I How to relate multiplication of irrational numbers to real world?

I'm aware of the axioms of real numbers, the constructions of real number using the rational numbers (Cauchy sequence and Dedekind cut). But I can't relate the arithmetic of irrational numbers to real world usage.

I can think the negative and positive irrational numbers to represent (measurement of) real life quantities that satisfy the axioms. This is similar to relating positive and negative rational numbers to real world usage.

However, I cant do so for multiplication of irrational numbers. Multplication of rational numbers takes the usual intepretation of "(a out of b) of something".

Any thoughts?
 

fresh_42

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The area of a circle or a rhombus inside a square are multiplications of irrational numbers.
 
I need more help :(
 

FactChecker

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In a physical length measurement, a particular length is rational or irrational depending only on what units you are using to do the measurement. There is no physical difference between a rational versus an irrational length. Suppose a rectangle is measured to have an irrational width and height. Then the area is the product of two irrational numbers in that choice of units.
 
In a physical length measurement, a particular length is rational or irrational depending only on what units you are using to do the measurement. There is no physical difference between a rational versus an irrational length. Suppose a rectangle is measured to have an irrational width and height. Then the area is the product of two irrational numbers in that choice of units.
I understand that. I'm seeking help in the interpretation of multiplication of irrational numbers in the real world. Something similar to the real world interpretation of multiplication of rational numbers, for instance, 1/2 times of 3 can be interpreted as divide 3 into 2 parts, and take 1 out of the 2 parts.
 

FactChecker

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I look at it as the limit of rational multiplications of the truncated representations, so I can not help further.
 

fresh_42

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Thea area of the pink square is ## \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}## which can be found in the real world. Likewise has the circle with a diameter of ##1## an area of ##\frac{\pi}{4}## and a circumference of ##\pi \cdot 1##. All are multiplications by irrationals.

I do not understand your problem. The way they came into use was, that there have been actually lengths which were not rational, the diagonals of squares. As they could be sides of other squares (see image), they had to be multiplied to get the area.

Algebraically it is as you said above: reals are equivalence classes of Cauchy sequences, and sequences carry a ring structure which turns out to be a field.
 

Klystron

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As a young rookie when I encountered a mathematical object difficult to visualize; i.e., apply to the "real world", I would fall back on set theory.

In posts #2 and 7 mentor @fresh_42 explains and discusses your question in what we could describe as geometric solutions. The last sentence of post #7 provides broad guidance on algebraic methods to solve these problems including operations on irrational (and complex) numbers:
Algebraically it is as you said above: reals are equivalence classes of Cauchy sequences, and sequences carry a ring structure which turns out to be a field.
If geometric objects and length relations provide insufficient real world linkage, consider examples of physical fields, such as an electromagnetic_field (EMF).
 

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