Square Root Of 2

Square Root Of 2 Since square root of 2 is a irrational number which we know or assume its decimal part is not a finite number or doesn t terminate how come we say that this infinite number not in terms of being a large number times itself is equals 2

Law of Exponents Rightarrow x a 1 2 is a positive root of x 2 a assuming a 0 Further by definition sqrt a is also a positive root of x 2 a If a 1 2 neq sqrt a then the quadratic x 2 a has 2 roots a 1 2 pm sqrt a contradicting the theorem that a nonzero polynomial over a field has no more roots than its degree begingroup So if the square root is defined to be the principal square root I do not need to write x for sqrt x 2 Because it s agreed that the principle square root is meant so sqrt x 2 x

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This happens clearly because the two square roots cancel each other out due to the fact that the squaring operator is the inverse of the square root operator and vice versa Now let s perform the same method on x In 2 2 of her essay on mathematical morality Eugenia Cheng includes the following example Why is it possible for an irrational to the power of an irrational to be rational Here is a nice little proof that it is possible Consider sqrt 2 sqrt 2 If

The irrationality of the square root of 2 follows from our knowledge of how Pythagorean triples behave specifically that for positive integers x y and z if x 2 y 2 z 2 then x is not equal to y But if the square root of 2 were rational then there would exist positive integers a and b such that a b the square root of 2 Then a 2 b 2 begingroup The square root of i is 1 i sqrt 2 Try it out my multiplying it by itself It has no special notation beyond other complex numbers in my discipline at least it comes up about half as often as the square root of 2 does that is it isn t rare but it arises only because of our prejudice for things which can be expressed using small integers endgroup

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Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow the largest most trusted online community for developers to learn share their knowledge and build their careers I am trying to prove the existence of the square root of 2 I have some steps with a very vague explanation and I would like to clarify

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My Math Resources Squares Square Roots Poster

Radicals How Come Square Root Of 2 Times Itself Equals 2

https://math.stackexchange.com/questions/4032857/how-come-square-…
Since square root of 2 is a irrational number which we know or assume its decimal part is not a finite number or doesn t terminate how come we say that this infinite number not in terms of being a large number times itself is equals 2

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Why The Square Root Of X Equals X To The One Half Power

https://math.stackexchange.com/questions/656198
Law of Exponents Rightarrow x a 1 2 is a positive root of x 2 a assuming a 0 Further by definition sqrt a is also a positive root of x 2 a If a 1 2 neq sqrt a then the quadratic x 2 a has 2 roots a 1 2 pm sqrt a contradicting the theorem that a nonzero polynomial over a field has no more roots than its degree

Get More Here!!