Properties of Real NumbersDate: 05/20/97 at 19:46:14 From: Sarah Mahlstedt Subject: Properties of real numbers I have a problem that is all about properties of real numbers. I don't get it at all! Help explain what they are! Particularly, what are associative properties, commutative properties, and the zero product property? Date: 05/22/97 at 19:59:18 From: Doctor Reno Subject: Re: Properties of real numbers Hi Sarah! These properties are simply fancy names for mathematical ideas that you probably already know very well. Let's look at them one at a time to learn more. ASSOCIATIVE PROPERTY There are two associative properties, actually, but don't let that upset you! One is the associative property of addition, and the other is the associative property of multiplication. These are sometimes also called "grouping properties," and you will see why soon. Let's say that you want to add 3 numbers: 25, 75, and 30. We can add these up two different ways. We can add the first two: 25 + 75 = 100. Then we can add that sum (100) to 30 and have a new sum of 130 (100 + 30 = 130). But we could also add them in this way: 75 + 30 = 105, and then 105 + 25 = 130. The answer is the same. All that the associative property of addition says is that we will always get the same answer no matter how we "group" numbers when we add them! That's it! But we knew that, didn't we? It seems obvious. Sometimes mathematics is like that...it gives big, fancy names to things that seem easy and that we already know. In the fancy language of mathematics, the associative property of addition says that if a, b, and c are any whole numbers, then (a + b) + c = a + (b + c). We have shown that to be true in my example above. The associative property of multiplication is written in the same fancy mathematical language: For any whole numbers a, b, and c, (a x b) x c = a x (b x c). Once again, we all know and understand this property. If we want to multiply three numbers together....let's say 3 x 5 x 8.....we can do this two ways: 3 x 5 = 15 15 x 8 = 120...which is to say that 3 x 5 x 8 = 120. OR... 5 x 8 = 40 40 x 3 = 120...which is to say that 3 x 5 x 8 = 120! The same answer! And that's all there is to the associative properties! We use these properties regularly to solve equations and arithmetic problems, but they are so familiar to us that we may not even realize that we use them! COMMUTATIVE PROPERTY The commutative properties are more fancy names for stuff you already know about math! And once again, there are two of them..one for addition and one for multiplication. These are the easier properties to remember. They simply say that if 2 + 5 = 7, then 5 + 2 = 7; and if 2 x 5 = 10, then 5 x 2 = 10. That's it! The fancy words go like this: If a and b are any whole numbers, then a + b = b + a. For multiplication, it says: For any whole numbers a and b, a x b = b x a. That seems pretty obvious, doesn't it? ZERO MULTIPLICATION PROPERTY OF WHOLE NUMBERS This property is the easiest one of all! All it means is that whenever you multiply a number by zero, you get zero for an answer! That's all! Of course, we have to say it differently in math...For any whole number a, a x 0 = 0 = 0 x a. That's all there is, Sarah! Relax and enjoy your math, and remember that we have to write these simple ideas in this fancy language in order to be exact and precise. Thank you for writing to Dr. Math, and be sure to write again if you have any further problems! -Doctor Reno, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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