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Q3 (3)   The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?  

              43. \overline{123456789}

As the decimal part of the given number is non-terminating and repeating, the number is rational but its denominator will have factors other than 2 and 5.

Q3 (2)   The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p / q what can you say about the prime factors of q? 0.120120012000120000

Since the decimal part of the given number is non-terminating and non-repeating we can conclude that the given number is irrational and cannot be written in the form  where p and q are integers.

Q3 (1)   The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p / q what can you say about the prime factors of q? 43.123456789

The denominator is of the form 2a x 5b where a = 9 and b = 9. Therefore the given number is rational and has a terminating decimal expansion.

Q2   Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

decimal expansions of  rational numbers are (i)  (ii)  (iv)  (vI)  (viii)  (ix) 

Q1 (10)   Without actually performing the long division, state whether the following rational
               numbers will have a terminating decimal expansion or a non-terminating repeating decimal
              expansion:  77 / 210 

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q1 (9)   Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
              expansion: 35 / 50 

The denominator is of the form 2a x 5b where a = 1 and b = 1. Therefore the given rational number will have a terminating decimal expansion.

Q1 (8)   Without actually performing the long division, state whether the following rational
              numbers will have a terminating decimal expansion or a non-terminating repeating decimal
             expansion: 6 / 15 

The denominator is of the form 2a x 5b where a = 0 and b = 1. Therefore the given rational number will have a terminating decimal expansion.

Q1 (7)    Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
            expansion: \frac{129 }{2 ^ 2 7^5 5 ^ 7 }

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q1 (6)   Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
             expansion:  \frac{23}{2 ^3 5 ^ 2 }

The denominator is of the form 2a x 5b where a = 3 and b = 2. Therefore the given rational number will have a terminating decimal expansion.

Q 1 (5)   Without actually performing the long division, state whether the following rational
              numbers will have a terminating decimal expansion or a non-terminating repeating decimal
            expansion: 29 / 343

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q 1 (4)    Without actually performing the long division, state whether the following rational
               numbers will have a terminating decimal expansion or a non-terminating repeating decimal
               expansion: 15 / 1600 

The denominator is of the form 2a x 5b where a = 6 and b = 1. Therefore the given rational number will have a terminating decimal expansion.

Q1 (3)   Without actually performing the long division, state whether the following rational
            numbers will have a terminating decimal expansion or a non-terminating repeating decimal
           expansion:  64 / 455 

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q1 (2)   Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
             expansion:   

              17 / 8 

The denominator is of the form 2a x 5b where a = 3 and b = 0. Therefore the given rational number will have a terminating decimal expansion.
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