To determine if a set of numbers forms a field, you need to check if it satisfies the two main properties of a field: closure under addition and multiplication, and the existence of additive and multiplicative inverses for every element. Additionally, you should also check if the field follows the distributive property.
Closure under addition and multiplication means that when you add or multiply any two elements in the set, the result will also be in the set. This property is important because it ensures that the set is closed and the operations will always give a valid result.
To prove the existence of additive and multiplicative inverses in a field, you need to show that for every element in the set, there exists an element that when added or multiplied with it gives the identity element (0 for addition, 1 for multiplication). This can be done through algebraic manipulation and solving for the inverse element.
No, the distributive property is one of the defining properties of a field. If a set of numbers does not follow this property, then it cannot be considered a field. It is important to check for the distributive property when proving that a set of numbers forms a field.
No, not all sets of numbers form fields. A set must satisfy all the properties of a field to be considered a field. For example, the set of natural numbers (1, 2, 3, etc.) is not a field because it does not have additive and multiplicative inverses for all elements. Only certain sets, such as the real numbers, rational numbers, and complex numbers, form fields.