Definition: Integers are whole numbers that can be positive, negative, or zero. They include all the counting numbers (1, 2, 3, …), their opposites (−1, −2, −3, …), and zero.
Properties:
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Closure under Addition: If you add two integers, the result is always an integer.
- Example: (2 + (-3) = -1), which is also an integer.
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Closure under Subtraction: If you subtract one integer from another, the result is always an integer.
- Example: (5 - (-2) = 7), which is also an integer.
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Closure under Multiplication: If you multiply two integers, the result is always an integer.
- Example: (4 \times (-3) = -12), which is also an integer.
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Closure under Division: Division of integers may or may not result in an integer.
- Example: (10 \div (-2) = -5), which is an integer. But (5 \div 2 = 2.5), which is not an integer.
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Additive Identity: The sum of any integer and zero is the integer itself.
- Example: (7 + 0 = 7).
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Additive Inverse: Every integer has an additive inverse, such that adding an integer to its additive inverse results in zero.
- Example: The additive inverse of (3) is (-3) because (3 + (-3) = 0).
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Associative Property of Addition: The grouping of integers does not affect their sum.
- Example: ((4 + 5) + 3 = 4 + (5 + 3) = 12).
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Associative Property of Multiplication: The grouping of integers does not affect their product.
- Example: ((2 \times 3) \times 4 = 2 \times (3 \times 4) = 24).
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Commutative Property of Addition: The order of integers does not affect their sum.
- Example: (3 + 6 = 6 + 3 = 9).
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Commutative Property of Multiplication: The order of integers does not affect their product.
- Example: (2 \times 5 = 5 \times 2 = 10).
Understanding these properties of integers is fundamental in arithmetic and algebraic operations involving integers. They help in performing calculations and proving mathematical statements.