ALGEBRA
Sec. 6
MathHands.com
Márquez
INVERSE AXIOMS
INVERSE AXIOMS: INVERSES
It is helpful to recall the definition of a binary operation. As we have stated it, a
binary operation is a mixing recipe for mixing two items. We used the color analogy,
and we return to it now to establish a meaningful notion of an inverses. Inverses mean
nothing without a specified binary operation. Once we specify, a binary operation,
and assuming there is an identity element, then we can define inverses. Two numbers
are inverses if when mixed together they yield the corresponding identity.
In the world of color, and color mixing, the identity, the do-nothing element is water.
Thus in the world of mixing colors two colors are inverses if when mixed together
they yield water. This raises a very natural question. What is the inverse of red, for
example? That is what is the color that mixed with red yields water? In fact, there is
no such color. This is a perfect place to highlight the great human spirit of invention
and audacity. There is no known color such that when mixed with red yields water,
but that does not stop us. We will invent a color, an imaginary color, and call it
special red. We will create out of thin air, special red, for the sole purpose of being
the mixing inverse of red, thus red mixed with special red yields water, just like that!
ADDITIVE INVERSES: [AINV.]
In the world of Adding with Numbers, the identity is zero. Thus two numbers
are additive inverses if added they yield zero, the additive identity. With that
in mind lets consider the number ’3’ and ask what number added to ’3’ yields
0? In the beginning, there was no such number. But beign the audacious,
inventious beings that we are, out of thin air, we invented and created a new
number and called it -3. Think about how unnatural -3 is... Have you ever
seen -3 dogs walking around? Nevertheless, we invented it for the sole purpose
of being the additive inverse of 3. This is percicely how we will define ’−3’ as
the additive inverse of 3. We will use ¡Ainv.¿ for short to refer to the definition
of negative numbers as additive inverses. Symbolically we will write
−3 + 3 = 0
< Ainv. >
3 + −3 = 0
< Ainv. >
OR
MULTIPLICATIVE INVERSES [MINV.]
In the world of MULTIPLYING with Numbers, the identity is 1. We will accept
that any number times one is still the same number. Moreover, two numbers are
multiplicative inverses if, when multiplied, they yield 1, the multiplicative identity.
With that in mind lets consider the number ’3’ and ask what number multiplied by
’3’ yields 1? In the beginning, there was no such number. But, again, being the
audacious, we conquered, we invented and created a new number and called it, one
third, we also invented a symbol to represent it, 13 . Also note 1/3 is just as natural
or unnatural as -3.... Have you ever seen 1/3 cat walking around? Nevertheless,
we invented it for the sole purpose of being the multiplicative inverse of 3. This is
percicely how we will define ’ 13 ’ as the Multiplicative inverse of 3. We will use ¡Minv.¿
for short to refer to the definition of this type of fraction as a multiplicative inverse.
Symbolically we will write
1
·3 =1
3
< M inv. >
1
=1
3
< M inv. >
OR
3·
c
2007
MathHands.com
ALGEBRA
Sec. 6
MathHands.com
Márquez
1
NOTE: We will assume every number, A, has a multiplicative inverse, namely A
, EXCEPT the number 0. We
will not assume that zero has a multiplicative inverse. Unless otherwise stated, we will assume all multiplicative
inverses here are for non-zero quantities.
Some questions to think about
1. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1
·4=1
4
2. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
13 ·
1
=1
13
3. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
(blah) ·
1
=1
blah
4. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
(x + 3) ·
1
=1
(x + 3)
5. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1
·e =1
e
6. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
(−1) ·
1
=1
(−1)
7. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1 1
− + =0
4 4
8. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
3 3
− + =0
4 4
c
2007
MathHands.com
ALGEBRA
Sec. 6
MathHands.com
Márquez
9. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
√
√
− x+1+ x+1=0
10. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
√
1
√
· (− x + 1) = 1
− x+1
11. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1
· (−1) = 1
−1
12. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
−1 + 1 = 0
13. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
0 = 3 + −3
14. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
0 = −31 + 31
15. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1 = stuf f ·
1
stuf f
16. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1=3·
1
3
17. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1 = stuf f ·
1
cat
c
2007
MathHands.com
ALGEBRA
Sec. 6
MathHands.com
Márquez
18. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
1 = cat ·
1
cat
c
2007
MathHands.com