Definition 1: Absolute Value
abs(a) = a if a is nonnegative or abs(a)=-a if a is negative.
So for example:
abs(5) = 5
abs(0) = 0
abs(-1) = 1
Now, let's look at some basic properties
Lemma 1: abs(ab) = abs(a)*abs(b)
Proof:
Case I: both a,b positive
abs(ab) = ab = abs(a)*abs(b)
Case II: both a,b negative
abs(ab) = ab = (-a)*(-b) = abs(a)*abs(b)
Case III: one negative, one positive
Assume a is positive, b is negative (since a,b are symmetrical, we can switch them as necessary)
abs(ab) = -ab = a*(-b) = abs(a)*abs(b)
QED
Lemma 2: -abs(a) ≤ a ≤ abs(a)
Proof:
Case I: a is nonnegative
-a ≤ a ≤ a
so
-abs(a) ≤ a ≤ abs(a)
Case II: a is negative
a ≤ a ≤ -a
so
-abs(a) ≤ a ≤ abs(a)
QED
Lemma 3: abs(a) ≤ b if and only if -b ≤ a and a ≤ b.
Proof:
(1) Assume abs(a) ≤ b
Case I: a is nonnegative
abs(a) = a
Since abs(a) ≤ b, it follows that a ≤ b and b is nonnegative
Since b is nonnegative and a is nonnegative, then it -b ≤ a.
Case II: a is negative
Since abs(a) ≤ b, it follows that -a ≤ b which is the same as -b ≤ a and therefore b must be nonnegative.
Since b is nonnegative, it follows that a ≤ b.
(2) Assume that -b ≤ a and a ≤ b.
Case I: a is nonnegative
Since a ≤ b, it follows that b is nonnegative
So abs(a) ≤ b.
Case II: a is negative
Since -b ≤ a, it follows that b ≥ -a.
Since a is negative, -a is positive, and we have:
abs(a) ≤ b.
QED
Lemma 4: Triangle Inequality
For all real numbers a,b
abs(a + b) ≤ abs(a) + abs(b)
Proof:
(1) For all real numbers a,b (from Lemma 1 above)
-abs(a) ≤ a ≤ abs(a)
-abs(b) ≤ b ≤ abs(b)
(2) Adding these two conditions together gives us:
-[abs(a) + abs(b)] ≤ a + b ≤ abs(a) + abs(b)
(3) Let c = a+b and d =abs(a) + abs(b)
(4) Using Lemma 3, we know that:
abs(c) ≤ d if and only if -d ≤ c and c ≤ d.
(5) But using step #2, we know that:
-d = -[abs(a) + abs(b)] ≤ c = a + b
and
c = a + b ≤ d = abs(a) + abs(b)
(6) So, using step #4 we get:
abs(c) ≤ d
which is equivalent to:
abs(a + b) ≤ abs(a) + abs(b)
QED
5 comments :
Dear Larry
I randomly stumbled across your blog while searching for places to link to for explanations of some basic mathematical concepts.
Generally, I like the range of topics you cover, but I do have some stylistic criticism / suggestion:
It seems as if you never use pictures in your entries, which is bad in my opinion. It gives the impression of mathematics as an extremely dry subject.
I'm commenting here because a posting on the Triangle Inequality is literally screaming at least for a picture of a triangle visualizing the statement of the triangle inequality. Even that simple addition would already raise the quality significantly.
Obviously I do not know your intentions in writing this blog, and I do know that creating pictures can be a time consuming task, so I'm not really complaining. We all live in a free world. All I'm saying is that I would link to some of your entries occasionally if the presentation style were more accessible.
So, this is really only meant as a heads up - do with it whatever you like.
Cheers,
Nicolai
Hi Nicolai,
Thanks very much for your comment!
I am excited that you take my blog so seriously that you have made your suggestion.
I agree with you that pictures would be a big improvement.
The challenge comes down to finding pictures that are not protected by copyright that will add to the blog.
I am not very artistic and that is the main reason that there are no pictures.
There are some wonderful math sites that include pictures including cut-the-knot.org, mathworld, and of course, Wikipedia.
I do use graphics for proofs about geometry. As I have time, I'll start adding more visualizations to my explanations.
Cheers,
-Larry
Has helped me, a beginner, with understanding so much, you wouldn't believe it. thanks
Thank you so much for Lemma 3.
All the proof I reviewed in the net they skip the Lemma 3 .
Does this hold true for complex numbers? having a good time visualizing this thanks for not posting pictures I don't want to be lazy and not use my imagination;)
Post a Comment