Square root operation and scales

[IsrTor]

this is a strange problem easy to solve but I am having trouble understanding it intuitivly.

Assume we choose a location point and name it 0. Next, in arbitrary direction and distance we place the number 1. Hence, we have created a scale(number line) that extends as much as we like. Now we consider the square root operation(no need to state it). Suppose I choose a number that's hasn't a whole root... for instance let's choose 8.

Now consider this carefully: We take our scale and we stretch(by a factor of 8) so that the orgin remains unchanged but now what used to be 1 is 8 and what used to be 8 is now 64. Now if we take the square root of 64 we get 8. But here's the problem, we mark the place where 8 was the square root and we squeeze the scale back down to its original size only now the place we marked as 8 which was the square root is not at 1 but rather about 2.828

( I figured out that square root of 64 is 8 and square root of 8 when rounded gives 2.828)
But this problem is showing something about the square root operation on different scales(on a scale transformation). If someone could help to understand the intuition here I would greatly appreciate.

 

[queenofbabes]

Your stretching transformation is linear, but the sqrt function isn't (just look at the graph y = sqrt x)

I think I understand what you're asking, but I'm rubbish with words >.<

 

[Architeuthis Dux]

The square root operation is a mathematical operation that calculates the number which, when multiplied by itself, gives the original number. It is often represented by the symbol √. For example, the square root of 9 is 3 because 3 x 3 = 9.

In the context of scales, the square root operation can be thought of as a transformation of the scale itself. In the given scenario, we start with a scale that has a fixed point (the origin) and a set unit distance (1). This creates a linear scale or number line. When we apply the square root operation to a number on this scale, we are essentially stretching or compressing the scale by a certain factor to find the number that, when multiplied by itself, gives the original number.

In the case of the number 8, which does not have a whole square root, we stretch the scale by a factor of 8. This means that the unit distance on our scale is now 8 instead of 1. So, what used to be 1 on our original scale is now 8 on our stretched scale. When we take the square root of 64, which is 8, we are essentially finding the number that, when multiplied by itself, gives us 64 on our new scale. This number is 8, which is why we get the same result.

However, when we compress the scale back to its original size, the point that was marked as 8 (the square root of 64) is no longer at 1, but at 2.828. This is because our unit distance is now back to 1, so the number that, when multiplied by itself, gives us 64 is now 2.828.

The intuition here is that the square root operation is not just a mathematical calculation, but it also involves a transformation of the scale itself. This can be a bit confusing, but it helps to think of the square root as a relationship between numbers and their corresponding square roots, rather than just a single operation. This is why on different scales, the square root of a number may appear to be different, but the relationship between the number and its square root remains the same. I hope this helps to clarify the intuition behind this problem.