Cardinal number: irrationals vs fractals

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Loren Booda
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How does the cardinal number for the set of irrational numbers compare to that for a fractal set?
 
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Cardinal number for irrationals is the same as for reals, i.e. c. Fractals (if I understand what you are driving at) form a continuous subset of the plane. Since the cardinality of points in the plane is also c, any continuous subset will have cardinality c.
 
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The cardinal number for the set of irrational numbers is uncountable, meaning that it cannot be put into a one-to-one correspondence with the natural numbers. This is because there are infinitely many irrational numbers between any two rational numbers, making it impossible to count them all.

On the other hand, the cardinal number for a fractal set can vary depending on the specific set being considered. Some fractal sets may have a finite cardinality, while others may have an infinite cardinality. This is because fractals are self-similar and can have infinitely many iterations, leading to varying sizes and complexities.

In general, the cardinal number for a fractal set is likely to be smaller than the cardinal number for the set of irrational numbers. This is because while fractals can have infinitely many points, they are still bounded and can be contained within a finite area. Irrational numbers, on the other hand, are unbounded and can extend infinitely in both positive and negative directions on a number line.

In summary, the cardinal number for the set of irrational numbers is larger than that for a fractal set, due to the uncountable nature of irrational numbers and the bounded nature of fractals.
 

What is a cardinal number?

A cardinal number is a type of number that represents the size or quantity of a set, group, or collection. It is often used to describe how many objects are in a set, without specifying the objects themselves. Examples of cardinal numbers are 1, 2, 3, 4, etc.

What are irrational numbers?

Irrational numbers are numbers that cannot be expressed as a simple fraction (ratio) of two integers. They are decimal numbers that continue infinitely without repeating a pattern. Examples of irrational numbers are pi (3.14159...), the square root of 2 (1.41421...), and the golden ratio (1.61803...).

What are fractals?

Fractals are geometric shapes that exhibit self-similarity at different scales. This means that when you zoom in on a fractal, you will see the same or similar patterns repeated. Fractals are created through mathematical equations and can be found in nature, such as in snowflakes and coastlines.

What is the difference between irrational numbers and fractals?

The main difference between irrational numbers and fractals is that irrational numbers are infinite decimal numbers that do not exhibit self-similarity, while fractals are geometric shapes that exhibit self-similarity at different scales. Additionally, irrational numbers are used to represent quantities, while fractals are used to describe geometric patterns.

How are irrational numbers and fractals related?

Irrational numbers and fractals are related in that some fractals, such as the Mandelbrot set, are created using complex numbers and can exhibit irrational behavior. Additionally, the concept of self-similarity in fractals is related to the infinite nature of irrational numbers. However, they are distinct mathematical concepts with different properties and applications.

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