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# 6.1 Absolute Value

6.1   Definition (Absolute values.) Recall that if is a real number, then the absolute value of , denoted by , is defined by

We will assume the following properties of absolute value, that follow easily from the definition:

For all real numbers with

For all real numbers , and all

and
 (6.2)

We also have

and

6.3   Theorem. Let and let . Then for all we have

and

Equivalently, we can say that

and

Proof:    I will prove only the first statement. I have

6.4   Definition (Distance.) The distance between two real numbers and is defined by

Theorem 6.3 says that the set of numbers whose distance from is smaller than is the interval . Geometrically this is clear from the picture.

I remember the theorem by keeping the picture in mind.

6.5   Theorem (Triangle inequality.) For all real numbers and
 (6.6)

Proof      For all and in R we have

and

so

Hence (Cf. (6.2))

6.7   Exercise. Can you prove that for all ?

Can you prove that for all ?

Remark: Let be real numbers with and .

Then

This result should be clear from the picture. We can give an analytic proof as follows.

6.8   Examples. Let

Then a number is in if and only if the distance from to is smaller than , and is in if and only if the distance from to is greater than . I can see by inspection that

and

Let

If , then is in if and only if , i.e. if and only if is closer to than to .

I can see by inspection that the point equidistant from and is , and that the numbers that are closer to than to are the positive numbers, so . I can also do this analytically, (but in practice I wouldn't) as follows. Since the alsolute values are all non-negative

6.9   Exercise. Express each of the four sets below as an interval or a union of intervals. (You can do this problem by inspection.)

6.10   Exercise. Sketch the graphs of the functions from to defined by the following equations:

(No explanations are expected for this problem.)

6.11   Exercise. Let be the functions described in the previous exercise. By looking at the graphs, express each of the following six sets in terms of intervals.

Let . Represent graphically on a number line.

Remark: The notation for absolute value of was introduced by Weierstrass in 1841 [15][Vol 2,page 123]. It was first introduced in connection with complex numbers. It is surprising that analysis advanced so far without introducing a special notation for this very important function.

Next: 6.2 Approximation Up: 6. Limits of Sequences Previous: 6. Limits of Sequences   Index
Ray Mayer 2007-09-07