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keiop3
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why is the sin(2*pi/7) non-constructible?
A non-constructible number is a real number that cannot be constructed using only basic arithmetic operations (addition, subtraction, multiplication, division) and the extraction of square roots. In other words, it cannot be expressed as a finite combination of rational numbers and square roots.
Non-constructible numbers are a subset of irrational numbers. While all non-constructible numbers are irrational, not all irrational numbers are non-constructible. Some irrational numbers, such as pi or e, can be expressed as infinite series or continued fractions, while non-constructible numbers cannot.
Yes, non-constructible numbers can be approximated using decimal expansions or continued fractions. However, these approximations will always be finite and will never be equal to the exact value of the non-constructible number.
Non-constructible numbers play a significant role in geometry and algebra. They help us understand the limitations of our basic arithmetic operations and the concept of constructibility in geometric constructions. They also have applications in number theory, where they are used to prove theorems about algebraic numbers.
Yes, there are several well-known examples of non-constructible numbers, such as the golden ratio, the square root of 2, and the cube root of 2. Other examples include algebraic numbers with irrational coefficients, such as √2 + √3, and numbers that cannot be expressed as the root of a polynomial with rational coefficients, such as π.