Methodology of Scientific Research

It is hard to see small changes

It is easy to see the difference between 10 and 15 things

It is hard to see the difference between 1000 and 1005 things

It is very hard to see the difference between 1000000 and 1000005 things

We observe proportions, not quantities

It is easy to see the difference between 10 and 15 things

It is easy to see the difference between 1000 and 1500 things

It is easy to see the difference between 1000000 and 1500000 things

“Difference” means “minus”

The expression $1005 - 1000$ is called the difference between 1005 and 1000

“Ratio” means “divided by”

The expression $\frac{1005}{1000}$ is called the ratio between 1005 and 1000

We observe ratios, not quantities

The value $\frac{1005}{1000}=1.005$ is small and hard to see, but $\frac{1500}{1000}=\frac{1500000}{1000000}=\frac{15}{10}=1.5$ is easy to see

Powers of 10

In a first approach, we approximate measurements by powers of 10

(after choosing some suitable units)

We even have names for some of them them:

deci, centi, milli, micro, nano, pico

deca, hecto, kilo, mega, giga, tera, peta, exa

Order of magnitude

The order of magnitude of a value is its power of 10

More precisely, is the integer part of the logarithm base 10

Same order of magnitude

Most of times we care about the ratio between values

We say that two quantities are in the same order of magnitude if their ratio is between 0.1 and 10

Middle points

Instead of going $10ˆ{-1}, 10ˆ{0}, 10ˆ{1}, 10ˆ{2}, 10ˆ{3}$ we can increment the exponent by 0.5 $10ˆ{-1}, 10ˆ{-0.5}, 10ˆ{0}, 10ˆ{0.5}, 10ˆ{1}$

Since $$10ˆ{0.5} = \sqrt{10}≈ 3.16…$$ we can say $0.1, 0.3, 1, 3, 10, 30, 100,…$

Upper and lower bounds

When estimating a value, we usually can guess that the real value is somewhere between two values

In other words, we guess lower and upper bounds $$L$$ and $$U$$

The width of this interval is $$U/L$$

The center of this interval is $$\sqrt{LU}$$

(this is called geometric mean)

How to calculate without electricity

Today we do most of our computations with calculators

In old times people used mechanical computers

The most common ones were slide rules

Homework

Learn how to use the slide rule