Dividing an irrational number by a nonzero rational number results in an irrational number. The proof by contradiction assumes that the result is rational, leading to the conclusion that the irrational number must also be rational, which contradicts the initial premise.
I have a question. I realize that two rational numbers added together equal a rational number and that a rational added to a irrational equal a irrational number; but how do I show what a irrational plus a irrational equal?
The discussion centers on the nature of irrational and transcendental numbers, specifically whether they can have repeating decimal expansions. It is established that numbers like pi, e, and √2 do not repeat, as they are proven to be irrational, meaning they cannot be expressed as fractions of integers. The argument is made that if a number were to repeat, it would have to be rational, which ...
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 ...
The discussion centers on proving or disproving the existence of a rational number x and an irrational number y such that x^y is irrational. Participants suggest that proof by contradiction is a common method, particularly for disproving existence. However, proving existence may be more straightforward through construction, especially if one can find a specific example. The Gelfond-Schneider ...
Alright, heading says it all. This is a nice problem heh.. I can see how to prove sqrt(5) is irrational. I think this method works up to the points where the fact 5 is a prime is used, (ie prime lemma) on 5 which doesn't work so well on 6! hehe Was thinking of maybe using product of primes...
As another illustration, I understand that if a sequence (s_n) given as s_n = \sqrt {2} + 1/n is a sequence of IRRATIONAL numbers converging to an IRRATIONAL limit.
The discussion centers on the relationship between irrational numbers and the measurement of physical quantities. It highlights that while irrational numbers, such as the square root of 2, exist mathematically, they cannot be precisely measured in practice due to the limitations of measurement tools. Participants note that measurements are approximations and often require rounding, which ...
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?
The Greeks recognized the existence of irrational numbers, such as √2, but struggled with their implications due to their philosophical beliefs that all numbers should be expressible as ratios of whole numbers. They often approximated irrational numbers for practical applications, using methods like the Babylonian method and the Euclidean algorithm to find approximations. Euclid's Elements ...
