Properties of Addition and Multiplication

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Properties of Addition and Multiplication

• Commutative
• a b b a
• a ? b b ? a
• Associative
• a (b c) (a b) c
• a ? (b ? c) (a ? b) ? c

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• Identity Element
• a 0 a 0 a
• a ? 1 a 1 ? a
• Distributive
• a ? (b c) (a ? b) (a ? c)
• a ? (b ? c) (a ? b) ? (a ? c)
• Multiplication by zero
• a ? 0 0 a ? 0

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• Closure
• If a and b are any two whole numbers, then
• a b (or a ? b) is a unique whole number.
• Examples of closure
• Is the following set closed under addition?
• 0, 1, 2, 3,
• Yes, because any two numbers we add together
  gives us another number in the set.

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• 2. Is the following set closed under addition?
• 0, 1, 2, 3
• No, because 2 3 5 and 5 is not in the set.
• 3. Is the following set closed under addition?
• 0,1
• No, because 1 1 2 and 2 is not in the set.

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• 4. Is the following set closed under
  multiplication?
• 2, 4, 6, 8,
• Yes, any two even numbers we multiply together
  gives us another number in the set.
• 5. Is the following set closed under
  multiplication?
• 0,1
• Yes
• 0 ? 0 0 1 ? 1 1 0 ? 1 0 1 ? 0 0

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The Division Algorithm

• a q ? b r 0 ? r lt b
• q quotient
• r remainder
• a dividend
• b divisor
• Note a ? b q (r/b)

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Models of Addition

• Union of Two Disjoint Sets
• Measurement Model

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Models of Subtraction

• Take-Away Set Model
• Missing Addend Model
• Comparison Model
• Measurement Model

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Models of Multiplication

• Repeated Addition
• Array Model ( Row by Column Model)
• Cartesian Product Model

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Models of Division

• Repeated Subtraction ( Division by Grouping)
• Partition Model ( Division by Sharing)
• Missing Factor Model

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Exponents

• a1 a
• am a ? a ? ? a (m factors)
• a0 1
• am ? bm (a ? b)m
• an ? am a(n m)
• (am)n a(m ? n)
• am / an a(m?n)
• am / bm (a/b)m