91Ó°ÊÓ

Understanding proof techniques is essential in real analysis, as it allows us to substantiate the truth of mathematical statements. The exercise given asks to show that \(1 /(a b)=(1 / a)(1 / b)\), given that \(a \eq 0\) and \(b \eq 0\). This is a classic example of proof by direct verification, which consists of taking an assumed true statement and, through valid algebraic manipulations, arriving at another true statement.

In this case, the proof starts by examining both sides of the equation individually. By processing each side independently and then demonstrating that their forms are equal, we are able to affirm that \(1 /(a b)\) is indeed equal to \(1 / a)(1 / b)\), under the given constraints. This method is uncomplicated yet powerful, because once the equality is established, it applies universally for all non-zero values of \(a\) and \(b\).

The topic of fraction multiplication is central to many areas of mathematics, including real analysis. Multiplying fractions is a straightforward process: you multiply the numerators (the top numbers) to find the new numerator, and multiply the denominators (the bottom numbers) to find the new denominator.

Using the exercise as an example, when we multiply \(1/a\) by \(1/b\), we multiply the numerators, \(1 \times 1\), to get \(1\), and multiply the denominators, \(a \times b\), to get \(ab\). Thus, \( (1/a)(1/b) = 1/(ab)\), which demonstrates the original claim. It’s critical for students to understand this rule, as it helps in solving equations involving fractions and allows for simplification of rational expressions.

Rational expressions are fractions that involve variables. They resemble ratios between two polynomials. The key to working with rational expressions is to recognize that the rules for fractions apply to them as well. For example, with \(1 / a\) and \(1 / b\), both are rational expressions.

When dealing with rational expressions, it's important to note the non-permissible values for the variables—values that would make the denominator equal to zero. In the provided exercise, the stipulation that \(a \eq 0\) and \(b \eq 0\) ensures that our rational expressions are well-defined. It is crucial when operating with these expressions to simplify when possible and always check the validity of our manipulations against the domain of the variables involved.