- #1
kmeado07
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Homework Statement
Prove that if x^2 is irrational then x must be irrational.
Homework Equations
The Attempt at a Solution
Maybe do proof by contradiction. I'm not really sure where to start.
kmeado07 said:so i let x= a/b
then obviously x^2 = a^2/b^2
im not sure how to continue to reach the contradiction
Irrational numbers are numbers that cannot be expressed as a ratio of two integers. They are non-terminating and non-repeating decimals. Some examples of irrational numbers are pi (3.14159...), e (2.71828...), and the square root of 2 (1.41421...).
There are various methods to prove that a number is irrational. One way is to show that the number cannot be written as a ratio of two integers. Another way is to use the proof by contradiction method, assuming that the number can be written as a ratio and then showing that it leads to a contradiction.
Yes, irrational numbers can be written in decimal form but they will be non-terminating and non-repeating. This means that the decimal representation will go on forever without repeating the same sequence of digits.
No, not all square roots are irrational numbers. For example, the square root of 4 is 2 which is a rational number. However, the square root of non-perfect squares (numbers that are not perfect squares) are irrational numbers.
The main difference between rational and irrational numbers is that rational numbers can be written as a ratio of two integers, while irrational numbers cannot. Rational numbers also have a finite or repeating decimal representation, while irrational numbers have a non-terminating and non-repeating decimal representation.