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- Can a polynomial have an irrational coefficient? - Physics Forums
A polynomial can indeed have irrational coefficients, as demonstrated by the example y = x^2 + sqrt (5)x + 1 The confusion arises from historical conventions where coefficients were often assumed to be rational, but modern mathematics recognizes polynomials over any field, including the reals and complex numbers In Algebra 2, the term "polynomial over the rationals" may not be explicitly
- irrational = rational - Physics Forums
Can someone prove that there exists x and y which are elements of the reals such that x and y are irrational but x+y is rational? Certainly, there are an infinite number of examples (pi 4 + -pi 4 for example) to show this, but how would you prove the general case?
- Showing that tan(1) is irrational - Physics Forums
(1 ∘) is irrational Homework Equations The Attempt at a Solution Suppose for contradiction that is rational We claim that this implies that is rational Here is the proof by induction: We know by supposition that the base case holds So, suppose that is rational Then , and this is the ratio of two rational numbers, and so is rational
- Square Root of an Irrational Number is Irrational - Physics Forums
The discussion revolves around proving that the square root of an irrational number is also irrational Participants suggest using proof by contradiction, where assuming the square root is rational leads to a contradiction regarding the nature of the original number They clarify that the contrapositive approach—showing that if the square root is rational, then the original number must be
- Irrational numbers arent infinite. are they? • Physics Forums
There are an infinite number of irrational numbers just as there are an infinite number of integers, rational numbers and real numbers However since reals are uncountable and rationals are countable then irrationals are uncountable meaning there are many more irrationals than rationals
- Constructing Lengths with Irrational Numbers - Physics Forums
The discussion centers on the nature of irrational numbers and their existence in the physical world, questioning how lengths like the hypotenuse of a right triangle can be considered valid when they are infinite and non-repeating Participants argue that real numbers, including irrationals, are abstract mathematical constructs rather than physical entities, with some asserting that perfect
- Proof that Log2 of 5 is irrational - Physics Forums
Homework Statement Prove that log2 of 5 is irrational Homework Equations None The Attempt at a Solution I just had a glimpse of the actual solution
- Irrational Numbers: Expressible as Infinite Summations?
Not all irrational numbers can be expressed as infinite summations, as there are uncountably many irrational numbers but only countably many mathematical expressions While any specific irrational number can be represented by a converging sequence, there exist irrational numbers that cannot be uniquely expressed in this manner due to the limitations of mathematical notation The pigeonhole
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