We all know that you can remove common terms from the top and bottom of a fraction. Did you realize that individual digits count as terms?
Disbelief! Then perhaps you need to see an example. Consider the fraction
If you cancel out the 6's, you are left with
Go ahead. Get out your calculator and try it. Sure enough 16/64 is 1/4.
But does this work in general? Let's try some more:
cancel the 9's to get
Looks good so far. But maybe it only works when the top number starts with a 1. Let's try
cancel the 6's to get
Again, it's correct.
Before you try this in real life, you might want to try a few more examples, like 89/91.
Did you know you can simply reverse the digits of a prime to get a new prime? Obviously, this won't help with 3, 5, and 7, since reversing the digits just leaves you with the same thing.
But let's carry on to two digits. Well, 11 is pretty boring. If we reverse the digits of 13, we get 31, which is certainly prime. And, to check ourselves, if we reverse the digits of 15 (not a prime), we get 51, which is 3 times 17, so it is not prime. Continuing, 17 gives 71, and 19 gives 91, so it all looks good.
Before you try this in real life, you should probably try another prime, say 23. (And you might want to go back and check your earlier results also. Just multiply the lucky number 7 by the unlucky number 13.)
It's not what you think! The reason there are no negative numbers is simply that -1 is just another way of writing 1. Watch, I can prove it. I'll even explain it as I go along.
Certainly, you'll have to allow me to start with
-1 = -1
Then, if I divide both sides by 1, I get
-1/1 = -1/1
Now, we know that -x/y = -(x/y) = (-x)/(y) = (x)/(-y). It doesn't matter where you put the minus sign. So, from that we get
-1/1 = 1/-1
And, if we take the square root of both sides, we get
root(-1/1) = root(1/-1)
But we can split the square roots out, so
root(-1) / root(1) = root(1) / root(-1)
Now, we can cross multiply (to get rid of the fractions), and get
root(-1) * root(-1) = root(1) * root(1)
But surely root(x) * root(x) = x. That's the definition of root(x), so
-1 = root(-1) * root(-1) = root(1) * root(1) = 1
Which leaves us with
-1 = 1
Which is what I told you originally. So you can see that there really are no negative numbers.
If you don't agree, try examining the proof closely. You can see I supported each step along the way.
For another example of misdirection, like the first example on this page, check out the bellboy problem, or the case of the missing dollar.