We can try to understand what the real numbers are by first understanding how they behave. Abstract algebra is an area of mathematics that looks at sets and operations like addition or multiplication defined on these sets to see what the properties of these systems are. For the real numbers, we believe the following statements to be true and have worked with these properties for years. We will consider these to be axioms and define a field to be a set together with operations called addition and multiplication for which the following statements are true. We shall denote the additive inverse of x as - x . 6.
Multiplication is both associative and commutative.
There is a field element called the multiplicative identity and denoted as 1, such that for every x in the field. Denote this element y as x -1 . For any x , y , z in the field, . We say that multiplication distributes over addition. Show that in a field the additive inverse is unique.