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
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
The expression \[1005 - 1000\] is called the difference between 1005 and 1000
The expression \[\frac{1005}{1000}\] is called the ratio between 1005 and 1000
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
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
The order of magnitude of a value is its power of 10
More precisely, is the integer part of the logarithm base 10
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
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 \[0.1, 0.3, 1, 3, 10, 30, 100,…\]
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)
Today we do most of our computations with calculators
In old times people used mechanical computers
The most common ones were slide rules
Learn how to use the slide rule