Question

The questions that follow are intended to provoke your thinking about some of the issues mentioned above. (a) Find three prominent open questions conjectures in mathematics, and record their precise state- ments on your paper (b) Let us accept CL as our working environment for this part of problem #3. Here, we will investigate some claims about the rationality irrationality of certain numbers. Recall that a real numberz is rutional if there exist integers m and n, with n 0 such that z·, A real number is ma ional if it is not rational. In what follows, we make reference to decimal expansions of real numbrs. Every real number can be represented as a (possibly unbounded) decimal, and some numbers have two decimal expansions. For example, 3.58-3.579 The Decimal Rationality Theorem A real number r is rational if and only if a decimal expansion of either terminates or repeats. Let e be the base of the natural exponential and natural logarithmic functions. Of course, from Calculus we know that e = lim [ 1 + about the existence of patterns within the unbounded decimal expansion of e. In particular, the following statemen, which we'll call P, is an open question: There exists a string of one hundred consecutive nines in the decimal expansion of e. Let's define a number as follows m It is known that e is irrational, but not much is known 0.333 if P is true and the first string of one hundred consecutive nines in e k3s begins at the kth position after the decimal if P is false, e. if Pis true Prove that there exist integers a and b, with bメ0, such that w = 응. Is your proof valid in BL? Why or why not?

Answer 1