Square root operation and scales

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SUMMARY

The discussion centers on the intuitive understanding of the square root operation, particularly when applied to a transformed scale. The user describes a scenario where a number line is stretched by a factor of 8, resulting in the square root of 64 being correctly identified as 8. However, upon compressing the scale back to its original size, the position of 8 shifts to approximately 2.828. This illustrates the non-linear nature of the square root function, contrasting with the linear transformation applied to the scale.

PREREQUISITES
  • Understanding of basic number line concepts
  • Familiarity with square root operations
  • Knowledge of linear transformations
  • Graphing functions, specifically the graph of y = sqrt(x)
NEXT STEPS
  • Explore the properties of non-linear functions, focusing on the square root function
  • Study linear transformations and their effects on number lines
  • Investigate the geometric interpretation of square roots
  • Learn about function scaling and its implications in mathematical operations
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Mathematicians, educators, students studying algebra, and anyone interested in the geometric interpretation of mathematical operations.

IsrTor
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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.
 
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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 >.<
 

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