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
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)
Irrational Numbers and Real Life I need 6 answers - Physics Forums Irrational numbers are useful within mathematics only, but for that exact reason they are useful in the real world They allow us to develop theories with useful concepts like derivatives, integrals, the various results of analytical geometry, the rules trigonometry etc
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
Can one use an irrational number as a base? - Physics Forums All numbers, including integers, rational numbers, and irrational numbers, are defined independently of any numeration system I kind of get that, but I am missing what a rational number is, I think If I take and lay straight a rope of unit length in the base 1 number line, I can then cut a rope that is the length of the circle enclosing the
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
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
Discrete Math- Irrational numbers, proof or counterexample - Physics Forums If r is any rational number and if s is any irrational number, then r s is irrational Homework Equations A rational number is equal to the ratio of two other numbers An irrational number can't be expressed as the ratio of two other numbers The Attempt at a Solution I said that this statement is false
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