1. Write the following in decimal form and say what kind of decimal expansion each has:
4. Express 0.99999… in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Let x = 0.99999... ...(1)
Multiply both sides by 10,
we have [∵ There is only one repeating digit.]
10 x x = 10 x (0.99999…)
or 10x = 9.9999 ...(2)
Subtracting (1) from (2),
we get 10x – x = (9.9999…) – (0.9999…)
or 9x = 9
or x = 9/9 = 1
Thus, 0.9999… = 1 As 0.9999… goes on forever, there is no gap between 1 and 0.9999
Hence both are equal.
5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of (1/17)? Perform the division to check your answer.
Since, the number of entries in the repeating block of digits is less than the divisor.
In 1/17, the divisor is 17.
∴ The maximum number of digits in the repeating block is 16. To perform the long division, we have.
The remainder 1 is the same digit from which we started the division.
Thus, there are 16 digits in the repeating block in the decimal expansion of 1/17. Hence, our answer is verified.
6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?
Let us look at decimal expansion of the following terminating rational numbers:
We observe that the prime factorization of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both.
Note: If the denominator of a rational number (in its standard form) has prime factors either 2 or 5 or both, then and only then it can be represented as a terminating decimal.
7. Write three numbers whose decimal expansions are non-terminating non-recurring.
8. Find three different irrational numbers between the rational numbers (5/7) and (9 ).
To express decimal expansion of 5/7 and 9/11, we have:
9. Classify the following numbers as rational or irrational: