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- f equals are added to equals, then the sums are equal according to the additive axiom.
- If a = b and c = d, then a + c = b + d
- If a = 3 and a = b, then b = 3. If c = 5 and c = d, then d = 5. Therefore,
- if a = b, that means 3 = 3 and if c = d, that means 5 = 5, so
- a + c = b + d in this case is 3 + 5 = 3 + 5
- multiplicative axiom states that if a = b and c = d, then ac = bd. In
- other words, if equals are multiplied by equals, the answer should be
- If a = b and c = d, then ac = bd
- Let's look at an example.
- If a = 6 and a = b, then b = 6. And if c = 8 and c = d, then d = 8.
- Therefore, we can say a = 6 and c = 8, so ac = (6)(8) or 6c = (6)(8).
- he commutative property tells us that the order in which two numbers are added does not affect their sum.
- a + b = b + a
- We can add 3 + 2 to get 5 or we can add 2 + 3 and get 5.
- The commutative property also works with multiplication.
- ab = ba
- We can multiply 3 x 15 and get 45 or we can multiply 15 x 3 and get 45.
- he associative property extends this by saying that the grouping of numbers under addition does not affect their sum.
- (a + b) + c = a + (b + c)
- For example, (8 + 2) + 5 = 8 + (2 + 5)
- Which gives 10 + 5 = 8 + 7 (both of which equal 15)
- we have a string of numbers such as 4 + 6 + 3 + 4 + 2 + 1, we could add
- them in order: 4 + 6 = 10 + 3 = 13 + 4 = 17 + 2 = 19 + 1 = 20.
- Or we could quickly see that 4 + 6 is 10 and 3 + 2 is 5 and 4 + 1 is 5 and then add 10 + 5 + 5 to get 20.
- distributive property of multiplication over addition tells us that the
- product of a number and the sum of two numbers is the same as the sum
- of the products of the first number and each of the others.
- a(b + c) = ab +ac
- For example, 5(3 + 8) = 5(3) + 5(8)
- In other words, 5(11) = 15 + 40
- Which is 55 = 55
- shows us that we can distribute the multiplier that is outside the
- parenthesis to each of the numbers inside the parenthesis.