Spivak Calculus Chapter 1 Problem 3.i

[Shawn Garsed]

Homework Statement

Prove the following: $a / b = ac / bc$, if $b, c \neq 0$.

Homework Equations

P1-12

The Attempt at a Solution

$a/b = a*b^{-1}$
$1 = c*c^{-1}$
$a/b*1 = (a*b^{-1})(c*c^{-1})$
$a/b = (a*c)(b^{-1}*c^{-1})$

Now, if $b^{-1}c^{-1} = (bc)^{-1}$, then the problem is easy to solve. However, you don't prove this until problem 3.iii and I'm assuming I can only use P1-12.

Some hints about how to proceed would be very helpful.

 

[Ray Vickson]

Homework Statement

Prove the following: $a / b = ac / bc$, if $b, c \neq 0$.

Homework Equations

P1-12

The Attempt at a Solution

$a/b = a*b^{-1}$
$1 = c*c^{-1}$
$a/b*1 = (a*b^{-1})(c*c^{-1})$
$a/b = (a*c)(b^{-1}*c^{-1})$

Now, if $b^{-1}c^{-1} = (bc)^{-1}$, then the problem is easy to solve. However, you don't prove this until problem 3.iii and I'm assuming I can only use P1-12.

Some hints about how to proceed would be very helpful.

What is the meaning of $a/b$; that is, if we say $a/b = r$, what are we actually saying? In other words, how could you re-write the relation between $a,b,r$ in another way?

 

[PeroK]

What about looking at $ac(bc)^{-1}bc$?

 

[Shawn Garsed]

What is the meaning of $a/b$; that is, if we say $a/b = r$, what are we actually saying? In other words, how could you re-write the relation between $a,b,r$ in another way?

How did I not see that, I feel so stupid.

$1 = b* b^{-1}$
$a = a *(b* b^{-1})$
$a = (a*b^{-1})*b$
$ac = ((a*b^{-1})*b)*c$
$ac = (a*b^{-1})*(bc)$
$(ac)*(bc)^{-1} = (a*b^{-1})*((bc)*(bc)^{-1})$
$(ac)/(bc) = a/b$

I think this is it.

 

[GFauxPas]

For someone unfamiliar with Spivak, what are the variables? Numbers?

 

[Shawn Garsed]

For someone unfamiliar with Spivak, what are the variables? Numbers?

They are real numbers.