- #1
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- TL;DR Summary
- Two rules give different answers for same expression
Dear all,
We learn students to work with significant figures. I guess we all know those rules of thumb. Now imagine I have an expression like L = 2x, where x is a length a the 2 is an amount, i.e. the value of a discrete 'function'. If x=0,72 m (2 significant figures), then L = 1,44 --> 1,4 (2 significant figures); we view the 2 as exact.
But if I write L=x+x, then L = 0,72+0,72=1,44, and I use the rule of thumb that I look at the same amount of decimals. What exactly is the origin of these contradicting accuracies?
Within error analysis, I can define a function z=f(x,y) and calculate the error dz as a function of dx and dy,
[tex]
dz = \sqrt{(\frac{\partial f}{\partial x} dx)^2 + (\frac{\partial f}{\partial y}dy)^2 }
[/tex]
If z(x,y) = x+y, then
[tex]
dz = \sqrt{(dx)^2 + (dy)^2}
[/tex]
and if z(x,y)=x*y, then
[tex]
dz = |x*y|\sqrt{(\frac{dx}{x})^2 + (\frac{dy}{y})^2}
[/tex]
If I take y=x in z(x,y) = x+y, I obtain z=x+x and hence
[tex]
dz = \sqrt{2}|dx|
[/tex]
while if I take y=2 (dy=0) in z(x,y) = x*y, I obtain z=2x and hence
[tex]
dz = 2|dx| > \sqrt{2}|dx|
[/tex]
Do these two different errors for the same mathematical expression make sense? Does it have it have to do something with different meanings to the expressions z=2x and z=x+x and errors canceling each other out such that the first error is smaller than the second? And is the answer to this question also the answer for my first question about significant numbers and those rules of thumb? Any references to this? Many thanks! :)
We learn students to work with significant figures. I guess we all know those rules of thumb. Now imagine I have an expression like L = 2x, where x is a length a the 2 is an amount, i.e. the value of a discrete 'function'. If x=0,72 m (2 significant figures), then L = 1,44 --> 1,4 (2 significant figures); we view the 2 as exact.
But if I write L=x+x, then L = 0,72+0,72=1,44, and I use the rule of thumb that I look at the same amount of decimals. What exactly is the origin of these contradicting accuracies?
Within error analysis, I can define a function z=f(x,y) and calculate the error dz as a function of dx and dy,
[tex]
dz = \sqrt{(\frac{\partial f}{\partial x} dx)^2 + (\frac{\partial f}{\partial y}dy)^2 }
[/tex]
If z(x,y) = x+y, then
[tex]
dz = \sqrt{(dx)^2 + (dy)^2}
[/tex]
and if z(x,y)=x*y, then
[tex]
dz = |x*y|\sqrt{(\frac{dx}{x})^2 + (\frac{dy}{y})^2}
[/tex]
If I take y=x in z(x,y) = x+y, I obtain z=x+x and hence
[tex]
dz = \sqrt{2}|dx|
[/tex]
while if I take y=2 (dy=0) in z(x,y) = x*y, I obtain z=2x and hence
[tex]
dz = 2|dx| > \sqrt{2}|dx|
[/tex]
Do these two different errors for the same mathematical expression make sense? Does it have it have to do something with different meanings to the expressions z=2x and z=x+x and errors canceling each other out such that the first error is smaller than the second? And is the answer to this question also the answer for my first question about significant numbers and those rules of thumb? Any references to this? Many thanks! :)