Multiplication - Wilfrid Laurier University Physics Labs
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Calculations with Uncertainties Recap Uncertainty Calculations - Multiplication Wilfrid Laurier University Terry Sturtevant Wilfrid Laurier University May 9, 2013 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Calculations with uncertainties Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Calculations with uncertainties When quantities with uncertainties are combined, the results have uncertainties as well. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Calculations with uncertainties When quantities with uncertainties are combined, the results have uncertainties as well. Following is a discussion of multiplication. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Calculations with uncertainties When quantities with uncertainties are combined, the results have uncertainties as well. Following is a discussion of multiplication. For the following examples, the values of x = 2 ± 1 and y = 32.0 ± 0.2 will be used. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant Multiplication by a constant with uncertainties Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. (A constant is a number with no uncertainty.) Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. (A constant is a number with no uncertainty.) What happens to the uncertainty? Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. (A constant is a number with no uncertainty.) What happens to the uncertainty? x =2±1 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. (A constant is a number with no uncertainty.) What happens to the uncertainty? x =2±1 → 4x can be as small as 4 × 1 = 4 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. (A constant is a number with no uncertainty.) What happens to the uncertainty? x =2±1 → 4x can be as small as 4 × 1 = 4 since 2 − 1 = 1 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. (A constant is a number with no uncertainty.) What happens to the uncertainty? x =2±1 → 4x can be as small as 4 × 1 = 4 since 2 − 1 = 1 → 4x can be as big as 4 × 3 = 12 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication by a constant - Example Suppose we have a number with an uncertainty, and we multiply it by a constant. (A constant is a number with no uncertainty.) What happens to the uncertainty? x =2±1 → 4x can be as small as 4 × 1 = 4 since 2 − 1 = 1 → 4x can be as big as 4 × 3 = 12 since 2 + 1 = 3 so 4x = 8 ± 4 = (4 × 2) ± (4 × 1) Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, 6 x The nominal value of x is here. (i.e. the value without considering uncertainties) Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, 6 x If we multiply by 1/2, both x and ∆x get smaller. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, So in general, ∆ (C x) = C ∆x Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, So in general, ∆ (C x) = C ∆x When multiplying by a constant, we multiply the uncertainty by the constant as well. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties What if both numbers have uncertainties? Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, z = (x = 2 ± 1) × (y = 32.0 ± 0.2) Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, z = (x = 2 ± 1) × (y = 32.0 ± 0.2) → z can be as small as 1 × 31.8 = 31.8 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, z = (x = 2 ± 1) × (y = 32.0 ± 0.2) → z can be as small as 1 × 31.8 = 31.8 since x can be as small as 1 and y can be as small as 31.8 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, z = (x = 2 ± 1) × (y = 32.0 ± 0.2) → z can be as small as 1 × 31.8 = 31.8 since x can be as small as 1 and y can be as small as 31.8 → z can be as big as 3 × 32.2 = 96.6 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, z = (x = 2 ± 1) × (y = 32.0 ± 0.2) → z can be as small as 1 × 31.8 = 31.8 since x can be as small as 1 and y can be as small as 31.8 → z can be as big as 3 × 32.2 = 96.6 since x can be as big as 3 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, z = (x = 2 ± 1) × (y = 32.0 ± 0.2) → z can be as small as 1 × 31.8 = 31.8 since x can be as small as 1 and y can be as small as 31.8 → z can be as big as 3 × 32.2 = 96.6 since x can be as big as 3 and y can be as big as 32.2 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties - Example If we multiply these numbers, z = (x = 2 ± 1) × (y = 32.0 ± 0.2) → z can be as small as 1 × 31.8 = 31.8 since x can be as small as 1 and y can be as small as 31.8 → z can be as big as 3 × 32.2 = 96.6 since x can be as big as 3 and y can be as big as 32.2 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 z can be as big as 3 × 32.2 = 96.6 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 z can be as big as 3 × 32.2 = 96.6 The nominal value of z is Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 z can be as big as 3 × 32.2 = 96.6 The nominal value of z is z = 2 × 32.0 = 64.0 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 z can be as big as 3 × 32.2 = 96.6 The nominal value of z is z = 2 × 32.0 = 64.0 So we can say z ≈ 64.0 ± 32.4 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 z can be as big as 3 × 32.2 = 96.6 The nominal value of z is z = 2 × 32.0 = 64.0 So we can say z ≈ 64.0 ± 32.4 and we see that ∆z ≈ 32.4 = Terry Sturtevant 1 2 + 0.2 32.0 64.0 = ∆x x + ∆y y z Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 z can be as big as 3 × 32.2 = 96.6 The nominal value of z is z = 2 × 32.0 = 64.0 So we can say z ≈ 64.0 ± 32.4 0.2 and we see that ∆z ≈ 32.4 = 12 + 32.0 64.0 = ∆x x + ∆y So in general, ∆ (xy ) = xy ∆x x + y Terry Sturtevant ∆y y z Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To summarize, z can be as small as 1 × 31.8 = 31.8 z can be as big as 3 × 32.2 = 96.6 The nominal value of z is z = 2 × 32.0 = 64.0 So we can say z ≈ 64.0 ± 32.4 0.2 and we see that ∆z ≈ 32.4 = 12 + 32.0 64.0 = ∆x x + ∆y So in general, ∆ (xy ) = xy ∆x x + y ∆y y z When multiplying numbers, we add proportional uncertainties. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To show this graphically, remember that the product of two numbers is the area of a rectangle with sides equal to the two lengths. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To show this graphically, remember that the product of two numbers is the area of a rectangle with sides equal to the two lengths. For illustration purposes, we’ll only show uncertainties in one direction. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties To show this graphically, remember that the product of two numbers is the area of a rectangle with sides equal to the two lengths. For illustration purposes, we’ll only show uncertainties in one direction. Just remember that uncertainties can be in either direction. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, x 6 This is the nominal value of x Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, x ∆x The maximum value of x includes ∆x. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, y This is the nominal value of y . Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, ∆y y The maximum value of y includes ∆y . Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, xy This is the nominal value of the area; i.e. xy . Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, max(xy ) This is the maximum value of the area; i.e. (y + ∆y ). Terry Sturtevant (x + ∆x) × Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, This is the difference between the nominal value of the area and the maximum value of the area. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, ∆y x This part of the difference has a size of x∆y Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, ∆x y This part of the difference has a size of y ∆x Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, ∆x ∆y This part of the difference has a size of ∆x∆y Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, ∆x ∆y Because this is relatively small, we’ll ignore it. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties Graphically, This is approximately the difference, and has a size of y ∆x + x∆y Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, If the uncertainty in xy is approximately equal to y ∆x + x∆y , or Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, If the uncertainty in xy is approximately equal to y ∆x + x∆y , or ∆ (xy ) ≈ y ∆x + x∆y Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, If the uncertainty in xy is approximately equal to y ∆x + x∆y , or ∆ (xy ) ≈ y ∆x + x∆y We can multiply both the top and bottom of the right side by xy so we get Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, If the uncertainty in xy is approximately equal to y ∆x + x∆y , or ∆ (xy ) ≈ y ∆x + x∆y We can multiply both the top and bottom of the right side by xy so we get ∆ (xy ) ≈ xy xy (y ∆x + x∆y ) Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, If the uncertainty in xy is approximately equal to y ∆x + x∆y , or ∆ (xy ) ≈ y ∆x + x∆y We can multiply both the top and bottom of the right side by xy so we get ∆ (xy ) ≈ xy xy (y ∆x + x∆y ) which becomes Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, If the uncertainty in xy is approximately equal to y ∆x + x∆y , or ∆ (xy ) ≈ y ∆x + x∆y We can multiply both the top and bottom of the right side by xy so we get ∆ (xy ) ≈ xy xy (y ∆x + x∆y ) which becomes ∆ (xy ) ≈ (xy ) ∆x x + ∆y y Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Multiplication by a constant Multiplication with Multiple Uncertainties Multiplication with Multiple Uncertainties So finally, If the uncertainty in xy is approximately equal to y ∆x + x∆y , or ∆ (xy ) ≈ y ∆x + x∆y We can multiply both the top and bottom of the right side by xy so we get ∆ (xy ) ≈ xy xy (y ∆x + x∆y ) which becomes ∆ (xy ) ≈ (xy ) ∆x x + ∆y y Remember that if x or y can be negative, we’ll need absolute value signs around the appropriate terms, since uncertainty contributions should always be given as positive numbers. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Recap Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Recap 1. When multiplying by a constant, the uncertainty gets multiplied by the constant as well. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Recap 1. When multiplying by a constant, the uncertainty gets multiplied by the constant as well. 4 × (2 ± 1) = (4 × 2) ± (4 × 1) = 8 ± 4 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Recap 1. When multiplying by a constant, the uncertainty gets multiplied by the constant as well. 4 × (2 ± 1) = (4 × 2) ± (4 × 1) = 8 ± 4 2. When multiplying numbers, we add the proportional uncertainties. Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Recap 1. When multiplying by a constant, the uncertainty gets multiplied by the constant as well. 4 × (2 ± 1) = (4 × 2) ± (4 × 1) = 8 ± 4 2. When multiplying numbers, we add the proportional uncertainties. 1 0.2 (2 ± 1) × (32.0 ± 0.2) = (2 × 32.0) ± (2 × 32.0) + 2 32.0 = 64.0 ± 64.0 (0.5 + 0.00625) = 64.0 ± 32.4 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Recap - continued Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi Calculations with Uncertainties Recap Recap - continued 3. Uncertainties in final results are usually expressed to one significant figure, so the above result becomes 64.0 ± 32.4 = 60 ± 30 Terry Sturtevant Uncertainty Calculations - Multiplication Wilfrid Laurier Universi
