Basic Algebra and Geometry Made a Bit Easier Lesson Plans:
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Lesson 78: Review of Mathematical Properties
In this easy lesson, we'll review properties of numbers that we've been working with all this time, and we'll present the formal names for them.
You know that when you have a group of numbers, it doesn't matter what order you add or multiply them. This is called the associative property of addition or multiplication. It does not apply to subtraction or division. We can represent this symbolically like this:
(a + b) + c = a + (b + c). It doesn't matter what group we
add first. For multiplication, we can write:
(ab)c = a(bc)
If we're only working with two numbers, it's called the commutative property. It doesn't matter which number we add or multiply first. We can write:
a + b = b + a
ab = ba
We've seen the distributive property of multiplication
over addition. If we have a problem of the form:
a(b + c), we can either do b+c first, and then multiply the result
by a, or, we can distribute the a, and do
ab + ac.
The identity property reminds us of what number
we can add or multiply by to not change the original number.
We can say that 0 is the identity element in
addition, since any number + 0 is the same number.
a + 0 = a.
We can say that 1 is the identity element in multiplication, since
any number times 1 is the same number. a x 1 = a.
The inverse property of addition means that if you add a number to its negative (its additive inverse), you get 0. For example if you have 3 and you want to get rid of it, you could add -3 (which is really subtracting 3), and you'd be left with 0.
The inverse property of multiplication is a bit trickier. Here we want to be able to take any number, and multiply it by something so that we end up with 1 (the identity element) for multiplication). What we actually do is multiply the number by its reciprocal. Here is how that works. Let's take the number 7. I'll write it as 7/1, which is allowed. The reciprocal is the fraction flipped upside down, or 1/7. If we multiply 7/1 x 1/7, we get 1/1, which is 1. That means that 1/7 is the multiplicative inverse of 7.
These properties will come up again and again in math, and sometimes you're expected to know the names of these properties. Make sure you memorize them.
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