Basic Real Number PropertiesDate: 07/31/97 at 06:21:26 From: Tuesday Subject: Basic Real Number Properties Hi! Dr. Math, I'm a grade seven student here in the Philippines, and we have this research paper and I really need more material about Basic Real Number Properties: associative, commutative, closure, identity, inverse, distributive. Thank you, Tuesday Date: 08/01/97 at 18:41:29 From: Doctor Tom Subject: Re: Basic Real Number Properties Hello Tuesday, To talk about these properties, you need to think not only about the numbers themselves, but the operations you perform on them. Let's just start with + (addition) and x (multiplication). The operations + and x are associative. That means that if you want to figure out what 3+4+5 is, you can start by adding 3 and 4 and then add 5 to that, or you can add 4 and 5 first, and then add 3 to that. It's usually written like this: (3+4)+5 = 3+(4+5) The parentheses tell you what to do first. This is true of all real numbers - there is nothing special about 3, 4, and 5, so you often see it written: (a+b)+c = a+(b+c) where a, b, and c are any real numbers. If you replace "+" by "x", exactly the same thing is true; you can multiply in any order. The other way to see why it is important to recognize that this is a property is to look for operations that are NOT associative. Is - (subtraction) associative? Let's check: is (10 - 9) - 1 = 10 - (9 - 1) ? No. On the left, we get 0 and on the right, we get 2. Is division associative? Check some examples. Think about the other properties in the same way. + and x are commutative. That means a+b = b+a and a*b = b*a. Subtraction and division are not commutative. Closure means that if you add any two real numbers you'll get a real number. Same if you multiply two numbers. Subtraction is closed, but division is not. You cannot divide by zero. The identity for addition is a number that can be added to any other number and not change the other number. So zero is the additive identity. Add zero to anything and it doesn't change. Similarly, 1 is the multiplicative identity. Multiply anything by 1 and it doesn't change. The additive inverse of a number is something you can add to the original number to get the additive identity. Additive inverses always exist. The inverse of 4 is -4 since if you add 4 and -4, you get 0, the additive identity. All numbers but 0 have a multiplicative inverse. The inverse of 7 is 1/7; the inverse of 92 is 1/92. But 0 has no inverse because you can't multiply anything by zero and get 1 - you always get zero. Finally, the distributive law shows the interaction between addition and multiplication. It states that: ax(b+c) = axb + axc. In other words, you can either add the b and c first, before multiplying, or you can multiply a by each of b and c, and add the results, and the final answer will be the same. The distributive law holds for any three real numbers a, b, and c. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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