Unit 1.6 Propagation of Errors (Multiplication and Division)
Multiplication and Division: z = x y
or
z = x/y
Derivation: We can derive the relation for multiplication easily. Take the largest values for x and
y, that is
z + Δz = (x + Δx)(y + Δy) = xy + x Δy + y Δx + Δx Δy
Usually Δx << x and Δy << y so that the last term is much smaller than the other terms and can be
neglected. Since z = xy,
Δz = y Δx + x Δy
which we write more compactly by forming the relative error, that is the ratio of Δz/z, namely
The same rule holds for multiplication, division, or combinations, namely add all the relative errors
to get the relative error in the result.
From now on, the GOOD method technique is always the same (no formulas to remember) so we
will omit it from the formula section.
BETTER
BEST
 x y

z  z 

 ... 
y
 x

2
Example: w = (5.42 ± 0.03) cm, x = (3.0 ± 0.2) cm. Find z = w x.
Solution:
z = w x = (5.42) (3.0) = 16.26 cm
2
GOOD
zmax = (5.45)(3.2)=17.44
zmin = (5.39)(2.8)=15.092
z-zmax=16.26-17.44=-1.18
z-zmin =16.26-15.092=1.168
We will uses z  1.2
2
2
 x   y 
z  z       ...
 x   y 
2
Therefore z=(16.3 ± 1.2) cm or z=(16 ± 1) cm is also acceptable
BETTER
 w x 
z  z 


x 
 w
 0.03 0.2 
 16.26 


 5.42 3.0 
 1.174
2
2
Therefore z=(16.3 ± 1.2) cm or z=(16 ± 1) cm is also acceptable
BEST
 w   x 
z  z 
  
 w   x 
2
2
2
 0.03   0.2 
 16.26 
 

 5.42   3.0 
 1.0877
2
2
2
Therefore z=(16.3 ± 1.1) cm or z=(16 ± 1) cm is also acceptable.
Example: w = (5.42 ± 0.03) cm, x = (3.0 ± 0.2) cm. Find
Solution:
z
w 5.42

 1.8067 cm2
x 3.0
GOOD
zmax = (5.45)/(2.8)=1.9464
zmin = (5.39)(2.8)=1.6844
z-zmax=16.26-17.44= -0.1397
z-zmin =16.26-15.092=0.1223
We will uses
z  0.1
Therefore z=(1.8 ± 0.1) cm
BETTER
 w x 
z  z 


x 
 w
 0.03 0.2 
 1.8067 


 5.42 3.0 
 0.1343
Therefore z=(1.8 ± 0.1) cm
2
2
z
w
.
x
BEST
 w   x 
z  z 
  
 w   x 
2
2
2
 0.03   0.2 
 1.8607 
 

 5.42   3.0 
 0.1245
2
Therefore z=(1.8 ± 0.1) cm .
2