A rational number is any number that can be expressed as a ratio of two integers (numbers without decimals or fractions), such as 1/2, 3/4, or 5/7.
The statement "x is a rational number" is a conditional statement, meaning it is only true if the condition (x ≠ 0) is also true. This condition states that x cannot equal 0 in order for the statement to hold true.
This statement means that the trigonometric function tangent of x is not a rational number. In other words, it cannot be expressed as a ratio of two integers.
We can prove this using a proof by contradiction. Assume that x is a rational number and x ≠ 0, and that tan(x) is a rational number. This would mean that tan(x) can be expressed as a ratio of two integers. However, we can show that this is not possible by using the fact that tan(x) = sin(x)/cos(x), and using the trigonometric identities for sin(x) and cos(x). This leads to a contradiction, proving that our assumption was false and therefore, tan(x) is not rational.
Yes, one example is x = π/4. This is a rational number (3.14/4), but tan(π/4) = 1, which is not a rational number.