How can any measure of a physical quantity be irrational?

[Kilo Vectors]

Hello

Aren't all irrational numbers having an infinitely long decimal component? If so, how can any measure of a physical quantity be irrational?

the decimal component is infinitely long..but the magnitude of the physical quantity surely isnt?

 

[jbriggs444]

Measured physical quantities (ignoring discrete counts) are not exact. They are compatible with a range of exact values and are, consequently, neither rational nor irrational.

 

[HallsofIvy]

The number you get and, perhaps, write down can't be irrational. But that is a problem with "measurement", not mathematics. If you, for example, draw a line segment, define its length to be "1", draw a perpendicular at one end of that line segment and, geometrically, mark of an equal length on this new segment, then the segment connecting the two endpoints will have length "square root of 2", an irrational number. But if you then measure that length (using a ruler with the original length as "1") you will NOT get exactly "square root of 2".

 

[jbriggs444]

The number you get and, perhaps, write down can't be irrational. But that is a problem with "measurement", not mathematics. If you, for example, draw a line segment, define its length to be "1", draw a perpendicular at one end of that line segment and, geometrically, mark of an equal length on this new segment, then the segment connecting the two endpoints will have length "square root of 2", an irrational number. But if you then measure that length (using a ruler with the original length as "1") you will NOT get exactly "square root of 2".

I would disagree mildly. It is a problem with notation together with a problem of measurement. You cannot measure accurately enough to distinguish between rational and irrational. And most [but not all] notations make it difficult to write down irrationals. But one can write down pi, for example -- I just did.

 

[HallsofIvy]

I don't think there really is a disagreement because what I meant to say was that a measurement, that is written down, can't be irrational. Certainly, you can write down $\pi$ or $\sqrt{2}$ but you cannot measure a line segment and get either of those.

 

[jbriggs444]

If there is a marker on your ruler labelled $\pi$ or $\sqrt{2}$, you can get those results. I agree that one can then argue that the thing written down is a rational multiple of those units.

 

[Kilo Vectors]

hello so in theory, if we have a triangle with two sides of 1 cm (theoretically they have been able to get its length to 1 m..) the hypotenuse when drawn should be root of 2 shouldn't it? except when we draw it, it won't be that because length is finite and it must not haave an infinite repeating decimal component.

Measurements are never "true" there is only the accepted value, a value that can be agreed to represent the magnitude of the physical quantity best. In theory though, assuming we can draw two lines of exactly 1 cm each perpendicular, then join them to make a triangle, we cannot obtain root of 2? but must round it off to some decimal places correct? isn't it wrong to say that it would be root 2 because of the never ending decimal component?! or is it correct because the decimal component gets smaller and smaller in the units of magnitude and so is negligible? (eg beyond 4 DP the difference is negligible, any statement like that?)

thank you for answers..

 

[HallsofIvy]

hello so in theory, if we have a triangle with two sides of 1 cm (theoretically they have been able to get its length to 1 m..) the hypotenuse when drawn should be root of 2 shouldn't it? except when we draw it, it won't be that because length is finite and it must not haave an infinite repeating decimal component.

You seem to be misunderstanding "finite" and "infinite repeating decimal". How a number is represented in a specific numeration system has little to do with the nature of the number itself. The number represented by "1" in base 10, if written in base 3 has an "infinite repeating component"- 0.11111...

Measurements are never "true" there is only the accepted value, a value that can be agreed to represent the magnitude of the physical quantity best. In theory though, assuming we can draw two lines of exactly 1 cm each perpendicular, then join them to make a triangle, we cannot obtain root of 2? but must round it off to some decimal places correct? isn't it wrong to say that it would be root 2 because of the never ending decimal component?!

No. If, "in theory", with two lines of exactly 1cm length, perpendicular, the distance between their endpoints is exactly $\sqrt{2}$. There is no reason we "must" round it off. The problem appears to be that you do not know what a "number" is! You are mistaking the decimal representation of a number for the number itself.

thank you for answers..[/QUOTE]

 

[Kilo Vectors]

You seem to be misunderstanding "finite" and "infinite repeating decimal". How a number is represented in a specific numeration system has little to do with the nature of the number itself. The number represented by "1" in base 10, if written in base 3 has an "infinite repeating component"- 0.11111... No. If, "in theory", with two lines of exactly 1cm length, perpendicular, the distance between their endpoints is exactly $\sqrt{2}$. There is no reason we "must" round it off. The problem appears to be that you do not know what a "number" is! You are mistaking the decimal representation of a number for the number itself.

thank you for answers..

[/QUOTE]

Ah ok Mr Ivy, thank you.

 

[Kilo Vectors]

Is there a good explanation for what a number is?

 

[jbriggs444]

Is there a good explanation for what a number is?

The type of numbers we are talking about in the case at hand are the "real numbers". The nature of the real numbers is covered in a mathematical field called "real analysis". https://en.wikipedia.org/wiki/Real_analysis.

Be warned that the term "real" is just a name. Do not take it as an indication that these numbers are really real in a physical sense. They're still just numbers.

 

[Kilo Vectors]

The type of numbers we are talking about in the case at hand are the "real numbers". The nature of the real numbers is covered in a mathematical field called "real analysis". https://en.wikipedia.org/wiki/Real_analysis.

Be warned that the term "real" is just a name. Do not take it as an indication that these numbers are really real in a physical sense. They're still just numbers.

Thank you Mr Briggs.

I am aware what real numbers, rational, irrational, integers are lol, or maybe not

 

[Mark44]

How a number is represented in a specific numeration system has little to do with the nature of the number itself. The number represented by "1" in base 10, if written in base 3 has an "infinite repeating component"- 0.11111...

The base-10 number 1, when written as a base-3 number, is also 1.

In base 2, which might be what you were thinking of, 1₁₀ can be written as either 1₂ or as a repeating binary fraction 0.1111... (base-2).

In base 3, 0.111₃... means $\frac 1 3 + \frac 1 {3^2} + \frac 1 {3^3} + \dots$, which can be shown to converge to 1/2.

 

[HallsofIvy]

Right. Thanks.

 

[Khashishi]

Are you talking about math or measurement? You never have a perfect triangle in real life, and you never can perfectly measure the hypotenuse using a ruler. But if you make the measurement very precisely, then you will get something very close to what math tells you.

 

[Kilo Vectors]

Are you talking about math or measurement? You never have a perfect triangle in real life, and you never can perfectly measure the hypotenuse using a ruler. But if you make the measurement very precisely, then you will get something very close to what math tells you.

math, in some weird way..actually I know some basics of measurement science or theory (?) but this is a problem of measurement it seems and my knowledge of numbers was not enough to understand when I asked the question.

I know that the absolute error of a value is the measured value - accepted value where accepted value is the magnitude that can be agreed to represent the magnitude of the physical quantity best.

I guess my question was answered in the first few posts? it is neither a rational or irrational number, even though in theory it should be? I am actually still a little confused but i will surely come back when I have read up more on it (been a little occupied)..my apologies, i struggle with mathematic sometimes.

But I think you would mean if you measure it accurately, and not precisely as acoording to me:

accuracy is the degree of agreement between the measured and accepted value...
precision refers to the reproduce-ability of one measurement, it is related to the number of random errors in experimentation (inversely).

sorry sir, i know its nitpicking but don't want to get confused as i do very easily

thank you for your answer,,i will surely be back shouldve done more reeading

i understand one cannot measure say, length to get an irratoinal number. I knew that beforehand, but it is a problem with measurement and must read some stuffs again.