# the diagonal of a 1x1 unit square is equal to 2.?

okay so i came across an interesting theory the other day. using only vertical and horizontal lines to reach from corner of a perfect square to its vertical and horizontal opposite you will always travel the same distance. in the case of a 1x1 square, by travelling from the top-left corner to the bottom right corner using only diagonal and horizontal lines you will travel a distance of 2. no matter what path you take from one corner to its opposite in this situation the distance will be 2. even if it were composed of 1,000,000,000 horizontal and vertical steps

right now you probably think this is irrelevant because the hypotenuse would be a diagonal line. but think about this, a diagonal line is essentially composed of countless horizontal and vertical lines. the theory above stated is correct even if an infinite number of horizontal and vertical steps were used.

if you were to open up microsoft paint right now and draw a diagonal line in the smallest size, the line is not diagonal at all just a composition of vertical and horizontal lines, every line we have ever drawn can be broken down into vertical and horizontal lines. even the thinnest line drawn by the sharpest pencil can be broken down into smaller pieces

Update:

at Luc M. read the details thoroughly before posting, you did not disprove anything. all you did was state the rule i am contradicting here. what im saying is that even though a line appears to be diagonal even in your example, you can break it down to an infinite number of horizontal and vertical steps.

the distance from the top-left corner of a 1x1 square to the bottom right corner no matter what path taken will always equal 2.

the only way to disprove it is to draw a diagonal line so small that it is impossible to break down into horizontal and vertical steps.

and since we generally dont work with lines of this size is in our everyday life wouldnt my theory be correct.

Update 2:

no line can be drawn that is so small that it cannot be broken down further, even into individual molecules is all i am saying, what you are saying is that we are not calculating the vertical and horizontal components but rather the diagonal. what you have just stated is no more of a disproof than luc m

Relevance

The vertical distance of the two points would = 1 unit and the horizontal would also = 1 unit therefore making the total distance equal to two units.

The displacement however is still root 2

So you could be correct about the DISTANCE of the two points but what Luc M is getting at is the displacement which would = 2^(1/2) [root 2]

P.S the "steps"

you speak of are pixels and if you were to look at the "steps" on our ink lines you would see they are just millions of microscopic ink blobs (circles not squares)

Source(s): Dictionary term of distance and displacement Pythagoras thearom
• Anonymous