# FeaturedB Are there really 4 fundamental math operations?

1. Feb 15, 2017

### mountains

It's strange to me that multiplication and division are considered fundamental operations.
It makes sense for me that addition is a fundamental operation but multiplication is just like a function or algorithm that takes several numbers and apply additions. This is true even for multiplication with real number.

2. Feb 15, 2017

### Staff: Mentor

First of all: addition and subtraction is the same operation, as multiplication and division is also the same operation. The latter are only the reverse of the former. Next, there are objects, which we can multiply but not add, so multiplication isn't necessarily short for multiple additions.

An algorithm are both, because it gives as a rule to make a new element out of two others.

3. Feb 15, 2017

### mountains

Can you give me examples of those objects that we can only multiply but no add?

4. Feb 15, 2017

### Staff: Mentor

This depends a bit on what I may use.

Regular matrices can be multiplied. If you add them, they might not be regular anymore. The most simple example of them is to stretch or compress a line. You can do two stretches and get a new stretching factor, which is a multiplication. If you add them, you might get only $\{0\}$ which isn't a line anymore (stretching by the same amount in opposite direction).

In general the successive application of two functions is written as a multiplication, e.g. permutations on an ordered set or rotations in geometry.

It's a bit problematic to give examples, because if you have only one operation, then it doesn't matter how you write it: $a+b\, , \,a \cdot b\, , \, a \circ b\, , \, a \diamondsuit b$ and so on. So the examples for multiplications only must either leave the set on which the operation is defined if added, as in my first example, or simply just usually written as a multiplication, as in the example with the permutations.

From a mathematical point of view, addition and multiplication are just binary operations with some properties. On sets like numbers it just happens that you can define both. There are also other "fundamental" operations like $\cap$ and $\cup$ on subsets, or multiplications that aren't commutative or have other strange properties like $a \cdot a = 0$ or $a\cdot a = a \neq 1$. All of them make some sense somewhere, physics and biology come to my mind.

5. Feb 15, 2017

### epenguin

But if the question is about ordinary numbers, ordinary/everyday/school airthmetic why would mountains (and I) be wrong?

6. Feb 15, 2017

### mathman

other than for integers, multiplication is more than repeated addition.

7. Feb 15, 2017

### Staff: Mentor

Nobody said anybody was wrong. In the end it comes down to what can be considered "fundamental". In a world without zero, multiplication might be the fundamental operation. I understand that addition is closely related to counting, and therefore appears to be more natural than the measurement of the size of a field, which multiplication would be needed for. However, as long as we don't define "fundamental", e.g. historical, it will be a matter of taste.

8. Feb 16, 2017

### epenguin

That is not some minor exception! It is in any case the one I brought up.
What is the simplest extension or generalisation of the integer concepts for which what you say is true?
If so, it seems to me some people have strange tastes.

I thought mathematics was supposed to be assuming the minimum and deriving the maximum. Why therefore assume laws of multiplication if you can prove them from those of addition?

9. Feb 16, 2017

### Staff: Mentor

$\mathbb{Z}[t]$ for a transcendental $t$ and $\mathbb{Z}[a]$ for an algebraic $a$
That's true without any doubt.
Because there are rings, which carry both structures at the same time. And the multiplicative structure in algebras are usually far from being the one in rings and fields of numbers. The notation $(A \cdot B)(v)=A(B(v))$ is certainly a convention, one that is induced from the fact, that $(A+B)(v)=A(v)+B(v)$ is also possible, although not always defined. But this convention is easier to grasp as if we wrote $A\diamondsuit B$ (and easier to write).

But I admit, if you consider
$$\textrm{ counting } \rightarrow (\mathbb{N},+) \rightarrow (\mathbb{N_0},+) \rightarrow (\mathbb{Z},+) \rightarrow (\mathbb{Z},+,\circ) \rightarrow \ldots$$
as a canonical way of development, then addition is "more fundamental" than multiplication, just because it comes first in this construction and very likely also in anthropology.

10. Feb 16, 2017

### _PJ_

What's the thoughts on this:

Addition is the only fundamental mathematical operation.
That's not to say that others are derived from this, in fact, they are not, however, addition is the only one that can always be empirically and experimentally verified absolutely in all cases.

The invention of multiplication also introduces caveats and exceptions and special cases for commutation or irrationality etc.

11. Feb 18, 2017

### bahamagreen

Empirical and experimental evidence indicates that the addition of velocities V1 and V2 must always fall short of c...

12. Feb 24, 2017

### sudhirking

To the original questions, are their really only 4 fundamental operations, there have been many responses. Some would say "4? Of course not at least 2. The inverse operations are just the normal operation on the inverse elements". And there were other comments on saying the counting addition is the most fundamental and others on saying on how velocities add are fundamental. To all this I might agree, except that the way hyperbolic tangents add (which is how the relativists add velocities) and they way counters add (which is how we normal humans add) are isomorphic to each other so there really shouldn't be a philosophical dispute towards their difference. They are of the same type of addition.

To this I wholeheartedly disagree. On many levels.
Firstly is it fair the statement addition is the only fundamental? Doesn't that sound perverse? What are we to mean most fundamental? To this question I like this answer.
I like this answer a whole lot.

Secondly, addition needs no empirical or experimental verification. Consider apples on a table. Remove an apple. Or put an extra apple. Our intuition, our mode of experience naturally "sees" numbers and their addition in this way. The making vigorous of this intuition is what integers are, and how their addition is defined. So are they empirical? Experimental? No I would say they are more the product of the way humans experience the natural world, then the phenomenology of the natural world.

But most importantly, thirdly, I disagree when you say the other operations are not derived from addition. And this is one is the important one. They can be. And this I feel is actually deep.

Okay so consider an abelian group. "Addition" refers to the product in this group. Why must the group be abelian? Good question. This too is deep, it leads to the question of where groups come from. But I ignore it. Anyways, consider this structure because it encapsulates the bare minimum for our notions of addition. Can we invent multiplication? Yup. What is multiplication? This is one deep point. It is a homomorphism from the group to itself. It is an endomorphism. This means in the intuition of multiplication, that they are maps from numbers to numbers that respect the distributive law. But there might be more multipliers then there are numbers because there may be more of these maps then the elements in the set. Because we don't want that, we pose a further restriction on our abelian group. It must be generatable from a single element (that is it must be 1 dimensional). Then the number of endomorphisms and the number of elements in the set are the same. Then we can assign each map to a number and when we multiply two numbers, we really look at one of the numbers as a map, and the other number as the input. The output is the multiplication. Or if this isn't vigorous enough then what you can do is look at the collection of the endomorphisms. They are isomorphic to the original group under the addition operator. However there is a further operation amongst functions that is not present amongst numbers. Composition (this too is deep. Any operation we are considering can be viewed as a composition). The composition between these functions define multiplication.

13. Feb 28, 2017

### epenguin

Sorry for coming back late. I think though we have lost the point of the original question. Why is multiplication and division, let's just say of integers for the moment, defined to be fundamental operation, independent of rather than deriveable from the laws or definition of ordinary addition of ordinary integers?

Is not the ordinary understanding of the meaning of multiplication n×m 'take n sets, each of m objects and and add them all up'? So it seems to me that all you need to understand at first is addition, or maybe not even understand it but have axioms for it, know how to do it. That given addition, you can define multiplication in terms of it, and that you can prove what I have seen given as axioms for multiplication. The commutative law for example.

With my definition above I am not like Wells' anti-hero Mr Polly, who could never remember whether it was six sevens or seven eights that made 56, and had no way to find out. Whereas I can find out using the above definition, I can put down a row of seven counters, do this eight times, I can make a nice rectangle of counters to be sure and so check I have the right number, 8, of rows, and then count up the total number I have, as long as I can count that high. I can do it with Hindu-Arabic numbers by moving counters so as to form rows of 10. I can use the principle to write a computer program for multiplication, and I suppose somebody has already done this and I am using it. If my counters a square they can fit together and form rectangles, I can calculate areas.

Is this not what everybody understands by multiplication?

(As for addition, this seems to me to be related to the concept of permanence. I can combine two heaps of potatoes or coins and consider that I still have the same number of these objects as before, then I can separate them in different ways into a different heaps and consider the total constant. I can recombine the heat and recover the original number I had. This concept is not innate I believe and is only grasped about age 4-5. This idea of permanence is fundamental to economic exchanges and money. It also comes up in chemistry where we consider atoms to be for practical purposes permanent. (I'm always explaining to students the stoichiometry that causes them such problems is only arithmetic that they have forgotten, true then division, ratios and proportions come into it.)

Is there anything wrong with this, anything I am overlooking? Wouldn't civilisation collapse without this understanding?

14. Feb 28, 2017

### Khashishi

It's arbitrary what we call a fundamental operation.

15. Feb 28, 2017

### epenguin

Being able or unable to derive its rules from those of a smaller system is not an arbitrary question.

16. Feb 28, 2017

### Khashishi

Is the "min" operation fundamental?
min(4, 6) = 4
Is the "modulo" operation fundamental?
7 mod 3 = 1
Is the number of combinations?
5 nCr 3 = 10
Why or why not? I assert that we call addition, subtraction, multiplication, division fundamental only because they are simple to learn and have been historically of high importance.

17. Feb 28, 2017

### Aufbauwerk 2045

I like the digital computer viewpoint. Express numbers in binary. Build a full adder from a few NAND gates. Subtract by adding negative numbers when you use 2's complement representation. Etc.

18. Feb 28, 2017

### epenguin

I see now what you mean by arbitrariness of calling an operation fundamental.
Not sure about the first one, but the second seems to require ordinary addition and subtraction, while the third requires ordinary multiplication. If so, then these operations are more fundamental than yours.
But if not, the OP's question still remains - just instead of calling the operations "fundamental" call them "the historically of high importance" operations, the question of whether they are independent awesome can be derived from others remains.

Last edited: Feb 28, 2017
19. Feb 28, 2017

### pwsnafu

In Peano, addition is defined through the successor operator. So, addition is not fundamental either.

20. Mar 1, 2017

### epenguin

Yes but a simple question was asked in #1 so how have we got to #20 and people insist on talking about something else? It begins to sound like embarrassment or evasion.

Can we derive multiplication from addition? was the question.

Whether we can derive addition from something else is a different question, on the face of it. Interesting, informing us what numbers really 'are', it would justify an article or a tutorial here, but on the face of it the question I asked would follow after, once the rules of the arithmetic are established. I am accepting for the moment that they are.