A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not equal to 0.
We can use proof by contradiction to show that the square root of 2 is irrational. Assume that the square root of 2 is rational, then it can be expressed as a fraction a/b, where a and b are integers with no common factors. Squaring both sides gives us 2 = a^2/b^2, which means that a^2 = 2b^2. This implies that a^2 is even, and therefore a must also be even (since the square of an odd number is always odd). This means that a can be expressed as 2c, where c is an integer. Substituting this into our equation gives us 2 = 4c^2/b^2, which simplifies to 1 = 2c^2/b^2. This means that b^2 must also be even, and therefore b must also be even. However, this contradicts our original assumption that a and b have no common factors, since they are both even. Therefore, our initial assumption is false, and the square root of 2 must be irrational.
The process for proving that a number is irrational varies depending on the specific number. However, in most cases, proof by contradiction is used. This involves assuming that the number is rational and arriving at a contradiction, which proves that the number must be irrational.
No, a number cannot be both rational and irrational. A number is either rational or irrational, based on whether it can be expressed as a ratio of two integers or not.
Rational and irrational numbers are both subsets of real numbers. Real numbers include all rational and irrational numbers, as well as integers, fractions, and decimals. Rational numbers can be expressed as terminating or repeating decimals, while irrational numbers cannot be expressed in this way.