Graphics Reference
In-Depth Information
60 On thegraph of
y
x
(qn 52) draw theline
y
b
, for some
positive
b
. For what values of
x
do you get
x
b
?
Use the equivalence of
a
b
with
b
a
with
a
b
(qn 6) to show that
b
a
b
implies
b
a
b
, and
deduce that
b
a
, and of
a
b
.
Conversely, show that if
a
b
, or max(
a
,
a
)
b
, then
b
a
b
.
b
a
b
implies
61 If
a
and
b
are two numbers of the same sign verify that
a
b
a
b
.
If
a
and
b
are two numbers of different sign verify that
a
b
a
b
.
From qn 54 wehave
a
a
a
and
b
b
b
for all
numbers
a
and
b
. Now useqn 10 to obtain
a
b
a
b
a
b
, and then qn 60 to prove that
a
b
a
b
for any two numbers
a
and
b
.
If onesubstituts
a
x
y
and
b
y
z
in this inequality, we get
x
z
x
y
y
z
.Now
x
y
is thedistancefrom
x
to
y
on the number line, so this inequality says that for any three
points on the number line, the sum of the distances between two
pairs cannot be less than the distance between the third pair, and it
is from this that thename
triangle inequality
stems.
62 Deduce from qn 61 that
a
b
a
b
.
Deduce also from qn 61 that
a
b
c
a
b
c
.
63 By putting
b
a
in thetriangleinequality,
provethat
c
a
c
a
,
and deduce that
a
c
a
c
c
a
.
c
Provefurthe r that
c
a
c
a
. This result can be
pictured by marking two points on the real line drawn on tracing
paper. If the paper is folded at 0 so that the line falls onto itself, the
left-hand side of the inequality is the folded distance between
a
and
c
, while the right-hand side gives the unfolded distance between
a
and
c
.
64
The triangle inequality in two dimensions
The strange name given to the inequality
x
y
x
y
is
more understandable in terms of the inequality for complex
numbers
z
w
z
w
. This result is the claim that for a
trianglein theArgand diagram, thesum of thelengths of two sids
is greater than the third. We prove this by considering the triangle
in
R
R
with vertices
O
(0, 0),
A
(
a
,
b
) and
P
(
p
,
q
).
