calculus-2014-05-21

DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse
. . . So you have uncovered a secret to performing the trick. But why does it work?
31. There is something special about the numbers in the upper left corners of each card, what is it?
If you were a born to a civilization with one finger on each of your two hands, or with just one
hand which had two fingers on it, you would likely count in a base-two number system. You would
also do this if you were a computer where the smallest units of information have just two states - on
and off. In such a system the “digits” are only 0 and 1 and are called bits, a portmanteau of the
words “binary” and “digit”.
The expanded, base-two representation of a number is then a number of the form:
and this number is written in base-two as:
a 0 + a 1 × 2 + a 2 × 2 2 + . . . + a n × 2 n
a n . . . a 2 a 1 a 0 .
32. What is the base-ten representation of the numbers whose base two representations is 1011?
33. If this number was the secret number in the trick above, what cards would it be on?
34. What is the base-ten representation of the numbers whose base two representations is 10010?
35. If this number was the secret number in the trick above, what cards would it be on?
36. Precisely describe how the trick above is related to the base-two numeration system.
37. If you were used to counting/representing numbers in base-two, would this trick seem very magical
to you? Explain.
4. Base-Two “Decimals”
In the previous trick you got some idea what it was like to represent whole numbers in base-two.
Is there an analogue of decimals in base-two? Sure, the expanded notation now simply uses powers
of two instead of powers of ten in the denominators:
a −1
2 + a −2
2 2 + a −3
2 3 + . . .
In Figure 2.3 four numbers on the real number line between 0 and 1 are represented by dots.
38. For the number represented by the right-most point, determine the expanded, base-two representation
so that the first 5 bits are correct. Explain how you know all of these bits are
correct.
39. Repeat 38 for the number represented by the point second from the right.
40. Repeat 38 for the number represented by the point third from the right.
41. Repeat 38 for the number represented by the point furthest to the left.
42. Given any point between 0 and 1 do you think that you could, with sufficient magnification and
sufficient time, determine its expanded, base-two representation to any specified (but finite)
number of bits? Explain.
43. If you can determine each number as precisely (but finitely) as desired, will the representation be
unique or is it possible to have one point to have two different expanded, base-two representations?
Explain.
5. Infinite Series
Just as with base-ten decimals, one can use infinitely many bits to represent numbers in base-two.
44. Using your experience from the previous section, what number is represented by the base-two
number 0.111111 . . .? What does this remind you of?
45. Write the base-two number 0.111111 . . . in expanded notation.
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