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
Square Root Of 2
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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 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
The square root of 2 is the positive number that when multipled by itself yields 2 The square root of 2 is an irrational number so it can t be represented in a finite sequence of digits When it s rounded off it s approximately 1 414 the square route of 2 will carry on forever but the longest people have found is 1 41421356237 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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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 Below are some particular calculations when the radius of the circle is 1 2 eqalign P 2 2 1 cdot sqrt 2 approx 2 82842712 cr P 3 2 2 cdot sqrt 2 sqrt2
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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
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
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