Assignment II Math for Poets 2007

Problem 1 consists of Problems 16,17,18 on page 47 of the book (1,2,3 on page 45 of the first edition). We repeat them here:

• What proportion of the first 1000 natural numbers have a 3 somewhere in them? For example, 135, 403 and 339 all contain a 3, whereas 402,677 and 8 do not.
• What proportion of the first 10,000 natural numbers contain a 3?
• Explain why almost all million-digit numbers contain a 3.

In your answer, explain as clearly as possible. The criterion is that fellow students, not following this class, should be able to read your text and understand your explanation by reading it (without having you to explain in the background). Communication of math ideas is something to be learnt.

In problem 2 we look at the numbers that consist of n consecutive ones, i.e. 11, 111, 1111, 11111, etc.

• Consider a number consisting of an even number of ones. Suppose it is larger than 11. Why can't the number be prime? (For example, in class we have seen the factorisation 1111 equals 11 times 101).
• Consider a number N consisting of n ones. Suppose n is not prime. Why can't the number N be a prime?
• Using the factorisation site, make a table of the first twenty numbers consisting of ones together with their factorisation.