Question

The rational number representation of 1.88888.... is​

Answer 1

Answer:

i am not

providing the correct answer but I am providing with example to so that you can make you by own.

Step-by-step explanation:

Since the decimal representations of a and b are non-terminating and non-repeating. So,

a and b are irrational numbers.

We observed that in the first two places of decimal a and b have the same digits. But in the third place of decimal a has a 1 whereas b has zero.

∴ a > b

Construction of a rational number between a and b : As mentioned above, first two digits after the decimal point of a and b are the same. But in the third place a has a 1 and b has a zero. So, if we consider the number c given by

c = 0.101

Then, c is a rational number as it has a terminating decimal representation.

Since b has a zero in the third place of decimal and c has a 1.

∴ b < c

We also observe that c < a, because c has zeros in all the places after the third place of decimal whereas the decimal representation of a has a 1 in the sixth place.

Thus, c is a rational number such that

b < c < a.

Hence , c is the required rational number between a and b.

Construction of an irrational number between a and b : Consider the number d given by

d = 0.1002000100001……

Clearly, d is an irrational number as its decimal representation is non-terminating and non-repeating.

We observe that in the first three places of their decimal representation b and d have the same digits but in the fourth place d and a 2 whereas b has only a 1.

∴ d > b

Also, comparing a and d, we obtain a > d

Thus, d is an irrational number such that

b < d < a.