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# 5.3 Existence of Roots

5.35   Definition (Graph.) Let be a function. The graph of is

5.36   Remark. If is a function from to , then graph is

You may find it useful to think of as points on a line, and as points in a plane and to represent the graph by a picture. Any such picture is outside the scope of our formal development, but I will draw lots of such pictures informally.

5.37   Definition (Sum and product of functions.) Let be a field, and let . Let be sets and let be functions. We define functions , , , and by:

where .

5.38   Remark. Let be a field, let be a set, and let , be functions with the same domain. Then the operations are binary operations on the set of all functions from to . These operations satisfy the same commutative, associative and distributive laws that the corresponding operations on satisfy; e.g.,
 (5.39)

Proof of (5.39). For all ,

Hence, . (Two functions are equal when they have the same domain, the same codomain, and the same rule.)

5.40   Definition (Increasing and decreasing.) Let be an interval in and let . We say
is increasing on if for all .
is strictly increasing on if for all .
is decreasing on if for all .
is strictly decreasing on if for all .

5.41   Remark. Since , we can reformulate the definitions of increasing and decreasing as follows:
is increasing on if for all .
is decreasing on if for all .

5.42   Exercise. Is there a function that is both increasing and decreasing? If the answer is yes, give an example. If the answer is no, explain why not.

5.43   Exercise. Give an example of a function such that is increasing, but not strictly increasing.

5.44   Exercise. Let and be increasing functions. Either prove that is increasing or give an example to show that is not necessarily increasing

5.45   Exercise. Let and be increasing functions. Either prove that is increasing or give an example to show that is not necessarily increasing.

5.46   Theorem. Let , let , . Then .

The proof is by induction, and is omitted.

5.47   Theorem. Let . Let for all in . Then is strictly increasing on .

Proof: The proof follows from induction on or by factoring , and is omitted.

5.48   Exercise. A Let be an interval in and let be a strictly increasing function on . Show that for each the equation has at most one solution in .

5.49   Theorem. Let and let in . Then there is a unique in such that

Proof: First I will construct a binary search sequence in such that

By completeness of , I'll have for some . I'll show , and the proof will be complete.

Let . Then

For , define

The proof that is a binary search sequence and that for all is the same as the proof given in example 5.16 for , and will not be repeated here. By completeness for some . Since , we have . It follows that

By the formula for factoring (cf. (3.78)), we have

for all . By Archimedean property 3 (cf corollary 5.28), it follows that , i.e .

Let . Since is strictly increasing on , it follows from exercise 5.48A that has at most one solution in and this completes the proof of the theorem.

5.50   Notation (.) If and , then the unique number in such that is denoted by , and is called the th root of . An alternative notation for is .

5.51   Exercise. A Let , let , and let .
a)
Show that .
b)
Show that if , then .

5.52   Definition (.) If and we define where , and . The previous exercise shows that this definition does not depend on what representation we use for writing .

5.53   Theorem (Laws of exponents.), For all and all
a)
b)
c)

Proof: [of part b)] Let where are integers and are positive integers. Then (by laws of exponents for integer exponents),

Also,

Hence, , and hence by uniqueness of roots.

5.54   Exercise. A Prove parts a) and c) of theorem 5.53.

5.55   Entertainment. Show that of the two real numbers

one is in , and the other is not in .

5.56   Note. The Archimedean property was stated by Archimedes in the following form:
the following lemma is assumed: that the excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area. The earlier geometers have also used this lemma.[2, p 234]

Euclid indicated that his arguments needed the Archimedean property by using the following definition:

Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.[19, vol 2, p114]
Here multiplied'' means added to itself some number of times'', i.e. multiplied by some positive integer''.

Rational exponents were introduced by Newton in 1676.

Since algebraists write etc., for etc., so I write for and I write etc. for etc.[14, vol 1, p355]
Here denotes the cube root of .

Buck's Advanced Calculus[12, appendix 2] gives eight different characterizations of the completeness axiom and discusses the relations between them.

The term completeness is a twentieth century term. Older books speak about the continuity of the real numbers to describe what we call completeness.

Next: 6. The Complex Numbers Up: 5. Real Numbers Previous: 5.2 Completeness   Index