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I am not quite sure what you mean with "derive" - what sort of derivation you will accept. If you take the square root of an integer, unless the integer happens to be a perfect square, you get an irrational number. And yes, there is proof of that. The can be found in most high school algebra books.
What else can I help you with?
Are there more rational than irrational numbers?
The answer requires a bit of mathematics, but goes like this:The product of any 2 rational numbers is a rational number.The product of any 2 irrational number is an irrational number.The product of a rational and an irrational number is an irrational number!Therefore simple logic tells us that there are more irrational numbers than rational numbers. There is a way to structure this mathematically, and I believe it is called an "Inductive Proof".Interesting !I'm going to say "No".I reason thusly:-- For every rational number 'N', you can multiply or divide it by 'e', add it to 'e',or subtract it from 'e', and the result is irrational.-- You can multiply or divide it by (pi), add it to (pi), or subtract it from (pi),and the result is irrational.-- You can take its square root, and more times than not, its square root is irrational.There may be others that didn't occur to me just now. But even if there aren't,here are a bunch of irrational numbers that you can make from every rational one.This leads me to believe that there are more irrational numbers than rational ones.-------------------------------------------------------------------------------------------------------There are infinitely many more irrationals than rationals; this was proved by G. Cantor (born 1845, died 1918). His proof is basically:The rational numbers can be listed by assigning to each of the counting numbers (1, 2, 3,...) one of the rational numbers in such a way that every rational number is assigned to at least one counting number;If it is assumed that every irrational number can be assigned to at least one counting numbers (like the rationals), then with such a list it is possible to find an irrational number that is not on the list; so is it not possible as there are more irrationals than there are counting numbers, which has shown to be the same size as the rational numbers, thus showing that there are more irrationals than rationals.
Define and give example of irrational numbers?
an irrational number is any real number that cannot be expressed as a ratio a/b, where a and bare integers, with b nonzero, and is therefore not a rational number.Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.[1]When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable By Paul Philip S. Panis
Is there a proof to show that if you multiply an irrational number times an irrational number you get a rational number?
There cannot be a proof since your assertion is not necessarily true. sqrt(2)*sqrt(3) = sqrt(6). All three are irrational numbers.
What is the history of irrational numbers?
There are uncountably infinite irrational numbers and their existence has been known for over 2500 years. In some cases, although their irrationality was suspected, rigorous mathematical proof took longer. Some notable events:sqrt(2): Pythagoreans (6th Century BCE).pi: Johann Heinrich Lambert (18th Century).e: Leonhard Euler (18th Century).In the late 19th Century, Georg Cantor proved that the number of irrationals is an order of infinity greater than the number of rationals.
Why is 3 square root of 2 is irrational?
sqrt(2) is irrational. 3 is rational. The product of an irrational and a non-zero rational is irrational. A more fundamental proof would follow the lines of the proof that sqrt(2) is irrational.
