Final answer:
A subfield F of the real numbers is defined by having closure, inverses, and distributing properties, not by the nature of its elements. These properties ensure the mathematical integrity necessary for a set to be considered a field within the real numbers.
Step-by-step explanation:
In the realm of real numbers, a subfield F is not defined by consisting of only irrational numbers, nor by containing only integers and their multiples, nor by containing finite decimal representations. Instead, a subfield F, such as the field of rational numbers, is defined by satisfying closure, having inverses for addition and multiplication, and adhering to the distributive properties. These defining characteristics ensure that the set is mathematically rich enough to be considered a field within the real numbers.
For instance, if we take a complex number A with real and complex parts (a + ib), where a and b are real constants, when we multiply it by its conjugate A* (a - ib), we obtain A* A = (a + ib) (a - ib) = a² + b². In this multiplication, the complex numbers effectively cancel out, leaving us with a sum of squares which are both real numbers, elucidating the closure property in the field of complex numbers as well.
Fractions also demonstrate the rules of a field, where you have addition, subtraction, multiplication, and division, except by zero, with the need for common denominators highlighting the concepts of closure and the existence of inverses in addition.
It is important to recognize that certain mathematical truths, such as axioms and properties, though universally true within mathematics, may not apply to other disciplines like chemistry or social science, as highlighted in a critique by LibreTexts™.
When calculating areas and volumes, understanding the underlying principles rather than just memorizing formulas is essential to bring order to the process. Mathematical concepts, like integer powers and fractions, are well defined and allow for calculations beyond simple multiplication, as seen with powers like 3¹.⁷, which cannot be written out as a simple repeated multiplication but can still be computed.