Final answer:
The proposition states that if the square of a real number is irrational, then the number itself must be irrational. To understand this, we must acknowledge that the square of a rational number is also rational. Therefore, if the square is not rational (irrational), the original number can't have been rational in the first place.
Step-by-step explanation:
The proposition in question is: For any real number r, if r2 is irrational, then r is irrational. This proposition is a statement about real numbers and their properties related to rationality and irrationality.
To understand this proposition, let's define what irrational numbers are. Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. This means they cannot be written as a simple fraction, and their decimal representations are non-repeating and non-terminating.
The proposition implies a conditional statement. If the square of a real number (r2) is determined to be irrational, then the original number 'r' itself must also be irrational. Let's break this down:
- If r were a rational number, it could be expressed as a ratio of two integers, let's say a/b where a and b are integers with no common factors other than 1, and b is not zero.
- When we square a rational number (in this case, (a/b)2 = a2/b2), the result is also a ratio of two integers, which makes it a rational number.
- The contrapositive of the original statement is also true: If r is rational, then r2 is rational. Therefore, if r2 is irrational, r cannot be rational; it must be irrational.
Hence, the proposition is a logical statement about the relationship between a number and its square concerning rationality.