- #1
bgwyh_88
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I came across this question. How do you show that √N is irrational when N is a nonsquare integer?
Cheers.
Cheers.
A number is considered irrational if it cannot be expressed as a ratio of two integers. This means that the number cannot be written as a fraction in the form of a/b, where a and b are integers.
To prove that the square root of a number, let's say n, is irrational, we can use a proof by contradiction. This means we assume that the square root of n is rational, and then show that this leads to a contradiction, thus proving our initial assumption wrong.
Yes, an example of a number whose square root is irrational is 2. The square root of 2 cannot be expressed as a fraction and has a decimal representation that goes on infinitely without repeating.
Proving the irrationality of the square root of a number is important because it helps us understand the nature of numbers and their properties. It also has applications in various fields of mathematics, such as number theory and geometry.
Yes, there are other methods to prove the irrationality of the square root of a number, such as using the fundamental theorem of arithmetic or the rational root theorem. However, the proof by contradiction method is the most commonly used and straightforward method for proving the irrationality of the square root of a number.