# Discussion: symmetry in mathematics

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My recent reading has sparked something in me by which I've found myself looking for examples of symmetry in social sciences and everyday life. Symmetry is what allows us to recognize other humans as humans and at the same time allows all humans to look different, tell the difference between a cat and a dog and know that your wallet after it has been moved by a third party to different location is still your wallet. The concept is pretty wide and today I, while pondering some stuff, started thinking about symmetry in mathematics, which raised a question that I don't have the mathematical background to answer, but I think would serve as a topic of some interesting discussion. Here's a couple of lemmas (I'm not sure all of them have ever been proven):

- Addition is symmetric to subtraction (10 + 10 = 10 - -10)
- (In mathematics) either addition or subtraction is obsolete in terms of functionality (any and all functions of one type can be substituted with a negated version of the other)
- Multiplication is not symmetric to division if division is be declared obsolete (the closest equivalent to 1 / 10 is 10 * (1 / 10), which still cannot be expressed without a division)
- Division is symmetric to multiplication while multiplication can declared obsolete (in simple terms, 10 * 10 = 10 / (1 / 10) )

While not particularly practical, I've been pondering whether the above points actually hold true and can be proven for any and all cases in mathematics. In particular, can division be removed from mathematics without math breaking down?

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My recent reading has sparked something in me by which I've found myself looking for examples of symmetry in social sciences and everyday life. Symmetry is what allows us to recognize other humans as humans and at the same time allows all humans to look different, tell the difference between a cat and a dog and know that your wallet after it has been moved by a third party to different location is still your wallet. The concept is pretty wide and today I, while pondering some stuff, started thinking about symmetry in mathematics, which raised a question that I don't have the mathematical background to answer, but I think would serve as a topic of some interesting discussion. Here's a couple of lemmas (I'm not sure all of them have ever been proven):

- Addition is symmetric to subtraction (10 + 10 = 10 - -10)
- (In mathematics) either addition or subtraction is obsolete in terms of functionality (any and all functions of one type can be substituted with a negated version of the other)
- Multiplication is not symmetric to division if division is be declared obsolete (the closest equivalent to 1 / 10 is 10 * (1 / 10), which still cannot be expressed without a division)
- Division is symmetric to multiplication while multiplication can declared obsolete (in simple terms, 10 * 10 = 10 / (1 / 10) )

While not particularly practical, I've been pondering whether the above points actually hold true and can be proven for any and all cases in mathematics. In particular, can division be removed from mathematics without math breaking down?

You're using the term "symmetric" in a way that it's not meant to be used. Symmetric means a totally different thing.

Addition and subtraction are inverse operations. That means that a + b - b = a

Multiplication and subtraction are inverse operations. That means that a * b / b = a

That's all there is to it, really. You're muddying the subject by throwing fractions in there and calling that division.

As to your question of whether we need division, the answer is yes. We also need subtraction. You state that: "(In mathematics) either addition or subtraction is obsolete in terms of functionality (any and all functions of one type can be substituted with a negated version of the other)"

I'd ask you how you went about finding the "negated version" of the other. What's it really mean to have a negated version? It means subtracting that number from zero. You could not have found it without subtraction. In the same way, you could replace "10 / 2" with "10 * 0.5". But how did you find 0.5? You had to use multiplication.

Each operation is expressible in terms of its inverse, but we need the first operation to figure out how to express it in terms of the second.

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My recent reading has sparked something in me by which I've found myself looking for examples of symmetry in social sciences and everyday life. Symmetry is what allows us to recognize other humans as humans and at the same time allows all humans to look different, tell the difference between a cat and a dog and know that your wallet after it has been moved by a third party to different location is still your wallet. The concept is pretty wide and today I, while pondering some stuff, started thinking about symmetry in mathematics, which raised a question that I don't have the mathematical background to answer, but I think would serve as a topic of some interesting discussion. Here's a couple of lemmas (I'm not sure all of them have ever been proven):

- Addition is symmetric to subtraction (10 + 10 = 10 - -10)
- (In mathematics) either addition or subtraction is obsolete in terms of functionality (any and all functions of one type can be substituted with a negated version of the other)
- Multiplication is not symmetric to division if division is be declared obsolete (the closest equivalent to 1 / 10 is 10 * (1 / 10), which still cannot be expressed without a division)
- Division is symmetric to multiplication while multiplication can declared obsolete (in simple terms, 10 * 10 = 10 / (1 / 10) )

While not particularly practical, I've been pondering whether the above points actually hold true and can be proven for any and all cases in mathematics. In particular, can division be removed from mathematics without math breaking down?

I think when you say "obsolete" you mean "redundant". The relationship between addition and subtraction is completely analogous to the relationship between multiplication and division.

The way we normally think about them in modern mathematics is that addition is the important operation. It so happens that every number x (for some notions of number, like integers or reals) has an opposite -x, i.e. a number than when added to the original gives zero as the result (x + (-x) = 0). For short, we normally write a - b instead of a + (-b), but subtraction is not really a separate operation.

The situation for multiplication is similar, except that we don't have a handy short way of expressing the inverse of a number (we don't write /x to denote the inverse of x). But the difference is purely notational.

If you are seriously interested in studying symmetry (although I don't agree that all the examples you gave are actually related to symmetry), you should try to learn some Group Theory.

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except that we don't have a handy short way of expressing the inverse of a numbe

x[sup]-1[/sup] ?

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[quote name='alvaro' timestamp='1313345650' post='4849050']
except that we don't have a handy short way of expressing the inverse of a numbe

x[sup]-1[/sup] ?

[/quote]

Yes, that is the usual notation. But I don't find it as concise as the unary `-'.

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Indeed, after some further Wikiing, the examples I brought aren't strictly related to symmetry; however, I do not agree that the process of finding the negative of a number necessarily involves subtracting it from zero. True, the process can be expressed that way, but it can also reliably be solved notationally regardless of the framework you're working in. You do not need to know the relationship between a positive and a negative number in order to deduce one from the other. What you cannot do without subtraction in this case is definitively prove that one is actually related to the other. Consider the binary version of negation: to convert a positive number to negative or vice versa, you simply flip a bit in the representation of the number (eg add or remove a minus in front of it) and can entirely ignore the underlying theory that enables mathematics to do the actual conversion. True, deep inside this would remain dubious as the process would yield the correct results based on an opaque transformation, but it would still work. The question, IMO, becomes more interesting when dealing with infinities.

PS - indeed, redundant is a better word to obsolete!

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Consider the binary version of negation: to convert a positive number to negative or vice versa, you simply flip a bit in the representation of the number (eg add or remove a minus in front of it) and can entirely ignore the underlying theory that enables mathematics to do the actual conversion.
What if I invent a new binary representation of numbers where the most-significant-bit identifies whether the stored number is actually the reciprocal of the regular interpretation?

Now we don't need division -- you simply flip a bit in the representation of the number (eg add or remove a "1/" in front of it) and can entirely ignore the underlying theory that enables mathematics to do the actual conversion.

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Actually, that is only true for one's complement representation of integers (and is not just binary). Neither does one's complement allow you to treat subtraction/addition as the same operation.

Two's complement on the other hand does work in allowing us to subtract numbers by adding one to the other; but all of this only works because we're working with modular arithmetic and is the same in any base, not just binary (where in general in base 'r' we call it the r's complement)

working in decimal under modulo 437 we have 5 + 432 = 0 (mod 437), so 432 is the additive inverse of 5, in the same way in binary, under modulo 1000[sub]b[/sub] (equiv. to 3 bit numbers in digital circuitry) the additive inverse of 101b is 11[sub]b[/sub] as 101[sub]b[/sub]+11[sub]b[/sub] = 0.

with r's complement you find the additive inverse by subtracting it from the size of the set. It just so happens that if the size is a power of 10 (in any base), then you can do the subtraction by flipping each digit and then adding 1; so in decimal under modulo 1000, the additive inverse of 437 is 562 + 1 = 563, or more commonly in two's complement with binary under mod 10^3, 101[sub]b[/sub] -> 010[sub]b[/sub] + 1 = 011[sub]b[/sub]

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