can't fail, but rather that the question of how it could fail, rather than simply asking whether it does fail, is an important question.

37. Originally Posted by KJW
Of course, I'm not suggesting that physical theories can't fail, but rather that the question of how it could fail, rather than simply asking whether it does fail, is an important question.
I think to answer that we need to ask ourselves whether the axioms of logic as we apply them to the universe and to our mathematical proofs are really the correct and suitable ones to describe the world around us, or whether they are merely a - potentially faulty, but seemingly fitting - construct of our own limited consciousness. Ultimately it boils down to this - what is the relationship between our logic and perception, and physical reality ? What does "physical reality" even mean ? Is it an absolute term, or is it something that is relative and observer dependent in itself ?

To illustrate this, consider a scenario where humans never develop eyes in the course of their evolution. Would the physical models we come up with to describe the universe around us be the same as the ones we have now ? Or how about a race of intelligent viruses, without any of the senses we have, and on much smaller scales - what kind of maths and physics would they develop ?

38. Originally Posted by Strange
Originally Posted by Hill Billy Holmes
Because I make it up myself doesn't make wrong.
The credo of cranks and ignoramuses.

39. Hey credo ,you have been following me around I think you must like me. The same thing when diptwats say Einstein( science JC ) is wrong or stupid those are the same ones that DIDN"T come up with relativity themselves. Credo, you would be hard pressed to come up with the 5th operation yourself ,or the 4th for that matter. 'jumpngehosefats' credo

40. Originally Posted by Hill Billy Holmes
Hey creod ,you have been folloeing me around I think you must like me. The same thing when diptwats say Einstein( science JC ) is wrong or stupid those are the same ones that DIDN"T come up with relativity themselves. Credo, you would be hard pressed to come up with the 5th operation yourself .or the 4th for that matter. jumpngehosefats credo
Would you like to try that again in English.

41. Thanks Mr. Strange it was a typo bad keyboard or buffer or something

42. Originally Posted by Markus Hanke
I think to answer that we need to ask ourselves whether the axioms of logic as we apply them to the universe and to our mathematical proofs are really the correct and suitable ones to describe the world around us, or whether they are merely a - potentially faulty, but seemingly fitting - construct of our own limited consciousness. Ultimately it boils down to this - what is the relationship between our logic and perception, and physical reality ? What does "physical reality" even mean ? Is it an absolute term, or is it something that is relative and observer dependent in itself ?
Because I'm a formalist, I don't see mathematics as the product of the human mind (except in a trivial sense). Rather, I see mathematics as something that can be done by machine. Indeed, I don't regard mathematics as rigorous unless it can be done by machine. As such, I don't see our logic as potentially faulty because the logic is what it is. For example, if and , then not because of some vague notion of what "equality" is, but because transitivity is built into the definition of .

Because I see the laws of physics as based on logical reasons rather than imposed from "above", questions such as "what does 'physical reality' even mean?" actually have a fairly simple answer... reality is what it is measured to be... nothing more. And given that a measurement is a relationship between physical objects, it is the relationship that is important and not the objects themselves. If one performs some experiment, applying measurements to the physical system, what are the possible outcomes, disregarding the laws of physics? If the actual outcome is not among this set of possible outcomes, that means our space of possible outcomes is too small and need to be enlarged. If the space is large enough, then what happens when we consider particular outcomes to be the same as other particular outcomes due to being described only from a different frame of reference? The space of distinct possible outcomes becomes much smaller than the original space of outcomes, implying the existence of constraints. But these constraints did not arise through means other than the symmetry of the space of descriptions. Mathematically, we see that connections are arbitrary, gauge transformations relate equivalent connections, curvature represents the distinct connections, and the Bianchi identity is the constraint on the curvature.

43. Originally Posted by KJW
Mathematically, we see that connections are arbitrary, gauge transformations relate equivalent connections, curvature represents the distinct connections, and the Bianchi identity is the constraint on the curvature.
True enough, I see your point

44. Originally Posted by Hill Billy Holmes
Hey credo ,you have been following me around I think you must like me. The same thing when diptwats say Einstein( science JC ) is wrong or stupid those are the same ones that DIDN"T come up with relativity themselves. Credo, you would be hard pressed to come up with the 5th operation yourself ,or the 4th for that matter. 'jumpngehosefats' credo
That's probably because it's nonsense.

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