5 = 3 + 4? The Staircase Paradox. Spot The Mistake 'Disproving' The Pythagorean Theorem

The Pythagorean Theorem says the square of the hypotenuse is equal to the sum of the squares of the sides. But this video says the hypotenuse is simply the sum of the sides. How is that possible? Can you figure out the mistake in the logic?

Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.

My Books

[jon]

No matter how small it is, the staircase will never become a straight line.

[gameguy]

It's not a paradox, even if there are infinitely many steps then there is an infinitesimal amount of distance saved through the diagonal of each tiny step, and if each of these units are added up infinitely we will get the desired result due to the fact infinite values don't have defined values, but a good question to ask people to see how they think.

[Hikarizu]

The Pi is 4 paradox works the same way. You take a square with a side of 2. Inside of it is a circle with a radius of one.(same center). Each corner of the square is like a one step staircase. You add more steps to make the staircase follow the curve of the circle. Eventually you end up with infinite small steps in the shape of a circle with that have the length 8. So 2*r*pi=8, but r=1. That makes 2*pi=8 and pi=4.

[chinareds]

What paradox? Obviously 5 = 7. Duh.

[JuniorNNL]

In the stair situation you're calculating the sum of the horizontal and vertical lenghts of the stair, which is the sum of the legs value. In the 3-4-5 triangle, you found the shortest distance between two points, which is a straight line. Not a paradox.

[dmacful]

Now, instead of calling it a staircase, we rename it a coastline... hmmmm

[NIKLASSOLVBERGOVERBY]

It's not a paradox. By making infinitely many steps, you're closing in on the triangle's area, not the length of the hypothenus.

[ariel_haymarket]

The paradox is easily resolved in the same way that constantly removing corners from a starting square that approaches a circle never makes it a circle and doesn't make pi = 4. Despite what it may appear, the more we zoom in, we will still see that it is not perfectly smooth (or a curve in the latter).

[uplink-on-yt]

For large enough values of 5, and/or small enough values of 7, this works out just fine.

[FLS]

It's the geometry that matters. A taxicab metric will always give 7 as the shortest path, but an euclidean one will give 5.

Reminds me of the coastline paradox.

[manfredwitzany]

Easy way to solve the puzzle:
Draw the straight line, which represents the limit to an infinite amount of stairs. Then draw one staircase strictly above and one strictly below this line. For both staircases you can calculate the total length as a function of stairs. You will see that both functions do not converge to the same value. Therefore this limit will not represent the value for the length of the line. If this would not be the case, we would have another more severe paradoxon: If you have a rectangle and draw lines from one edge to the opposite edge with the rule that the line should be monotonously rising, then all of these lines would have the same length, namely the sum of both sides of the rectangle. Hopefully this is not the case.

[NoelVerhoevenGplus]

The'paradox' lies in the definition of the word 'length', which in all three cases actually means the length of the path you follow to get from top left to bottom right. In all three cases it was calculated correctly, and results in the distance traveled according to the staircase drawn. Of course these are all different lengths.

[alexatg]

That's the total horizontal distance+vertical distance, not the distance between two pt

[vinodkumar-wmoq]

See children, I've got a new trick for you!
I have here a piece of rope. Now I'm gonna keep folding it until it
Children- woaaaa!!!!

[xaxi]

This reminds me of the coastline paradox. The paradox occurs because of fractals. Even if infinitely small, the shape of the staircase is still preserved, which makes it different from a straight line. This is could also be related to the butterfly effect (even if not related too much to the chaos theory) since little things such as infinitely small steps on a staircase increase drastically it's total length.

[thatweirdphoneguystickman]

It's because the angles are different. In the steps, you are just dividing the 3 and the 4 into proportions that equals 3 and 4 respectively, while with the slope, it is a straight line, meaning it would have a shorter distance, and would not be divided evenly because it does not have right angles. If the steps were awkward and sloped at the end instead of being right angles, the value would still be less than 7, because you aren't dividing into even fractions.

[hjdbr]

when I asked why such paradoxes exist to my professor in University, he gave me the simplest explanation I could have received: length is not a continuous property. If one shape approaches another, its perimeter will not necessarily approach the other shape's perimeter. You can easily check that using the arc length formula. The area, on the other hand, is continuous and when a shape approaches another its area will always approach the other one's area too.

[mayankmotwani]

No matter how small it is
Perp. and base can't merge into the hypotenuse

[sen]

The problem is only in the definition of "length of the staircase".
What do you mean by that? Is it the sum of the downward and horizontal components? Then it is 7 always. If it is the distance between the initial and the final positions then it is always 5.
No matter whether it is a wedge or a staircase.

[mrkaw]

I found this psuedo-paradox when trying to figure out how to draw a diagonal straight line on a computer when you can't subdivide pixels. In drawing a circle and counting the pixels you actual come up with a 'value' for pi that is equal to 4.