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Properties of Real Numbers - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When analyzing data or solving problems with real numbers, it can be helpful to understand the properties of real numbers. These properties of real numbers, including the Associative, Commutative, Multiplicative and Additive Identity, Multiplicative and Additive Inverse, and Distributive Properties, can be used not only in proofs, but in understanding how to manipulate and solve equations.

We're now going to talk about a number of properties that we can use with real numbers and these properties can be used when we are dealing with an operation on the same level. So what I mean by that is like strictly addition, strictly multiplication. Okay? And, all these properties tend to have two definitions, we have one for addition, and one for multiplication.
So first let's start with a associative and basically what associative is saying is if you have a string of operations that are all the same, your order doesn't matter. So what we could do is, if we're dealing with 1+2+3 we could add 1+2 first, and then add 3 so we'd get 1+2 is 3 plus 3 would be 6, or we could do 2+3 first which would be 5+1 you get 6 as well. So order is not going to matter as long as they are on the same sort of playing field.
The same thing holds true for associative. For multiplying all the way across, we could do 4x5 first and then multiply by 6 or 5x6 and then multiply by 4, you're going to end up with the same thing.
Okay. Another property is commutative, and basically what commutative is saying is that your order the numbers are listed doesn't matter either. So if we are adding 3+7, it's going to be the same thing as 7+3. Likewise with multiplication, 10 times 4 is equal to 4 times 10. You can easily try those out for yourself.
The next property is what's called the identity and basically the identity property is going to be the number that you either add or multiply to the number to keep it the same. So if you are dealing with addition, everything on this side is addition by the way, you are going to add zero. Any number plus zero stays the same whereas your identity property, anything times one, your number will remain the same. So identity of multiplication is just times one.
For the inverse property, is basically the number that you want to either add or multiply to a number to get back to the identity. So if you are adding you want to add in the opposite the negative number of whatever you're dealing with. So if you're dealing with four you want to add in negative four negative four is the inverse of four. Likewise with multiplication, you want to get back to your identity so you want to multiply by the reciprocal the one over whatever you're dealing with so one ninth is the inverse of 9 because when you multiply them together you get one.
Okay, so everything up until now has had an addition definition and a multiplication definition, they're pretty much the same but basically we're dealing with addition first multiplication. The last one is a combination of the two and it's called the distributive property which is basically if we are multiplying a number into a sum we can distribute that number in. So what we have here is four times quantity 7+2 is the same thing as four 4x7+4x2. You can just distribute that 4 in, okay?
So just a number of properties that come up pretty much on a daily basis when we're dealing with Math you may have seen a number of these in concept just to make sure we know the names as well as we go forward.

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