Elucubrations on positve, negative & imaginary numbers

[studiot]

1 hour ago, michel123456 said:

To me

1. multiplication has a very strong relationship with orthogonality.

2. Multiplication has also a relation with units (when you multiply meters with meters, you get m^2 which is different from m, when you multiply seconds with sec., you get s^2 which is also different, etc.) To me, it is bizarre to multiply length by length & still obtain a length. 

Although you seem to have stopped bothering with my offerings, I suggest you think more deeply here.

The product of Force x Distance is a grand counterexample to both your thoughts.

Force times parallel distance gives work or energy.

Force times perpendicular distance gives moment or leverage.

A compartmented box of 6 oranges x 10 oranges does not contain 60 square oranges

(Have you ever seen a square orange?)

It contains 60 oranges.

 

Edit I will leave it to Eise to explain why

(2 + i)(5 - 3i) does not give you an area, but another complex number.

(2 + i)(5 - 3i) = (10 + 5i -6i -3i²) = (13 - i)

 

Edited March 1, 2019 by studiot

[Eise]

1 hour ago, michel123456 said:

1. multiplication has a very strong relationship with orthogonality.

That really doesn't need to be. 

Every child at the party gets two pieces of chocolate. There are 4 children. The answer of the multiplication is definitely not 8 children.chocolate. The answer is  that there are 8 pieces of chocolate. So the dimension of this multiplication a.children x b.piecesOfChocolate is just piecesOfChocolate. So you must take care to transfer the concept of dimensions as used in physics (m², m/s², etc) to other situations.

1 hour ago, michel123456 said:

2. Multiplication has also a relation with units (when you multiply meters with meters, you get m^2 which is different from m, when you multiply seconds with sec., you get s^2 which is also different, etc.) To me, it is bizarre to multiply length by length & still obtain a length.

In mathematics we are just talking numbers, without any dimensions. What we expect of consistent number systems is that every operation defined on the elements of the number set is again an element of the number set. If you force dimensions on everything, then this would be impossible. 2 x 2 = 4 should be rewritten somehow like (2.units x 2.units = 4.units²), and so '4' would not be a member of the set of simple numbers. That makes no sense.

So with imaginary numbers: they are dimensionless. So e.g. the 'length' of an imaginary number is just a real number, with which you can calculate as any other number.

Also do not forget: the rules of the calculational operations in C must be defined in such a way that we get the well known rules in R when we apply them to elements of R, because R is a subset of C.

[michel123456]

4 hours ago, Eise said:

That really doesn't need to be. 

Every child at the party gets two pieces of chocolate. There are 4 children. The answer of the multiplication is definitely not 8 children.chocolate. The answer is  that there are 8 pieces of chocolate. So the dimension of this multiplication a.children x b.piecesOfChocolate is just piecesOfChocolate. So you must take care to transfer the concept of dimensions as used in physics (m², m/s², etc) to other situations.

In mathematics we are just talking numbers, without any dimensions. What we expect of consistent number systems is that every operation defined on the elements of the number set is again an element of the number set. If you force dimensions on everything, then this would be impossible. 2 x 2 = 4 should be rewritten somehow like (2.units x 2.units = 4.units²), and so '4' would not be a member of the set of simple numbers. That makes no sense.

So with imaginary numbers: they are dimensionless. So e.g. the 'length' of an imaginary number is just a real number, with which you can calculate as any other number.

Also do not forget: the rules of the calculational operations in C must be defined in such a way that we get the well known rules in R when we apply them to elements of R, because R is a subset of C.

Ok but you were using Pythagoras, which is geometric.

 

[Eise]

24 minutes ago, michel123456 said:

Ok but you were using Pythagoras, which is geometric.

Uh.. yes. The 'length' of the vector is just a real number. In complex numbers it is called the 'absolute value'. 

I think you should study e.g. the article in Wikipedia: it is really wonderful how it all fits together. 

[studiot]

1 hour ago, michel123456 said:

Ok but you were using Pythagoras, which is geometric.

I didn't mention pythagoras, but I did mention oranges in a box.

Furthermore the compartments form a prime example of the situation where area is a meaningless concept.

It doesn't matter how big or small the compartments are (so long as they will contain an orange)

and it doesn't mater what the area oft he box is.

There are still 10 rows of 6 oranges making 10 x 6 = 60 oranges in all.

Needless to say we are also considering 60 compartments in the box, whatever its dimnensions in metres.

Edited March 1, 2019 by studiot

[michel123456]

On 3/1/2019 at 5:51 PM, studiot said:

I didn't mention pythagoras, but I did mention oranges in a box.

Furthermore the compartments form a prime example of the situation where area is a meaningless concept.

It doesn't matter how big or small the compartments are (so long as they will contain an orange)

and it doesn't mater what the area oft he box is.

There are still 10 rows of 6 oranges making 10 x 6 = 60 oranges in all.

Needless to say we are also considering 60 compartments in the box, whatever its dimnensions in metres.

If I put oranges on the X and oranges on the Y the result of multiplication will still be oranges.

But if I put cm on X & cm on Y, measuring each compartment 10cm, I get 600 cm^2 (and not 600 cm).

What is the rule? When do I know which way is correct & which way is wrong (except from common sense).

 

[studiot]

3 hours ago, michel123456 said:

If I put oranges on the X and oranges on the Y the result of multiplication will still be oranges.

But if I put cm on X & cm on Y, measuring each compartment 10cm, I get 600 cm^2 (and not 600 cm).

What is the rule? When do I know which way is correct & which way is wrong (except from common sense).

 

 

That is a very good question.

I would have to invite others reading this to offer suggestions because I can't think of a rule to offer.

However here is another situation where x*y does not give the correct area.

Consider a roofer working out tiles for this roof 5 metres wide by 15 metres long

If he says the area of tiling is 5 x 15 and orders this amount he will be short, although x * y * = 5 * 15.

[michel123456]

Is there any other physical quantity that produces a square that is something, like m^2?

Does a sec^2 mean anything? Or a kg^2?

[Eise]

13 hours ago, michel123456 said:

If I put oranges on the X and oranges on the Y the result of multiplication will still be oranges.

No. They would be oranges². Which simply does not make sense. One can multiply a number of oranges with a dimensionless number. Or, if every orange weighs 200g, you can ask for the total weight (e.g. 6 oranges x 200g = 1.2 kg oranges). I confess I get a little lost here: I think the way you apply mathematics on concrete situation is not always trivial. You always make an abstraction if you transform a real situation into a mathematical description. E.g., if you are interested in transport weight, you can abstract from the fact that you are talking oranges. 6 apples of 200g weigh just as much as 6 oranges of 200g. However, if you are producer of orange juice, it is of importance that you will get 1.2kg oranges, and not apples.

Mathematics in this sense is just about numbers. As said before, meaningful operators on any number set should give you a number of that set again. What they stand for depends on how you translated a concrete situation in some number calculation.