Integers: Definition and Examples

Definition:
Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. They are represented as:

$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$

• Positive integers: $1, 2, 3, \dots$
• Negative integers: $-1, -2, -3, \dots$
• Zero ($0$) is a special integer that is neither positive nor negative.

Properties of Integers:

1. Closure: The sum, difference, and product of two integers are always integers.
2. Associativity: Addition and multiplication of integers are associative.
   $(a + b) + c = a + (b + c)$
   $(a \times b) \times c = a \times (b \times c)$
3. Commutativity: Addition and multiplication of integers are commutative.
   $a + b = b + a, \quad a \times b = b \times a$
4. Identity Element:
   • Additive identity: $0$ because $a + 0 = a$
   • Multiplicative identity: $1$ because $a \times 1 = a$
5. Inverse Elements: For every integer $a$, there exists $-a$ such that $a + (-a) = 0$.

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Examples of Operations on Integers:

1. Addition: $-5 + 7 = 2$
2. Subtraction: $6 - (-3) = 6 + 3 = 9$
3. Multiplication: $(-4) \times 3 = -12$
4. Division: Integer division may not yield an integer. For instance, $7 \div 3 = 2.33$, not an integer.

Would you like detailed proofs for any of these properties or more examples?

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Related Questions:

1. What are the differences between integers and natural numbers?
2. How do integers behave under division?
3. Can integers be irrational numbers?
4. What role does zero play in integer operations?
5. What are practical uses of integers in real-life problems?

Tip: Always remember that negative integers are ordered such that $-5 < -3$ even though $5 > 3$.