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Questions and Answers
What is the result of adding two integers with the same sign?
What is the distributive property of multiplication over addition?
What is the additive inverse of 5?
What is the result of multiplying two integers with different signs?
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What is the associative property of addition?
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What is the result of subtracting two integers with the same sign?
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What is the commutative property of multiplication?
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What is the result of dividing two integers with the same sign?
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What is the subtraction of integers defined as?
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Study Notes
Operations on Integers
Addition
- Commutative property:
a + b = b + a - Associative property:
(a + b) + c = a + (b + c) - Distributive property:
a + (b + c) = a + b + a + c - Additive inverse:
a + (-a) = 0 - Examples:
-
2 + 3 = 5 -
(-2) + 3 = 1
-
Subtraction
- Defined as addition of the additive inverse:
a - b = a + (-b) - Not commutative:
a - b ≠b - a - Not associative:
(a - b) - c ≠a - (b - c) - Examples:
-
5 - 2 = 3 -
3 - (-2) = 5
-
Multiplication
- Commutative property:
a × b = b × a - Associative property:
(a × b) × c = a × (b × c) - Distributive property:
a × (b + c) = a × b + a × c - Multiplicative identity:
a × 1 = a - Examples:
-
2 × 3 = 6 -
(-2) × 3 = -6
-
Division
- Defined as the inverse operation of multiplication:
a ÷ b = a × (1/b) - Not commutative:
a ÷ b ≠b ÷ a - Not associative:
(a ÷ b) ÷ c ≠a ÷ (b ÷ c) - Examples:
-
6 ÷ 2 = 3 -
(-6) ÷ 2 = -3
-
Rules for Operations on Integers
- When multiplying or dividing two integers with the same sign, the result is always positive.
- When multiplying or dividing two integers with different signs, the result is always negative.
- When adding or subtracting two integers with the same sign, the result has the same sign.
- When adding or subtracting two integers with different signs, the result has the sign of the integer with the larger absolute value.
Operations on Integers
Addition
- Commutative property holds:
a + b = b + a - Associative property holds:
(a + b) + c = a + (b + c) - Distributive property holds:
a + (b + c) = a + b + a + c - Additive inverse exists:
a + (-a) = 0 - Examples of addition:
2 + 3 = 5,(-2) + 3 = 1
Subtraction
- Defined as addition of the additive inverse:
a - b = a + (-b) - Not commutative:
a - b ≠b - a - Not associative:
(a - b) - c ≠a - (b - c) - Examples of subtraction:
5 - 2 = 3,3 - (-2) = 5
Multiplication
- Commutative property holds:
a × b = b × a - Associative property holds:
(a × b) × c = a × (b × c) - Distributive property holds:
a × (b + c) = a × b + a × c - Multiplicative identity exists:
a × 1 = a - Examples of multiplication:
2 × 3 = 6,(-2) × 3 = -6
Division
- Defined as the inverse operation of multiplication:
a ÷ b = a × (1/b) - Not commutative:
a ÷ b ≠b ÷ a - Not associative:
(a ÷ b) ÷ c ≠a ÷ (b ÷ c) - Examples of division:
6 ÷ 2 = 3,(-6) ÷ 2 = -3
Rules for Operations on Integers
- Same sign → positive result in multiplication and division
- Different signs → negative result in multiplication and division
- Same sign → same sign result in addition and subtraction
- Different signs → sign of the larger absolute value result in addition and subtraction
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Description
Understand the properties and rules of addition and subtraction on integers, including commutative, associative, and distributive properties.
