MAT 335 Homework #1
Due: Monday, 21 January
Please hand in to the math office (SS 4072). Late penalalty: -15%
per day
Please hand in solutions to the problems that have a * (like 2*). You may work in groups of up to two.
Please check these questions regularly as sometimes errors are corrected and the choice of problems to
be handed in may be changed.
Last updated: 13 January, 11pm
Some useful formula
- (1)
- (1a) Show that
- [5/7]2 = .101(101) (Notation: (abc) means abcabcabc....)
- [5/8]3 = .12(12)
- (1b) What is [1/6]2? (Hint; it begins with .00 and then there is a repeating
pattern of length 2.) What is [1/6]3?
- (1c) Find [23/27]3 and [7/5]3 to 4 places (i.e., the first 4
digits in their ternary expansions).
- (2*)
- (2a) Indicate on a sketch of the interval [0,1] the subinterval of all numbers whose
ternary expansion begins with .201 (see Figure 2.10 in the text).
- (2b) Indicate on your sketch the point that has ternary expansion .111(111). What is
this number?
- (3)
Use the identification between the Cantor set and [0,1] discussed
in class (and in the text on page 75) to find the numbers in the
Cantor set that are matched with the following numbers from [0,1];
0, 1, 3/4, 1/8, 3/8, 1/6.
- (4)
Show that every point in the Cantor set C
is an accumulation point of points from the set E of
end points of intervals removed during the construction of C
(E = {0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, ...}).
That is, show that for any p in C and any d>0 (small number),
there is a q in E such that |p-q|< d. [This tells us that the end points "cluster"
around the points in the Cantor set.]
(Hints: Note that if q in E, then the ternary expansion of
q ends in infinitely many zeros
and also contains no 1's. And that if
x and y are two numbers in [0,1] whose ternary expansions agree to the
nth place, then |x-y|<1/(3n). )
- (5*)
Show that the 'Cantor middle-fifths' set has the same
properties as the Cantor middle-thirds set (the one we discussed in class).
So show that
the Cantor middle-fifths set (1) has length 0, (2) has at least as many points
as the interval [0,1], (3) is totally disconnected - is "dust" (i.e., contains no intervals),
and (4) is self-similar. The proofs of these assertions should be similar to
the proofs in class, and at the same level of mathematical rigour.
(The construction of the Cantor middle-fifth set removes the
middle fifth open interval of each subinterval that occurs at that
stage. Note that in the nth step 4n-1 intervals are removed;
in contrast, only 2n-1 intervals were removed in the nth step of the
construction of the Cantor middle-thirds set.)
(Hint: For (2), consider [x]4 and [x]5.)
- (6)
- (6a) Which numbers in [0,1] have two binary expansions? Two ternary expansions? Give
examples of numbers that have exactly one binary expansion. Give examples of numbers that have
exactly one ternary expansion.
- (6b) Show that all numbers in [0,1] of the form y = k/2n, for k between 0 and 2n,
have
a binary expansion that ends in zeros. Show that all numbers in [0,1] of the form
x = m/3n, for m between 0 and 3n, have ternary expansions that end in zeros.
- (7*)
Show that the length of the 'square' von Koch curve is infinity. (The
'square' von Koch curve is constructed in a similar way as the von Koch curve, but
squares if size (1/3)^n are added to each side at each step, instead of
triangles.)