Irrational + irrational = rational - Physics Forums Assume that the difference between a rational number and an irrational number is an irrational number The sum of any such two irrational numbers would be a rational number All we have to do now is prove the assumption, which I think is easier edit: I just saw tongos' proposition, which I think is better than mine
Irrational number to an irrational power - Physics Forums But as for the original question, this is a well-known proof that's often cited as a classic non-constructive proof The proposition is that an irrational raised to an irrational power can be rational So we consider ##x = \sqrt{2}^\sqrt{2}## Either ##x## is rational or irrational If it's the former, our work is done
Constructing Lengths with Irrational Numbers - Physics Forums By the way, abstract mathematics is inconsistent in that they treat irrational numbers as being both, infinite decimal expansions, and precise calculus limits In other words, Cantor's famous diagonal proof that the set of real numbers has a larger cardinality than the set of natural numbers depends on the infinite expansion of decimal numbers
Factoring irrational equations - Physics Forums However, if "0" does not appear, then the number in the brackets is not a value of x, so you must choose from the list of numbers (the ±1, ±5 3, ±1 3, ±5) However, after going through every possible number in that list, none ever got to "0" which means that the equation does not have any rational numbers (so the answer must be irrational)
Square Root of an Irrational Number is Irrational - Physics Forums By contraposition, if a is irrational then √a is irrational QED There's no need to look at the inverse of "if a is irrational then √a is irrational" and find a contradiction because the two statements "if √a is rational then a is rational" and "if a is irrational then √a is irrational" are one and the same, just worded differently
Proving Root n is Irrational: Perfect Square Affects Proof - Physics Forums n divides q^2, therefore n divides q Now n is a common factor for p q But we know that p q are co-prime Hence our assumption is wrong, root n is irrational 2) if n is a composite Let's say it is product of two primes c1 c2 The proof remains the same for any number of primes
Proving Irrationals Are Dense in the Reals - Physics Forums Some previous results that I'm using is that a rational plus an irrational is irrational, and that a rational multiplied by an irrational is also irrational So since I knew that for any r that is an element of Q, and x, y that are elements of R, then x < r < y Or that the rational numbers are dense in R So consider an irrational number v
Help with 2 Proofs: Prove Show Uniqueness - Physics Forums Show that if r is an irrational number, there is a unique integer n such that the distance between r and n is less than 1 2 Be sure to argue the uniqueness of n Homework Equations None that I know of The Attempt at a Solution I have absolutely no idea on where to even start for this problem
Is the Square Root of Pi Irrational? - Physics Forums A question in my book says to prove that pi is irrational, I found a proof which I'm happy with and found a similar one on the web however on the solutions they have done: assume √π is rational i e [tex] \sqrt{\pi} = \frac{p}{q} p,q \in \mathbb{Z} [ tex] [tex] \pi = \frac{p^2}{q^2}, p^2,q^2 \in \mathbb{Z} ∴ \pi \mathrm{is\ rational} [ tex]
Rational sequence converging to irrational - Physics Forums "for any irrational number there exists a sequence of rational numbers that converges to it", and it doesn't have a proof for it, just saying that it is a consequence of the fact that ##\mathbb{Q}## is dense in ##\mathbb R## Any ideas how to proceed for a rigorous proof?