Physics equations/01-Introduction/A:percentErr

Suggests that big/small is a quick and dirty way to calculate uncertainty and error

Calculating percent difference
The difference between two numbers depends on units used a difference of 0.3 meters a difference of 30 cm.  But the percent difference is universal.The purpose of this exercise is to show that a quick way to compare two numbers is to divide the large number (BIG) by the small number (SMALL):

$$\frac{BIG}{SMALL}=1+\underbrace{\frac{BIG-SMALL}{SMALL}}_{percent\; difference}$$

a) Your boss doesn't believe that
 * $$\frac{BIG}{SMALL}=1+\frac{BIG-SMALL}{SMALL}$$

Use simple high school algebra to convince him/her in a way that reflects your understanding of mathematics, as well as your communication skills.

b)There are actually several ways to calculate percent difference between two numbers. Given any two numbers, x and y, let BIG denote the larger (assuming both are positive), SMALL denote the smaller, AVE denote the average, (x+y)/2. And finally, it is often the case that one of the two values is the correct, or TRUE value, while the other was measured, which we may denote as MEAS.  For example:
 * 9.8 is the TRUE value of g
 * 10.4 is the MEAS (measured) value of g
 * 10.4 is also BIG
 * 9.8 is also SMALL

Perform the calculation using these numbers to four significant digits and show that to excellent approximation,

$$\frac{MEAS-TRUE}{TRUE}\approx\frac{BIG-SMALL}{TRUE}\approx\frac{BIG-SMALL}{AVE}\approx\frac{BIG-SMALL}{SMALL}$$

Of these the first is always considered correct. But the last is very handy because it can be found by calculating $$BIG/SMALL$$ and subtracting one. On a calculator it is a single operation, $$\frac{104}{98}=1.0612$$, from which we conclude that there is a 6% difference.

c) Repeat the calculation using 9.8 and 11.4 (with 9.8 being true)

d) Repeat using 9.8 and 16.4 (with 9.8 being true)

e) What can you say about when it is necessary to carefully define how the percent difference between two numbers was calculated?

f) One advantage of the $$BIG/SMALL$$ method is that you can say that they differ by a factor of $$BIG/SMALL$$ when the ratio is uncomfortably large. You are writing a report and your conclusion in involves the numbers 2.8 and 1.5. In a one sentence conclusion to your report you want to compare these numbers using both a percent and a ratio. Finish this sentence:


 * To appreciate the difference between 2.8 and 1.5, ...