The Integers

Definition

The integers are a set of numbers that includes all positive whole numbers, negative whole numbers, and zero. This set is denoted by the symbol $$ ext{Z}$$ and is fundamental in number theory and algebra, serving as a foundational building block for various mathematical structures, including integral domains and fields. The integers exhibit properties such as closure under addition and multiplication, and they are the primary examples of integral domains, where multiplication is commutative and there are no zero divisors.

5 Must Know Facts For Your Next Test

1. The integers include the set of positive numbers, negative numbers, and zero, represented as $$ ext{Z} = \\ \ ext{...,-3,-2,-1,0,1,2,3,...}$$.
2. Integers form an integral domain because they satisfy the properties of closure, associativity, commutativity for both addition and multiplication, and they have additive inverses.
3. There are no zero divisors among integers; if the product of two integers is zero, at least one of the integers must be zero.
4. The set of integers is infinite in both the positive and negative directions but is countable, meaning you can list them in a sequence.
5. Integers are used to define more complex algebraic structures such as fields when considered alongside rational numbers or real numbers.

Review Questions

• How do the properties of integers illustrate their role as an integral domain?
  • Integers demonstrate the key properties required for an integral domain by being closed under addition and multiplication, having no zero divisors, and allowing for additive inverses. For example, the sum or product of any two integers is always an integer. Since there are no pairs of non-zero integers whose product equals zero, this reinforces their classification as an integral domain.
• Compare the set of integers with a field, highlighting what makes integers distinct from fields.
  • While the set of integers forms an integral domain due to its properties like closure under operations and no zero divisors, it does not qualify as a field because not every non-zero integer has a multiplicative inverse within the integers. For example, 1/2 is not an integer; hence integers cannot satisfy the field condition where every non-zero element must have an inverse. This distinction is crucial for understanding how different algebraic structures function.
• Evaluate how understanding integers can help in the study of more complex mathematical structures like fields.
  • Understanding integers provides a crucial foundation for studying more complex structures such as fields because it highlights essential algebraic properties like closure, associativity, and the absence of zero divisors. By analyzing how integers operate within an integral domain context, one can appreciate how these properties extend to fields, where additional requirements like multiplicative inverses are introduced. This evaluation helps bridge simple arithmetic concepts to more advanced algebraic ideas.

Related terms

Integral Domain: A commutative ring with no zero divisors where the product of any two non-zero elements is non-zero.

Field: A set equipped with two operations (addition and multiplication) satisfying certain properties, where every non-zero element has a multiplicative inverse.

Zero Divisor: An element in a ring such that when multiplied by another non-zero element results in zero.