Final answer:
To verify if two expressions are equivalent, one can test different values for each expression. Approximations indicate similarity but not exact equivalence. Understanding when to use exact or approximate values is crucial in mathematics.
Step-by-step explanation:
To determine whether two expressions are equivalent, one can test these expressions using different values and see if they produce the same results. For example, if you were to consider the expression given as Example A.1.1, we are looking at an approximation (π ~ √10) rather than an equality. As the example suggests, while a calculator may provide precise values, the symbol ~ denotes a similarity, indicating that the values are close, but not necessarily equal.
Another fundamental concept is that if you perform the same operation on both sides of the equals sign, as mentioned in the statement, the expression should retain its equality. However, when it comes to approximations or the concept of equivalence, it’s not just about operations on an equation, but also about the inherent properties of the numbers or expressions in question.
A critical part of understanding mathematics is recognizing when approximations are suitable and when exact values are required. Engaging with skeptical hypotheses can build stronger intuition for these scenarios, as the example regarding Moore's argument about the existence of hands suggests. Such exercises can deepen one's grasp of mathematical logic and reasoning.