Are there really 4 fundamental math operations?

[Randy Beikmann]

I say you are doing exactly that. Say, to avoid extra issues, you are doing it 'by hand' . Then you are making the multiplication faster by some tricks - using the Hindu-Arabic decimal place system plus having made the ivestment of learning by rote all the multiples up to n, m = 10 at least. But you can only justify or discover the tricks, and only work out the multiplcation tables by the addition definition, how else?

I definitely can (and do) think of it as 15 pounds for the first square inch, and for the second and third, but in doing that I have already multiplied the units out. I don't immediately see how to turn the multiplication of units into addition of units.

 

[epenguin]

As per prescription I add the second and get 30, to that I add the third 15 and get 45.
I've told you what I have done.
I can do it with an abacus.
You've only told me you've multiplied, not how.
I don't think you can multiply with an abacus without seeing you're doing a sequence of addtions.

 

[anorlunda]

But a rotation can be seen as repeated addition with carry. In binary, shifting x left is the same as x+x = 2x.

Good one. But I wasn't thinking of that kind of rotation, but rather the kind of rotation like multiplication by $e^{i\theta}$

 

[Baluncore]

If I make an equilateral-triangle into a right-angle-triangle by stretching one side to make its length √2, how can I lengthen that side by repeated addition ?

 

[Randy Beikmann]

As per prescription I add the second and get 30, to that I add the third 15 and get 45.
I've told you what I have done.
I can do it with an abacus.
You've only told me you've multiplied, not how.
I don't think you can multiply with an abacus without seeing you're doing a sequence of addtions.

Your statements don't address my comments. I agreed from the start that, if using pure numbers, multiplication can be seen as repeated addition. The issue I have is when applying the mathematics, and the numbers are accompanied by units. Baluncore gave another similar example above.

 

[Mark44]

Your statements don't address my comments. I agreed from the start that, if using pure numbers, multiplication can be seen as repeated addition.

How about $e^2$ times $\pi$? What's the repeated addition that produces this product?

 

[Biker]

How about $e^2$ times $\pi$? What's the repeated addition that produces this product?

I mean what a calculator does is approximate an value for these numbers then you get a pure number plus fraction. Or I can think of it as there is an answer to this if you get more and more closer to the values until the rest becomes negligible.

 

[Mark44]

I mean what a calculator does is approximate an value for these numbers then you get a pure number plus fraction. Or I can think of it as there is an answer to this if you get more and more closer to the values until the rest becomes negligible.

If you approximate $e^2$ as 7 and $\pi$ as 3, you can certainly write 7 * 3 as 7 + 7 + 7, or 21. It gets harder if you approximate the two numbers as 7.4 and 3.1. How do you add 7.4 to itself 3.1 times? You can't express the product 7.4 * 3.1 purely as an addition.

You might say, well, we can look at 74 * 31, and then add calculate 74 + 74 + 74 + ... + 74, where there are 31 terms being added. However, the answer will by too large by a factor of 100, so you're then going to have to divide the answer by 100. So even with this trick, we're still not able to do 7.4 times 3.1 purely as an addition problem.

 

[Biker]

If you approximate $e^2$ as 7 and $\pi$ as 3, you can certainly write 7 * 3 as 7 + 7 + 7, or 21. It gets harder if you approximate the two numbers as 7.4 and 3.1. How do you add 7.4 to itself 3.1 times? You can't express the product 7.4 * 3.1 purely as an addition.

You might say, well, we can look at 74 * 31, and then add calculate 74 + 74 + 74 + ... + 74, where there are 31 terms being added. However, the answer will by too large by a factor of 100, so you're then going to have to divide the answer by 100. So even with this trick, we're still not able to do 7.4 times 3.1 purely as an addition problem.

I don't get why you can't divide or I can just look at it like this: add 7.4 three times and then find out what is 1/10 of 7.4.. 0.74 If I add this 10 times it gives me 7.4
(7.4 + 7.4 + 7.4 + 0.74) = 22.94
You can still think division as repeated addition.. I mean what could multiplication be if not repeated addition.

However, What I can't answer is the physical intuition behind multiplication
F = ma
It is strange to think of F = (m+m+... ) ... a times. Certainly it is strange, But the idea about a force that it gets scaled as the mass double or triples which is addition.
I can't really give an answer to that.

 

[jbriggs444]

I don't get why you can't divide or I can just look at it like this: add 7.4 three times and then find out what is 1/10 of 7.4.. 0.74 If I add this 10 times it gives me 7.4

You can only divide if you have defined what division means. Division is normally defined in terms of multiplication. We are still working on getting you to tell us how you define multiplication for decimal fractions.

Yes, you can proceed to define multiplication for terminating decimal fractions in terms of multiplication for integers combined with some book-keeping actions on the position of the decimal point. And yes, you can define multiplication for arbitrary real numbers in terms of the limiting behavior of terminating decimal fractions which more and more closely approximate the non-terminating decimal fractions which correspond to the real numbers you are multiplying. In the end, that definition has as much to do with dotting the i's and crossing the t's and making sure the definition works as it does with the repeated addition.

 

[pwsnafu]

I mean what a calculator does is approximate an value for these numbers then you get a pure number plus fraction. Or I can think of it as there is an answer to this if you get more and more closer to the values until the rest becomes negligible.

And what do you do about the non-computable real numbers?

 

[Mark44]

I don't get why you can't divide or I can just look at it like this: add 7.4 three times and then find out what is 1/10 of 7.4.. 0.74 If I add this 10 times it gives me 7.4

But the claim is that multiplication is repeated addition, and only addition.

 

[epenguin]

If I make an equilateral-triangle into a right-angle-triangle by stretching one side to make its length √2, how can I lengthen that side by repeated addition ?

I am not understanding the construction you mean. But you can multiply any length at all by any whole number in a ruler and compass construction.

 

[epenguin]

Your statements don't address my comments. I agreed from the start that, if using pure numbers, multiplication can be seen as repeated addition. The issue I have is when applying the mathematics, and the numbers are accompanied by units. Baluncore gave another similar example above.

Well you are making a distinction between numbers and the things that they are numbers of.

Units such as pounds weight just examples of things. We can apply arithmetic operations to things of the same concrete kind, like weights, like apples, to somewhat less concrete things like days of the week but still 7 × 4 = 28 as for any other things, letters of the alphabet which are actually not concrete things but there are still 26 of them, or to dollars which are surely not concrete material things at all though they have concrete representations (and here we may be close to the origins of arithmetic). We can forget about what kinds of things we are dealing with whilst we are calculating arithmetic, which is a great advantage.

So I was thinking that it is like things are nouns and numbers are adjectives, when we say five apples.
And that when we say 5 + 8 = 13 we are really saying
5 (things of a kind) + 8 (things we recognise as of the same kind) = 13 (things of that kind)
but we just compactify notation and write 5 + 8 = 13.
The things that we can apply the arithmetic operations to have to have a kind of permanence as I mentioned earlier, and also they have to be recoverable as long as we do not discriminate between the individual things of a kind, so that our system must include
5 + 8 = 13 → 13 = 5 + 8

(It just struck me that sometimes in multiplication you multiply your units as in area, and sometimes you don't as in your example. I don't know whether this is a difficulty for you for me.)

Anyway, if anyone denies that multiplication is repeated addition please explain to me how without repeated addition they can construct a multiplication table like the ones you find in children's books.

 

[AgentCachat]

How does tetration, pentation, etc, fit into the "fundamental operation" debate? Seems to me they suggest there is no fundamental operation.

 

[AgentCachat]

Is Zero considered to a member of the set of natural numbers by advanced students of math?
Intuitively it strikes me that the absence of a thing is not included in idea of 'a quantity of things'.
For example you cannot accuse somebody of stealing zero amount of cash.

You can "accuse" a check writer of having zero amount of cash if they in fact have no money in the bank.

 

[INT-986]

The notion of fundamental operations in mathematics seems to depend entirely on questions about the foundations of mathematics that I can't claim to have an answer to.

For example, if logicism can be made viable, then the fundamental operations are only those logical operations required to deduce any given mathematical truth.

 

[rootone]

You can "accuse" a check writer of having zero amount of cash if they in fact have no money in the bank.

OK, but that means negative numbers also are in the set of natural numbers.
You can miss a bus, but you cannot miss a bus that wasn't there in the first place

 

[_PJ_]

Surely if something is derived (or can be derived) from another, then this rules out that something as a fundament and may enforce the 'fundamentality' of the other.
¿?

 

[Baluncore]

Surely if something is derived (or can be derived) from another, then this rules out that something as a fundament and may enforce the 'fundamentality' of the other.

What if a can be derived from b, which can be derived from c, and c can be derived from a. What is fundamental there?

Unfortunately, there can be many definitions of the term “fundamental”.

 

[rootone]

What if a can be derived from b, which can be derived from c, and c can be derived from a. What is fundamental there?

My guess would be that a,b, and c are the same thing and the distinction depends on the state (or choice maybe?),of the observer.

 

[_PJ_]

What if a can be derived from b, which can be derived from c, and c can be derived from a. What is fundamental there?

Unfortunately, there can be many definitions of the term “fundamental”.

No. I must disagree; there is one definition of fundamental.

In your ecample, neither a nor b nor c are fundamental

I fear this discussion heading into murky waters of Kurt Gödel :)

Maybe nothing is fundamental, and mathematics is a magically stable house of cards on a foundation of imagination supported by matjematical elephants!

 

[rootone]

The concept that if I have two apples, and I will give one to you if you give me an apple back tomorrow seems like a fairly safe axiom.

 

[pwsnafu]

Surely if something is derived (or can be derived) from another, then this rules out that something as a fundament and may enforce the 'fundamentality' of the other.
¿?

If that's your definition of fundamental, then addition of natural numbers is not fundamental. We define addition as $a+S(b) = S(a+b)$ where $S$ is the successor operator.

 

[_PJ_]

If that's your definition of fundamental, then addition of natural numbers is not fundamental. We define addition as $a+S(b) = S(a+b)$ where $S$ is the successor operator.

who's "we" and why is the current mode of pregerred definition assumed to have any real objective arbitration over the nature of fundament?

I'm not arguing- I fully appreciate your most certainly more quaölified expertise - only I don't believe that answers to such questions can be so definite or absolute.
I have on reflection, changed my premature mindset concerning addition already since my first post in this topic, though.

 

[_PJ_]

incidentally, please be awate that in the quoted, tecent post, I made no cöaim as to what my definirion of fundamental was, only provided one example of what it clearly ISN'T.
There's a huge difference

 

[Baluncore]

No. I must disagree; there is one definition of fundamental.

I agree that there can be only one definition in anyone field. But here we are discussing multiple different fields. Mathematics requires internal consistency, while a post-modernist analysis allows for many interpretations in many different fields, or schools of thought. For example; teaching arithmetic to children, or the assumptions that underly formal mathematics.

 

[newjerseyrunner]

I think what is fundamental depends on context. For example, you say that you can't derive addition from more fundamental functions.

I can do binary integer addition with XOR, AND, and SHR . To me, there are three fundamental operators: AND, OR, and NOT.

So you can define your own axioms given your specific domain. If your domain is integers, then addition is the only axiom. If you're representing your integers as binary strings, addition is derived from more fundamental logic.