Final answer:
In the set of rational numbers Q, the well-defined operations include addition, subtraction, multiplication, and division, except by zero. To perform these operations, especially addition and subtraction, a common denominator is necessary, while multiplication and division follow their respective standard rules.
Step-by-step explanation:
The operations on numbers that are well defined in the set of rational numbers, denoted as Q, include addition, subtraction, multiplication, and division (excluding division by zero). Rational numbers are any numbers that can be expressed as the quotient or fraction ⅛ of two integers, with the denominator not equal to zero.
When dealing with rational numbers, the basic principle to use in working with addition and subtraction are finding a common denominator before combining the numerators. For multiplication, the general rule is to multiply the numerators together to find the new numerator and the denominators together to find the new denominator. In division, we multiply by the reciprocal of the divisor.
Understanding these operations with rational numbers relies on the principles of equality and the properties of arithmetic operations. For instance, any fraction with the same non-zero number in the numerator and the denominator is equivalent to 1. Furthermore, while dealing with powers, rational numbers behave as per the usual rules of exponents. Even when the powers are not integers, it is mathematically defined, for example, the notion of rational exponents like 3¹.⁷. These concepts extend beyond intuitively understanding fractions and follow specific algebraic rules allowing for all kinds of calculations within Q.