Q4. The number we construct must be between 0 and 1, so it will start with \"0.\

Question

Q4. The number we construct must be between 0 and 1, so it will start with "0.". We construct the number one decimal place at a time. To form the first digit after the decimal, go to the first digit after the decimal of the first number. If that digit is a 0 write a 1. If that digit is anything other than 0, write a 0. Proceed by going to the second digit of the second number to get the next digit. Construct the number for the list above. Q5. Could you construct such a number for any list of the reals in the unit interval? Explain. What does that tell you about our ability to build a list of the numbers in the unit interval?

Explanation / Answer

4) Let a list of reals be given. Starting with zero we write

0.0

0.1 or 0.0 acording to the first digit after decimal in the 1st no. of the list. Then

0.11 or 0.10 or 0.01 or 0.00 according to the 2nd digit after decimal in the 2nd no. of the list.

Continuing in this way we find a number which is different from each of the numbers in the list. I will elaborate it with an example. Let { 0.95786354, 0.10563418554, 0.2364764, 0.755036575 } be a list consisting of 4 reals.

Then my construction is as follows:

1st digit-- 0.0 ( I have written 0 after decimal because in the first number of the list the first digit after decimal is 9 i.e. different from 0.)

2nd digit-- 0.01 ( for 2nd digit after decimal in the 2nd number is 0)

3rd digit -- 0.010 (for 3rd digit after decimal in 3rd number is 6 i.e. different from 0)

4rth digit-- 0.0101 ( for 4rth digit after decimal in 4rth number is 0)

so finally we constructed a number 0.0101 which is different from each number of the list.

In this way you can construct the number for a given list.

5) So if a list is prepared in the above fashion writing numbers one after another then we can construct a number for that list. and its explanation is same as above. The reason is that the intermediate number (whose n th digit after decimal corresponds to n th after decimal of the nth number in the list ) constructed is already different from the previous n-1 numbers and it is to be made different from the coming number in the list. So we are assured of getting a new number always. This tells us that it is impossible to arrange reals of the unit interval in an order because if there is any order then we can construct a number and disturb the order in the above fashion.

This in other words is also known as the uncountability of the unit interval.

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