No more right-side left-side!
October 23, 2008 / updated September 8, 2011
Katherine E. Stange
Consider the following theorem:
[Theorem] For any real numbers a, b, q and r, the following relationship holds:
a − b q r
b
=
−
a
r
q
a
Let’s look at some left-side right-side proofs. What defines left-side right-side
isn’t that there are two sides, but that you start with what you wish to prove and
deduce something you know. The first one is standard.
Right-side left-side ‘proof ’ #1. We begin with what we want to show:
b
a − b q r
=
−
what we want
a
r
q
a
a − b b q r
+ =
add something to both sides
a
a
r
q
qr
a−b+b
=
simplify
a
rq
qr
a
=
simplify
a
qr
1=1
we’re done!
The second one seems ok too, but there’s a problem with this one. It’s hard
to say exactly what the problem is, though, since each step seems ok.
Right-side left-side ‘proof ’ #2. We start with what we want to show:
a − b q r
b
=
−
what we want
a
r
q
a
q
a−b
r
b
0×
=0×
−
multiply something on both sides
a
r
q
a
0=0
we’re done!
Can you see what’s wrong? This method of proof could be used to prove
something false! For example
1
A no-good nonsense non-proof that 2 = 3.
2=3
0×2=0×3
0=0
what we want
multiply something on both sides
we’re done!
That last one illustrates the danger of the left-side right-side philosophy. The
problem in this case is that each step wasn’t ‘reversible’. But in the direction we
went (down the page), everything seemed legitimate. Starting with what you
want to prove isn’t a way to prove things, it’s a way to get confused!
Here’s how you turn a right-side left-side proof into a real proof by using the
same simplification steps. Now we start with what we know and deduce the thing
we wish to prove. Try to see how the same steps from ‘proof’ #1 are used in this
argument:
A real proof. First, we simplify the left side:
a−b b
b
a−b+b b
a b
b
a−b
=
+ − =
− = − =1−
a
a
a a
a
a
a a
a
and the right side:
qr b
qr b
b
− =
− =1−
rq a
qr a
a
Thus, the two expressions are equal, and we have shown that
b
a − b q r
=
−
a
r
q
a
Right-side left-side has nothing to do with right and left sides. Two sides are
ok, as long as you start with what you know. Here’s a perfectly good, if kind of
strange proof (it’s strange because each step is hard to predict). It’s just the first
proof, turned upside down!
2
Another real proof. We know that 1 = 1, so we can desimplify this equation:
1=1
qr
a
=
a
qr
a−b+b
qr
=
a
rq
a−b b
q r
+ =
a
a
r
q
a−b
q
r
=
−
a
r
q
a − b q r
=
−
a
r
q
this is self evident!
de-simplify
de-simplify
de-simplify
b
a
b
a
add something to both sides
this is what we want
Try doing that with the right-side left-side proof of 2 = 3 and you’ll see what’s
wrong with it!
A nonsense no-good non-proof that 2 = 3.
0=0
0×2=0×3
2=3
this is self evident!
de-simplify
Uh-oh! Can we really cancel zeroes?
Frankly, when you have an equation to prove, usually the easiest thing to do
is just to rewrite the equation in the form ST U F F = 0 and try to simplify the
ST U F F to get the 0. In this case:
The easiest proof.
b
a − b q r
−
+
a
r
q
a
a−b
b
=
−1+
a
a
a−b+b
−1
=
a
a
= −a
a
=1 − 1
=0
3
simplify
simplify
simplify
simplify
simplify