Can we represent pi as a dedeking cut in the rationals

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In summary, the conversation discusses the construction of real numbers from rational numbers and the possibility of representing pi using Dedekind cuts. The speaker also mentions that they found the answer through a thread on a forum.
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Hi , I only recently read the construction of reals from rationals.

I could grasp that [itex]\sqrt{2}[/itex] can be represented as the set of rationals given by
{x[itex]\in[/itex] Q : x2 < 2 } . As we know this set is defined purely in terms of Q.

Is there a dedekind cut representation for pi as well ?

I read somehwhere that not all reals can be defined. But since pi is defined , what would it's dedeking cut representation be?
 
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What is a Dedekind cut?

A Dedekind cut is a mathematical concept used to represent irrational numbers as a partition of rational numbers. It was introduced by German mathematician Richard Dedekind in the late 19th century.

Can all irrational numbers be represented as Dedekind cuts?

Yes, all irrational numbers can be represented as Dedekind cuts. This is because Dedekind cuts are defined as the partition of rational numbers into two sets, with one set containing all numbers less than the irrational number, and the other set containing all numbers greater than or equal to the irrational number.

Why is pi represented as an irrational number?

Pi is represented as an irrational number because it cannot be expressed as a finite or repeating decimal. It is an infinite decimal that continues on without pattern, making it impossible to represent as a fraction of two integers.

How is a Dedekind cut different from a decimal representation of a number?

A Dedekind cut is a representation of a number as a partition of rational numbers, while a decimal representation is a representation of a number as a finite or repeating decimal. Dedekind cuts are typically used to represent irrational numbers, while decimal representations are used for both rational and irrational numbers.

What is the significance of representing pi as a Dedekind cut?

Representing pi as a Dedekind cut allows us to understand and work with the number in a new way. It also helps us to understand the relationship between irrational and rational numbers, and how they can be represented and compared using mathematical concepts.

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