associative a+(b+c) = (a+b) + c
commutative a+b = b+a
identity : there is a number 0 with
a+0=0+a = a
inverse a + -a = -a + a = 0
associative a(bc) = (ab)c
identity there is a number 1 with
a*1 = 1*a = a
inverse If a not zero then there is an 1/a such that a(1/a) = (1/a)a = 1
BOTH (the biggie!!)
distributive a(b+c) = ab + ac
thank u soooooooooooooooo much, i really needed to understand this.
Associative property of Multiplication:
(x*y)*z = x*(y*z)
Associative property of Addition:
(x + y) + z = x + (y + z)
x*y = y*x
x + y = y + x
Distributive Property :
x*(y + z) = x*y + x*z
Identity property: There exists a number 1 such that:
1*x = x for all x.
Zero property: There exists a number 0 such that:
0 + x = x for all x
For every x there exists a number -x, such that:
x + (-x) = 0
For every x not equal to zero there exists a number x^(-1), such that:
x*x^(-1) = 1
There can only be one 0. Proof suppose there were two numbers 0 and 0' that both satisfy the property that the zero element has to satsify, then:
0 + 0' = 0'
because 0 is a zero element.
But because 0' is also a zero element, you also have:
0 + 0' = 0
This means that 0 = 0'
(-1)*x = -x
Let's check of (-1)*x satisfies the criterium of being the inverse (relative to addition) of x:
x + (-1)*x =
1*x + (-1)*x =
(1 + (-1))*x = (use that -1 is the inverse relative to addition of 1)
0*x = 0
So, we can conclude that (-1)*x is the inverse of x relative to addition, which means that (-1)*x = -x
thank u soooooo much!