Properties of AlgebraDate: 5/8/96 at 19:2:31 From: Justin March Subject: Properties of Algebra What are the properties used for algebra? What about the "distributive property"? Date: 10/16/96 at 19:16:47 From: Doctor Lani Subject: Re: Properties of Algebra Dear Justin, When we're working with real numbers (integers, fractions, decimals, roots, and numbers like Pi), we like to say that certain things will always be true. When we find these things, we give them the name "property." The example you give, the distributive property, is something that works for *any* set of real numbers. Because of this, properties are often stated with variables, which say that you can put any number in and that I'll use a, b, and c in this answer. Now, for some examples: Some properties true for multiplication and addition: THE COMMUTATIVE PROPERTY: a + b = b + a a * b = b * a Basically, this says that you can switch the numbers that you are adding or multiplying and your answer stays the same: 2 + 3 = 3 + 2; 57 * 22.3 = 22.3 *57 THE ASSOCIATIVE PROPERTY: (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Since parentheses mean "do this first," this property just tells you that you can add or multiply in any order without changing your answer: (2 + 3) + 6 2 + (3 + 6) \ / \ / = 5 + 6 = 2 + 9 = 11 = 11 Sometimes students mix up these names. I just think of the words: "commute" means going from one place to another and "associate" means hanging out with. Numbers commute when they switch places and they associate in parentheses. Also, students sometimes think that properties are silly or obvious. Well, test these properties with subtraction and division! Now, finally, I'll explain distribution. The official name is: DISTRIBUTION OF MULTIPLICATION OVER ADDITION: a * (b + c) = a*b + a*c This one is best explained by a couple of examples. (1) Have you ever had to multiply something like 4 * 52 in your head? If so, you might have used the distributive property! Like this: 4 * 52 = 4 * (50 + 2) = 4 * 50 + 4 * 2 = 200 + 8 = 208 It's easier to multiply 4 by 50 and 4 by 2 and then just add the answers. If you know that trick, you know the distributive property! (2) Why does it work? Well, let's look at another example: (a) 3 *(2 + 5) = (2 + 5) + (2 + 5) + (2 + 5) This is true because multiplying by 3 means adding the number three times. (b) (2 + 5) + (2 + 5) + (2 + 5) = 2 + 5 + 2 + 5 + 2 + 5 This works because the associative property tells me it doesn't matter which order I add in, so I don't need the parentheses. (c) 2 + 5 + 2 + 5 + 2 + 5 = 2 + 2 + 2 + 5 + 5 + 5 This is true because the commutative property tells me I can move stuff around without changing the answer. (d) 2 + 2 + 2 + 5 + 5 + 5 = 3*2 + 3*5 This is true because adding a number three times is the same as multiplying by 3. If you look at the first line of (a) and the last line of (d) you'll see that 3*(2 + 5) = 3*2 + 3*5. This is called a proof. It should convince you that the distributive property works! There are a lot more properties (identity, reflexive, distribution of multiplication over subtraction...). You can find a good list in any algebra book. Good luck! -Dr. Lani The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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