Physical Meaning of Mathematical Quantities

[EngWiPy]

Hello,

What is the physical meanining of rational and irrational numbers? In the same sense, what is the physical meanining of complex quantities?

Thanks in advance

 

[Pinu7]

Haven't you learned in kindergarten,

"If Billy started out with five apples, but then gave two of them to Anne, how much apples does Billy have now?"
That is why we use integers, for rational numbers we partition a pie amongst friends.

Irrational numbers were used to describe the length of the hypotenuse of a triangle.

Happy?

 

[HallsofIvy]

There is no "physical" meaning. Mathematics is not physics.

 

[Elucidus]

The only numbers that one could argue have "physical meaning" are the whole numbers 0, 1, 2, 3, etc. I have 1 tree in my backyard, I am instructing 4 courses this semester, I own 5 pets, and 0 rockets, etc.

All other numeric constructions, although useful, do not have a "phyiscal meaning". If you take a piece of chalk and break it in half, you have two pieces of chalk (not a fraction). What would be the "physical meaning" of the number -5?

The more outre a number gets, the more it is removed from a physical sense. Complex numbers, quarternions, octonions, etc all are very abstract and it would be extremely difficult to fabricate a reality for them.

--Elucidus

 

[HallsofIvy]

The only numbers that one could argue have "physical meaning" are the whole numbers 0, 1, 2, 3, etc. I have 1 tree in my backyard, I am instructing 4 courses this semester, I own 5 pets, and 0 rockets, etc.

One can argue that even whole numbers are not necessarily "physical". Counting depends upon being able to distinguish things. Some things, like elephant, can be counted easily but, say, slime mold. Can you always say that there are a specific number of slime mold in a yard?

All other numeric constructions, although useful, do not have a "phyiscal meaning". If you take a piece of chalk and break it in half, you have two pieces of chalk (not a fraction). What would be the "physical meaning" of the number -5?

The more outre a number gets, the more it is removed from a physical sense. Complex numbers, quarternions, octonions, etc all are very abstract and it would be extremely difficult to fabricate a reality for them.

--Elucidus

 

[Hootenanny]

The only numbers that one could argue have "physical meaning" are the whole numbers 0, 1, 2, 3, etc. I have 1 tree in my backyard, I am instructing 4 courses this semester, I own 5 pets, and 0 rockets, etc.

You gave these numbers a physics meaning when you associated them with objects, such as trees. The numbers on their own have no physical meanings. For example, the number -5 has no physical meaning, as you correctly assert. However, if I were to say -5^oC, that certainly has a physical meaning. Equally, the natural numbers do not have a physically meaning until one them with associates a physical quantity.

 

[EngWiPy]

You gave these numbers a physics meaning when you associated them with objects, such as trees. The numbers on their own have no physical meanings. For example, the number -5 has no physical meaning, as you correctly assert. However, if I were to say -5^oC, that certainly has a physical meaning. Equally, the natural numbers do not have a physically meaning until one them with associates a physical quantity.

But there is a strong relationship between physical and mathematical models, where the latter used to describe the former in a more convenient way. For example, one can say that -5 m is traveling of an object in an opposite (presumed) direction, the same is applied of -5 Ampere, where the minus sign here describes direction.

 

[dx]

The real line exists independently of any algebra that you introduce on it. The distinction between rational and irrational points is not a property of real numbers, but a property of the algebra. It is a distinction which is induced by the algebra. So when you ask what the physical meaning of rational and irrational numbers is, you are essentially asking what it means to multiply real numbers. You can define this in the usual way of course, but whether it is a useful definition or not depends on the context. For example, one use of real numbers is to describe points on a rigid rod. Now what does it mean to multiply two points on a rigid rod? Nothing; it is not a useful notion here, and therefore the distinction between rational and irrational is irrelevant in this case. Even in physical situations where the usual multiplication is meaningful, I don't know of any cases where the particular distinction between rational and irrational has any physical interest.

 

[Hootenanny]

But there is a strong relationship between physical and mathematical models, where the latter used to describe the former in a more convenient way. For example, one can say that -5 m is traveling of an object in an opposite (presumed) direction, the same is applied of -5 Ampere, where the minus sign here describes direction.

Numbers, whether real, imaginary, complex, rational, irrational, natural or otherwise are not 'mathematical models', they are simply mathematical objects. Granted, it is likely that the concept of numerals and numbers originated with the need to quantify a group of physical objects. However, our current definition of numbers is completely independent of any physical interpretation. As I said earlier and dx also comments, one can associate a physical meaning with any set of numbers, but that doesn't mean that that, or any other physical interpretation, is an inherent property of that set of numbers.