What is the way to get to any fraction in the fewest number of steps if you start at zero and each step you can either:

add 1
subtract 1
take the negative inverse

Question

What is the way to get to any fraction in the fewest number of steps if you start at zero and each step you can either:

add 1
subtract 1
take the negative inverse

kainui · Answer

For example you can get to 1/2 several ways:

Start at 0, add 1, add 1 again to get 2. Then take the negative inverse to get -1/2. Then add 1 again to get +1/2 and we're done. This is 4 steps.

Alternatively we can subtract 1, subtract 1, and then we'll have -2. The negative inverse is then +1/2 so this would be the better method since this was only 3 steps.

However the situation does not seem so easy, suppose we want something like 3/4?

vishweshshrimali5 · Answer

Well we can achieve it in this way:

Add one * 4
Negative inverse
Add one

kainui · Answer

We could add 
1*3,
 negative inverse,
 subtract 1 
negative inverse.

Also 6 moves.

vishweshshrimali5 · Answer

Yup

kainui · Answer

But maybe there's some other way that's trickier to do it in 5 moves? But in general is there an analytic solution to this kind of problem you think?

vishweshshrimali5 · Answer

May be we can find out something.

vishweshshrimali5 · Answer

Okay let me check it out

kainui · Answer

I'm getting some ideas, one sec let me think about them a little more.

vishweshshrimali5 · Answer

Well we can find out a general way for fraction of the form (n-1)/n

vishweshshrimali5 · Answer

Add one * n
Negative inverse
Add one

n+2 steps

vishweshshrimali5 · Answer

This one is a general method

kainui · Answer

True, that's a good start. But we could also look at fractions like 7/2 or 19/42 which are not quite so easily done by this.

vishweshshrimali5 · Answer

Yeah. But why are you always eager to find out difficult questions Kainui ? (kj)

vishweshshrimali5 · Answer

*just kidding

kainui · Answer

Hahaha! =) Actually this is connected to knot theory.

vishweshshrimali5 · Answer

Well never read that before ;)

But lets continue our journey

vishweshshrimali5 · Answer

Okay for something like 3/2:

Subtract one *3
Negative inverse
subtract one
Negative inverse

6 steps

kainui · Answer

Haha yes. You might actually appreciate it and be able to use it to come up with something. it's not very complicated. |dw:1405044097984:dw| Twist one way to add 1, twist the other way to subtract 1. Rotate the diagram by 90 degrees in either direction to get the negative inverse.