190Tutorial5

• This tutorial was discussed in a group consisting of Stephen, Nick, and Myself

a) Consider the number with the decimal expansion 0.12345678910111213..... Is this rational or irrational? What about the number 0.246810121416....? Is this rational or irrational? Make sure you justify your answer in detail.

Both of these numbers are irrational. This is because the decimal part of the number does not repeat. Decimals which repeat, even to infinity, can be written as a fraction of two integers and are thus rational.

The non-repetitive nature of the number 0.123456789101112... can easily be shown with the first occurrences of unique sequences. The sequence "10" Occurs for the first time at the 10th decimal place. The sequence "100" occurs for the first time at around the 190th decimal place.

Due to the nature of both sequences, a new multiple of 10 will only occur after all smaller multiples of ten have appeared. This patter continues and therefore there is no repetition. Both numbers are irrational.

b) Is 0.99999999.... the same as 1? Why or why not? Justify your answer in detail.

0.999... is the same as 1. The difference between them is infinitely small.

Consider the fraction 1/9.

9 * 1/9 is clearly equal to 9/9 which can be simplified to 1/1 = 1.

When writing 1/9 as a decimal however, we get 0.111111...

If to this number we add 0.11111... we get 0.222222...

If, to 0.222222... we add another 7 lots of 0.111111... we get, 0.999999... . What we have done however is add 9 lots of 1/9.

Thus, it is clear that 9/9 = 1 and 9 * 1/9 = 0.9999.... since 9/9 = 9*1/9 , 1 = 0.999999...