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

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• LittleRookie
In summary: EMFs can be described as vector fields with real numbers as points. Physical fields are analogous to sequences in that they have a ring structure, and physical quantities can be related to one another through operations on fields. For instance, the electric potential at a point can be related to the electric field at that point, and the electric field can be related to the electric potential.
LittleRookie
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 can't do so for multiplication of irrational numbers. Multplication of rational numbers takes the usual intepretation of "(a out of b) of something".

Any thoughts?

The area of a circle or a rhombus inside a square are multiplications of irrational numbers.

I need more help :(

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.

FactChecker said:
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.

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

member 587159

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 and hmmm27
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:
fresh_42 said:
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).

## 1. How can I use multiplication of irrational numbers in real life?

Multiplication of irrational numbers can be used in real life situations such as calculating the area of a circle or the volume of a sphere. These calculations involve multiplying the irrational number pi (π) with the radius or diameter of the circle or sphere.

## 2. Can you provide an example of how multiplication of irrational numbers is used in the real world?

One example is calculating the distance traveled by a car with a constant speed of 60 miles per hour for 2.5 hours. This can be represented by the equation 60 x 2.5 = 150 miles, where the irrational number 2.5 represents the time traveled.

## 3. How is multiplication of irrational numbers related to geometry?

In geometry, multiplication of irrational numbers is used to calculate the lengths of sides and areas of shapes with irrational measurements, such as square roots. For example, the Pythagorean theorem uses multiplication of irrational numbers to find the length of the hypotenuse of a right triangle.

## 4. Why is it important to understand how to relate multiplication of irrational numbers to real world?

Understanding how to relate multiplication of irrational numbers to real world situations is important because it allows us to solve practical problems and make accurate calculations in various fields such as science, engineering, and finance. It also helps us better understand the concept of irrational numbers and their significance in mathematics.

## 5. Are there any real world applications where multiplication of irrational numbers is used in combination with other mathematical operations?

Yes, multiplication of irrational numbers is often used in combination with addition, subtraction, and division in real world applications. For example, in finance, compound interest calculations involve multiplying an irrational number (interest rate) with the principal amount and then adding the resulting amount to the principal. In physics, the equation for work (W = F x d) involves multiplying an irrational number (force) with a distance.

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