To determine if can be obtained using a finite number of operations (where denotes exponentiation) starting from (or ), we analyze the set of numbers generated by these operations, denoted .

Key Observations:

1. Closure of : is constructed inductively, starting with (since is generated...

Short answer: This is not known. With the operations you allow (starting from N or Q and using only finitely many applications of +, −, ·, ÷, and exponentiation), it is an open problem whether one can obtain π, and it is also open whether π is a root of any nontrivial equation built from these same operations.

More context

• The set of numbers you can produce this way is countable (there are onl...

Denote by E the set of all real numbers one can build starting from ℚ (or ℕ) and applying only a finite number of the five operations +, −, ×, ÷, ^
(we allow exponentiation x^y at any stage as soon as x,y are in E). Clearly E is a countable sub‐field of ℝ closed under taking arbitrary real exponents of positive elements. One checks immediately that all algebraic numbers lie in E, and by ...

Constructing π from Natural Numbers with Basic Operations

This is an interesting question about whether π can be expressed using only a finite combination of natural numbers and basic operations.

Analysis of the Problem

Let's define as the set of all real numbers that can be constructed from using a finite number of operations from (where $...

Write down all “constants’’ you can make starting with the rational numbers and, one step at a time, applying one of the five operations

  x+y , x−y , x·y , x⁄y ( y≠0), x^y .

After n steps you obtain a single real number that can be written by a finite formula involving the symbol ^ once for every time you used the last operation.
Put

 E₀ = ℚ ,  E_{n+1}= the set of all numbers that can be o...

First, I need to understand what kind of numbers I can create with these operati...

Let

 𝔈 = the smallest subset of ℂ that
  • contains ℚ, and
  • is closed under the five operations
   x+y , x−y , x·y , x/y (y≠0) , x^y .

In other words 𝔈 is the set of all numbers that can be written by starting with rational constants and writing a finite “algebraic–exponential term’’ such as

 (((3/2)-5) ^ (7+√2)) / (4^(4^(1/3))) .

Your question is whether π∈𝔈, and, if not, wh...

The answer to your main question is no - you cannot obtain π using a finite number of operations {+, -, ·, ÷, ^} on ℕ (or ℚ).

Here's why:

Main Result

The set of numbers obtainable through finite applications of {+, -, ·, ÷, ^} starting from ℕ (or ℚ) forms what's called the field of elementary numbers. This field contains:

1. All rational numbers
2. All algebraic numbers that can be ...

This is an excellent question that connects number theory, algebra, and the theory of transcendental numbers.

Main Question: Can be obtained this way?

The short answer is no.

Let's break down why. The set of numbers you can generate is a subset of what are known as the elementary numbers.

1. The Set of Constructible Numbers: Let's call the set of numbers you can obta...