Video Transcript
In an experiment, a quantity 𝑥 is found to have a value of four plus or minus 0.1. What is the uncertainty in 𝑥 squared?
We’re told that the value of this quantity 𝑥 is four plus or minus 0.1. This number 0.1 is considered the uncertainty in 𝑥. Our question asks us to solve for the uncertainty in 𝑥 squared. If we write out 𝑥 times 𝑥, it would look like this. We can see that 𝑥 squared is equal to four times four, or 16. But there’s also an uncertainty associated with this result.
To find out what that is, let’s note that, in general, if we have two numbers — we’ll call them 𝑎 and 𝑏 — with associated uncertainties 𝜎 𝑎 and 𝜎 𝑏 and if a third number 𝑐 is equal to the product of 𝑎 and 𝑏, then the uncertainty in that product — we’ll call it 𝜎 𝑐 — is equal to 𝑏 times the uncertainty of 𝑎 plus 𝑎 times the uncertainty of 𝑏. When we consider multiplying 𝑥 by itself, we could think of the first four here as 𝑎 and the first 0.1 as the uncertainty in 𝑎 and then the second four as 𝑏 and the second 0.1 as 𝜎 sub 𝑏.
According to our equation then, the uncertainty in 𝑥 squared is 𝑏 — that’s four — times 𝜎 sub 𝑎, that’s 0.1, plus 𝑎 — that’s also four — times 𝜎 sub 𝑏, 0.1. Four times 0.1 is equal to 0.4. And if we add 0.4 to 0.4, we get 0.8. This is the overall uncertainty in our product 𝑥 squared.
