Additive Identity and Other PropertiesDate: 01/03/99 at 19:50:17 From: Elizabeth Subject: Properties I don't understand properties. Can you explain the identity of zero, the commutative property, and the multiplicative properties? Date: 01/04/99 at 12:40:21 From: Doctor Peterson Subject: Re: Properties Hi, Elizabeth. I'm not sure which multiplicative properties you mean; we talk about the multiplicative property of equality, about the commutative and associative properties of multiplication, and about the distributive property of multiplication over addition, and you might mean any of those. I hope my explanation of the others will help you. Write back if you need more. We call zero the additive identity because when you add it to anything, the result is identical to the original number - you haven't changed it. This means that n + 0 = 0 + n = n for any number n There's no other number for which you can say this, so zero is very special. Similarly, there is a multiplicative identity. It is a number you can multiply anything by and not change it. What is that? It will be the number X in this statement (where "*" means multiply): n * X = X * n = n for any number n. The commutative property means that it doesn't matter which order you do things in. For addition, this means that a + b = b + a for any numbers a and b. You can do the same thing with multiplication. These are things you've probably understood for years, but haven't thought of as "properties." You just knew that 5+6 and 6+5 are the same. They are important because not everything you can do has an identity, or is commutative. Suppose we picture operations like addition and multiplication as machines with two input pipes and one output pipe: a b in in | | +-+-+-+ | | | + | | | +--+--+ | out a+b Commutativity says that if you switch the two input hoses, the same thing will come out. That's true for addition. It's also true if the machine is a paint mixer that takes blue and yellow paint, or yellow and blue paint, and makes green. But it's not true for every machine you can imagine. Maybe the left pipe has to take water and the right air. It probably wouldn't work right if you switched them! When we list properties like these, we're reminding ourselves of all the ways in which addition "works nicely," all the things we can depend on when we work with numbers. That's what a property is. You might want to look through our archives. You can search for words like "commutative" and find lots of helpful explanations like this: Meanings of Properties http://mathforum.org/dr.math/problems/colleen7.23.97.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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