# Class 3.2: Orders of magnitude

# Methodology of Scientific Research

## Andrés Aravena, PhD

### March 3, 2022

## 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