2.4 Distance and absolute values
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This video is a short preparation
for the formal definition of limit.
I want to explain how to use
absolute values and inequalities
to think about distances,
both using geometry and algebra.
This will be necessary
to understand the definition of "limit."
First, let's recall
the basics of absolute value.
The best way
to define absolute value algebraically
is by breaking it into two cases:
the absolute value of real number x (|x|)
is itself (x) when it's positive,
and minus itself (-x) if it's negative.
We need to include 0
in one of the two cases;
it doesn't matter which one.
If you need to solve an equation
or an inequality algebraically,
normally, the best way
is to break it into cases
by using this definition.
But it's also helpful
to think of absolute value geometrically.
And geometrically, I'm going to say
that the absolute value of x
is the DISTANCE between x and 0.
And more generally,
if I have two real numbers x and a,
the absolute value of x – a (|x – a|)
is the distance between x and a.
a couple of properties of absolute value
that we know and use often:
absolute value of the product
of two real numbers
is the product of their absolute values.
But the same is NOT true for the SUM.
Instead, inequality #2 here
is the triangular inequality.
That comes in handy very often.
Now, let's move
to the most important part of the video.
What do I want to do
with absolute values?
When we try
to write a definition of limit,
we often need to talk
about numbers being close to each other.
And the way we do that
is with inequalities like this one:
"the absolute value of x – a (|x – a|)
is less than delta (δ),"
where ALL the variables are real numbers,
and δ is positive.
My goal is to explain
many equivalent ways
to write this expression.
First, let's interpret it.
According to what I said that
the absolute value means geometrically,
this inequality means
that the distance between x and a
is smaller than δ.
So whenever you want to think
that the distance between x and a
is smaller than δ,
you can EQUIVALENTLY write it
as |x – a| < δ. But this inequality includes an absolute value, and algebraically, sometimes this is hard to handle. We can get rid of the absolute value by doing the following: saying that the absolute value of a number is smaller than δ is the SAME thing as saying that the number is between -δ and δ. So these two lines are entirely equivalent. But in the last one, I now have two inequalities instead of one, but the absolute value is gone. And that normally makes the algebra easier. I can also try to isolate x by adding a to all the terms. So saying that x - a is between -δ and δ is equivalent to saying that x is between a - δ, and a δ. And once I have it this way, I like to think of this picture: that x is in the interval between a - δ, and a δ, not including the endpoints. What do we need to remember out of all of this? This is how we are going to try to formalize what it means for numbers to be close. And every time you see one of these inequalities, you can immediately convert it to the other equivalent inequalities, or to the geometric interpretation, or to this picture. And if you are able to do that; if you're able to move back and forth between all these equivalent statements, and you understand WHY they are equivalent, then you are ready to learn the formal definition of limit.