Q1. Suppose a jogger wants to use her ten fingers to count laps as she circles a

Question

Q1. Suppose a jogger wants to use her ten fingers to count laps as she circles a track. Each finger can be in two different states to represent a binary digital. How many laps can she conveniently count? Briefly justify your answer.

Q2. How many trinary (base 3) digits are required to represent numbers in the following ranges?

(a) The integers from 0 to 255 inclusively.

(b) The integers from 0 to 4,095 inclusively.

(c) The integers from 0 to 1,234,567 inclusively.

Q3. An expedition to Mars found the ruins of a civilization. The explorers were able to translate the mathematical equations: 5x

Explanation / Answer

1. Each finger can give 2 values, hence it can act as a bit. The number of non negative values we can have with 10 bits = 2^10 = 1024. Hence we can count 1023 laps and 1 for the "zero" lap. answer is 1023.

2.

a) ceiling(log(256)/log(3)) = 6 trinary digits.

b) ceiling(log(4096)/log(3)) = 8 trinary digits.

c) ceiling(log(1234568)/log(3)) = 13 trinary digits.

3. Let n be the base of the number system (n has to be greater than or equal to 9, since we know they have 8 as a number). Then the quadratic equation becomes:

5x^2-5nx+(1n^2+2n+5) = 0.

Dividing by 5 gives
x^2-nx+ (n^2+2n+5)/5 = 0.

We know that x=5 and x=8 are solutions to the quadratic.

Thus, (x-8 )(x-5) = x^2-nx+(n^2+2n+5)/5
So x^2-5x-8x+8*5 = x^2-(5+8 )x+8*5 = x^2-nx+(n^2+2n+5)/5
So n = 5+8 = 13.

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