Bill met a couple of friends from the “House of Rational Numbers, Q”. They claimed to be unharmed by The Reciprocus. Bill challenged both of them to prove their claim. These occupants went to the temple, entered and exited the temple,with no effect on them! The Reciprocus had no effect on these occupants!

How many occupants could be in that group? (Enter the numerical value)

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Solution

Consider any rational number, a. Its reciprocal would be $\frac{1}{a}$ .

Since the number and the reciprocal are equal, we must have $\frac{1}{a}$ = $a$

Multiplying both sides with a, we get $\frac{1}{a}$ $\times a$ = $a \times a \Rightarrow 1 = a^{2}$

Can you think of any number who’s square is 1? Logically, we know that one such number is 1.

What about -1? As we know, (-1) $\times$ (-1) = 1 (the product of two negatives make a positive), -1 is also possible.

The two occupants are 1 and -1.

Hence, there are 2 occupants who belong to that group.

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Similar questions

Bill heard that in “The Hall of Operations”, there was another ancient temple, named “The Reciprocus”.

Legend says that when an occupant, say “a”, of the “House of Rational Numbers, Q” enters the temple, it exits the temple as a different occupant, “b”,belonging to the same house. Bill observed that, these occupants “a” and “b”, when they enter the hall of multiplication, they exit the hall as “1”.

Bill realized that the numerical counterparts of a and b (Earth 1) are ___________ of each other.

Bill heard that the temple, The Reciprocus, does not allow one occupant of the house to enter. Which occupant is that?

(Enter the numerical value)

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Bill noticed that it was easy to find occupants of the “House of Rational Numbers, Q” if they “dressed up” in their decimal form. He wanted to understand if he could locate the occupants of the house on Numerus Linum if they were in their “ $\frac{p}{q}$ ” form. Bill remembered interacting once with the occupant “ $\frac{2}{9}$ ” before. He thought of locating this on the line. Bill knew that this occupant lived between occupants “0” and “1” – both of which had markings on the line. To locate “ $\frac{2}{9}$ ”, Bill first mentally divided the part on Numerus Linum between 0 and 1 with required number of markings – which was “x”. Then he navigated to division number “y” after 0 to reach “ $\frac{2}{9}$ ”.

What is the numerical value of (x+y)?

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Bill took his time to admire the “House of Rational Numbers, Q”. He observed an interesting feature of the occupants on Earth 2.

The physical size of the occupants depends on their numerical value, i.e. if “a” and “b” are two occupants of the house, the sizes of “a” and “b” depend on the numerical values“a” and “b”. If a $>$ b, then “a” would appear larger and if a $<$ b, “a” would appear smaller compared to “b”

Bill noticed TWO occupants of the house which were equal in size. Who could these occupants be? Choose all the correct possible pairs.

Bill and “0” were having their introductions when “0” claimed that it could make any number from Room “W” become “0” if “new” numbers (occupants) from a different room could interact with the numbers of “W”. Bill thought of verifying this and opened the door of Room “W”. He was surprised to find a huge set of occupants similar to those in Room “W” but they carried a “negative” sign along with them. The moment these numbers entered “Room W”, the room changed again and the signboard now read “Room I”.

(Integers are represented by the letter I)

Now, amongst the new occupants who came in, which is the greatest?

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