Brain Teasers
Disappearing Numbers
For every number that exists between 0 and 1 (say for example 2/3) there exists exactly one other number called its reciprocal (in this case 3/2) that is bigger than 1.
Also, for every number that exists between 0 and 1 (say 0.5) there is exactly one number that exists, found by adding 1 (1.5) that must lie between 1 and 2.
Since every number between 0 and 1 corresponds to exactly one number between 1 and 2 and at the same time corresponds to a number bigger than 1 then there must be the same number of numbers that lie between 1 and 2 as there are bigger than 1.
That means that all of the numbers bigger than 1 lie between 1 and 2 so there aren't any numbers bigger than 2.
This is clearly false, so where is the error?
Also, for every number that exists between 0 and 1 (say 0.5) there is exactly one number that exists, found by adding 1 (1.5) that must lie between 1 and 2.
Since every number between 0 and 1 corresponds to exactly one number between 1 and 2 and at the same time corresponds to a number bigger than 1 then there must be the same number of numbers that lie between 1 and 2 as there are bigger than 1.
That means that all of the numbers bigger than 1 lie between 1 and 2 so there aren't any numbers bigger than 2.
This is clearly false, so where is the error?
Hint
Reversing the reasoning will give you the same correspondence. Every number bigger than 1 has a reciprocal less than 1 and every number between 1 and 2 has a corresponding number less than 1 found by subtracting 1. No error here.Answer
The problem lies in how many numbers there are. There are an infinite number of numbers between 0 and 1 and hence an infinite number of numbers between 1 and 2 and also greater than 2. Surprisingly there can be different sized infinities so you cannot say that two infinite quantities are equal. (Take 1 away from infinite and you still have infinite).Hide Hint Show Hint Hide Answer Show Answer
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Comments
Bravoooo Jimbo, Bravoooo!
"Since every number between 0 and 1 corresponds to exactly one number between 1 and 2 and at the same time corresponds to a number bigger than 1 then there must be the same number of numbers that lie between 1 and 2 as there are bigger than 1. " - This statement makes no sense. It breaks rules of math and logic.
If all items in "A group" are in "B group" and all items in "B group" are in "C group" does NOT mean that there are the same number of items in "C group" as there are in "B group" NOR does it mean that "B" and "C" have the same number of items.
You might have been able to say that since there are an infinite number that corresponds..., and an infinite number that are bigger than 1, then all numbers are between 0 and 1; and then that would be wrong for the reason you stated (You can't compare infinty.)
If all items in "A group" are in "B group" and all items in "B group" are in "C group" does NOT mean that there are the same number of items in "C group" as there are in "B group" NOR does it mean that "B" and "C" have the same number of items.
You might have been able to say that since there are an infinite number that corresponds..., and an infinite number that are bigger than 1, then all numbers are between 0 and 1; and then that would be wrong for the reason you stated (You can't compare infinty.)
Fasil, you seem to know what you are talking about but you need to re-read the teaser. There is no "belong to" group. Each (every) number in group A (as you say) corresponds or matches exactly 1 item in group B, not "belongs to".
Now 1->2 and 2->4 and 3->6 etc means that every integer correspond to exactly one even number for example but common sense dictates that there are more integers than there are even numbers (ie all of the odd numbers). But the teaser proposes in another setting that you consider this. How many even numbers are there? How many integers are there? Since they are infinite sets then you cannot say how many elements each set has. Yes, it breaks the rules of set theory because the teaser asks you to apply common sense which fails with infinite sets.
Now 1->2 and 2->4 and 3->6 etc means that every integer correspond to exactly one even number for example but common sense dictates that there are more integers than there are even numbers (ie all of the odd numbers). But the teaser proposes in another setting that you consider this. How many even numbers are there? How many integers are there? Since they are infinite sets then you cannot say how many elements each set has. Yes, it breaks the rules of set theory because the teaser asks you to apply common sense which fails with infinite sets.
You know what, you're right. I misread it and my analogy should have read - If all items in "A group" correspond to all items in "B group" and all items in "A group" correspond to all items in "C group" then it follows that "B group" has the same number of items in "C group". - Which appears correct logically, but is wrong for the reason you and I stated. Infinity can't be compared.
Excellent math but it doesn't fit site definition of "math brain teaser" because it doesn't "require computations to solve."
2 thirds is less than 1
3 halves are more than 1
.5 < 1
1.5 > 1
No numbers greater than 2.
3 halves are more than 1
.5 < 1
1.5 > 1
No numbers greater than 2.
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