190Assignment3
1. (Mindscape 28, Chapter 2.2 of text). Suppose we have a pair of baby rabbits – one male and one female. As before, the rabbits cannot reproduce until they are one month old. Once they start reproducing, they produce a pair of bunnies (one of each sex) each month. Now, however, let us assume that each pair dies after three months, immediately after giving birth. Create a chart showing how many pairs we have after each month from the start through month nine.
Month # First Month Second Month Third Month Total 1 1 1 2 1 1 2 3 2 1 1 4 4 4 2 1 7 5 7 4 2 13 6 13 7 4 24 7 24 13 7 44 8 44 24 13 81 9 81 44 24 149
2. Find a sequence of 5 consecutive numbers, none of which is a prime. Find a sequence of 50 consecutive numbers, none of which is a prime. (Hint: what about 50x49x48x.....x3x2x1).
There are no prime numbers between 19609 and 19661. This is a gap of 52 numbers.
This was quickly found on http://www.research.att.com/~njas/sequences/b000040.txt.
3. (Mindscapes 9, 11, Chapter 2.7 of text). Find an irrational number between 0.0001 and0.000100001. Find a rational number between 0.0001 and 0.00010001. Generalise this argument to show that, between any two numbers (either rational or irrational) there is always both a rational and an irrational number.
An example of a number between 0.0001 and 0.00010001 would be 0.000100005. This is rational as it can be written as a fraction of two integers i.e. 100005/1000000000.
The process used to find this number, which is true for any two distinct rational or irrational numbers, is to take the number which is exactly half way between them. In other words, sum the two numbers and divide them by 2. If both numbers are rational, this will find a rational number between them. If one is irrational it will find an irrational number between them.
The statement "between any two numbers (either rational or irrational) there is always both a rational and an irrational number" is incorrect. It does not allow for the situation where both numbers are exactly the same. This statement is however true for any two distinct numbers.
4. (Mindscape 20, Section 3.2) Suppose you take a line that is 10 cm long, cut it in half, then take the left piece and cut it in half, and then take the leftmost piece and cut it in half, and so on, without ever stopping. How many different pieces of the line would you end up with? Would the set of all pieces have the same cardinality as the set of all natural numbers? Justify your answer.
The number of pieces of string would have the same cardinality as all natural numbers. This is because you could construct a one-to-one pairing with the numbers and half of string i.e. The first half of string would correspond to the number 0, half of the remaining string would correspond to 1, half of the left over string again would correspond to the number 2 and so on to infinity.