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If we estimate a square root using the so-called Babylonian method, the result is always overestimated and the reason obvious: we are ignoring the quadratic component of the solution. However, if we approximate a square root use a quick linear interpolation, the result is always underestimated. For example, approximating √55 would give us 111/15 = 7.4 when the true value is 7.416... (The Babylonian method would give 7.428...) As one commentator alluded to, this is because the root function is half-parabolic, but that really only explains the what. (The parabola is a visual representation of the function.)

I am trying to have a better intuition of the mathematical nature of the function, perhaps in a geometric sense. An equivalent question might be why the same operation applied to a square would underestimate the y.

If we estimate a square root using the so-called Babylonian method, the result is always overestimated and the reason obvious: we are ignoring the quadratic component of the solution. However, if we approximate a square root use a quick linear interpolation, the result is always underestimated. For example, approximating √55 would give us 111/15 = 7.4 when the true value is 7.416... (The Babylonian method would give 7.428...) As one commentator alluded to, this is because the root function is half-parabolic, but that really only explains the what. (The parabola is a visual representation of the function.)

I am trying to have a better intuition as to why the function is concave curvilinear rather than linear. I know that it is, but I thought that for a simple function like the square root, there might be a geometric intuition as to why it is. An equivalent question might be why the same operation applied to a square would underestimate the y.