Fun with Formulas !
Werner Joho
Paul Scherrer Institute (PSI)
CH5232 Villigen, Switzerland
4.July 2003
(updated 1.10.2010)
1
W.Joho
Introduction
Formulas can be fun. They often can be made to look simple,
transparent and thus beautiful (in the spirit of Einstein and
Chandrasekhar).
This can be achieved with some simple rules and a few tricks
of the trade.
This paper is a sample of some simple formulas, collected
during my career as a physicist.
The following material was presented (but not published)
at a seminar talk given at the CERN Accelerator School on
Synchrotron Radiation and Free Electron Lasers,
Brunnen, Switzerland, 2-9 July 2003
(this file is available on the WEB with google:
„JOHO PSI“)
2
W.Joho
Content
• philosophy for formulas
• capital growth
• new interpretation of Ohm‘s law
• logarithmic derivatives
• the relativistic equations of Einstein
• the magic triangle formed by the logarithmic
derivatives of the relativistic parameters
• Alternative Gradient Focusing, constructed by hand
• binomial curves everywhere,
approximation of a variety of functions, like
beam profiles, the fringe field of magnets,
the flux and brightness of synchrotron radiation etc.
• simple representation of phase space ellipses
• how to win money with statistics !
• design of beautiful tables with a Hamiltonian
3
W.Joho
philosophy for formulas
simplify formulas, they should look „beautiful“
formula should indicate the proper dimensions
use units of 1'000 (cm should not exist in formulas!)
choose right scales for plots (e.g. logarithmic)
example:
c = 3 ·108 m/s ?? better is:
c = 0.3·109 m/s or 300 m/ms or 0.3 mm/ps !!
for comparison of electric forces: q·Є (kV/mm)
with magnetic forces: q·v·B = q·β·c·B
c = 300 (kV/mm)/T !!
m0 = 4p 10-7 Vs/Am ?? better is:
m0 = 0.4p mH/m = 0.4p T/(kA/mm)
4
W.Joho
how
in in
electro-dynamics
how to
toavoid
avoidakward
akwardnumbers
numbers
electrodynamics
electron mass: m = 9.11·10-31 kg (?) => forget it !
use mc2  eUo ,
Uo = .511 MV
o = 8.854·10-12 As/Vm (?) => forget it !
use mo o = 1/c2
with mo = 0.4 p mH/m
introduce impedance Zo :
Zo 
1
m
= 0 c = 30 
4p 0 c 4p
( 29.9792458 )
Alfven current IA (used in space charge calculations):
IA = 4pomc3/e (?) => forget it ! (similarly for "perveance")
use instead “Ohms-law” :
IA = Uo/Zo = 511 kV/30 = 17 kA
charged particle in a magnetic field Bo
=> Larmor-frequency
electron:
o =
e
= 1.76·1011 C/kg (?)
m
e
c2
use
=
= 28 GHz/T
2p m 2p U 0
e B0
m
=> forget it !
(15.25 MHz/T for protons)
5
W.Joho
use logarithmic derivatives !
Example: Magnet Weight W of a Cyclotron:
W  W0 ( B  )3max
(W.Joho, Aarhus 1986, CERN Accelerator school)
=> for a plot : take logarithmic scale both for (B ) and W
=> take logarithm and then derivative
dW
d (B )
3
W
(B )
=> 1% change in (B ) max gives 3% change in weight
or
pˆ =   
dp d  d 


p


6
W.Joho
Einstein
Einstein triangle
triangle
in relativistic equations use dimensionless quantities for
velocity, energy and momentum on a democratic basis!
=> Einstein triangle and magic triangles for logarithmic derivatives
velocity:
v=c
total energy:
E =  E0 , E0 = mc2 = eU0 (0.511 MeV for electron)
 = (1-2)-1/2 ,
momentum:
p = ~p E0/c = ~p mc
~p   
“Pythagoras”- connection: E2 = E02 + (pc)2
give the Einstein triangle and derivatives
1  ~p 2   2
~p d~p   d
1
  2 1

d   3d
2
7
W.Joho
„Magic Triangles”
(W.Joho)
with logarithmic derivatives of
relativistic parameter
 ,  , ~p  
d~
p d d


~
p


d~
p
2 d

,
~
p

d

2 d
~
p
,

d
~
d
p
2
 ~

p
,  , ~
p are treated equally  democracy!
d

2
d~
p
~
p
~
p2

multiplication
factors form
inverse triangle
d
2

8
W.Joho
trigonometric functions for
relativistic formula !
 
1
(1   2 )
1
if   sin  , then  
cos 
~
p    tan  (" momentum" )
2) mildly relativistic case: use angle 
the following table gives some easy reference values for
quick estimates or for calculations with a pocket calculator!
(with use of Pythagoras triangle with sides 3,4,5)

 = sin
cos
300
36.90
450
53.10
600
0.5
0.6
0.71
0.8
0.87
0.87
0.8
0.71
0.6
0.5
9
= 1/cos p = 
= tan
1.15
0.58
1.25
0.75
1.41
1
1.67
1.33
2
1.71
W.Joho
highly relativistic case
highly relativistic case
 >> 1 ,   1
=> use angle 
  cos  1 – 2 / 2
1  
1/  sin  
1
2 2
~p   = 1/tan  1/
a) race to the moon between electron and photon:
electron "looses" by
L = (1 - ) L 
1
L
2 2
SLS:
ESRF:
 L
50m
E 2 [GeV ]
2.4 GeV, L = 8 m
6 GeV, L = 1.4 m
LEP II: 100 GeV, L = 5 mm
b) race over one undulator period u : if electron is just one or n
wavelengths  behind photon (slippage) => positive interference
L =  =
1
2n 2
u (1 + K2 / 2)
K = 0.0934 B[T] u [mm]
detour due to slalom in B-field !
10
W.Joho
Undulator Radiation
produced by an electron beam
of energy E = mc2
K-edges
* = 2‘500 mm
*
 2

ESRF
SLS
* = 1.5 mm
u 
2
K
*
2 2
  1 
    (property of undulator)
2n 
2

11
W.Joho
AG-focusing
simple example of alternative gradient focusing:
 FODO-lattice with thin lenses (focal length f)
if L = 2f => construction is possible by hand !
it takes 6 periods to get a 3600-oscillation
i.e. the phase advance/period is  = 600
exact solution with transfer matrices gives

L
sin 
2 4f
for L = 2f =>  = 600 (graphic example)
for L = 4f =>  = 1800 (instability !)
12
W.Joho
magnetic fringe field with binomial
1
x
B( x) 
, u
,
N 1/ S
(1  u )
xL
xL  gap , N  7 , S  3 ,
 3 free parameter
origin of
for
fit
x  at x (80% field )  gap
inverse :
1
u  ( S  1)1/ N
B
magnet edge
1.0
B
0.8
N=7, S=-3
0.6
0.4
0.2
0.0
0
1
2
13
3
u
4
W.Joho
Flux Spectrum of
synchrotron radiation
spectral flux F of electrons with energy E and current I
from a bending magnet with magnetic field B.
F = 2.46 ·1013 E[GeV] I[A] G1(x)
(photons/(s ·mrad ·0.1% bandwidth)
x=ε/εc , ε = photon energy , εc = critical photon energy
G1 x = x

x
K5 3 x dx
1
G1
~ 2.1x
1
3
0.1
c eV = 665 E 2 GeV B T
50%
0.01
~ 1.3 x e – x
0.001
0.001
0.01
0.1
x = c x
14
1
10
W.Joho
Flux-Spectrum of Synchrotron Radiation
from Bending Magnet with Field B
G1 x = x

x
K5 3 x dx
1
x
g ( x )  [(1  ( ) N ] S
xL
Fit with G1(x) = A x1/3 g(x) ,
fit of binomial g(x) with 8 data points to ±1.5%:
A = 2.11 , N = 0.848 , xL = 28.17 , S = 0.0513
G1 = normalized Flux
1
G1
0.1
0.01
0.001
0.001
0.01
0.1
15
1
x
10
W.Joho
general binomial curves
general binomial curves
F(x) = Fmax y(u) ,
u  x/xL , y(0) = 1
3 general cases :
short range : s > 0
inverse
(interchange n and s !):
y  (1  u n )1/ s
,
(0  u  1)
u  (1  y s )1/ n
,
(0  y  1)
long range : s< 0
inverse :
y  (1  u n )1/ s
,
(0  u  )
u  ( y s  1)1/ n
,
(0  y  1)
exponentials : (limit s = 0)
inverse :
y  exp( u n )
,
(0  u  )
u  ( ln y )1/ n
,
(0  y  1)
16
W.Joho
Classification of binomial curves
reference points for binomial curve
Y
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.841
top
mid
0.0625
0
bottom
0.5
1
Xtop
1.5
2
Xmid
2.5
Xbottom
3
X
1
x
binomial : y ( x)  [1  sign ( s) ( ) N ] S
xL
any 3 reference points will give a fit for
N, S and xL. But the chosen points
top, mid and bottom allow a convenient
classification in the (A,B)-Diagram
1.0
square
B
Classifica tion of binomials
in (A, B) - Diagram
short range (s>0)
y top  0.51/4  0.841
Gaussian
0.5
y mid  0.5
y bottom  0.54  0.0625
long range (s<0)
exponentials (s=>0)
0.0
0
0.5
A
1
17
A
x top
x mid
, B
x mid
x bottom
W.Joho
properties
of of
binomials
properties
binomials
1. y(u) = (1  un)1/s is monotonically decreasing, but
transformations like F(x) = A x y(u)
allow representations of functions which are not
monotonic (e.g. Flux- or Brightness- curves)
2. inverse function exists : y  u , n  s
3. 4 free parameter:
1. Fmax gives scaling in y
2. xL gives scaling in x , (u=1)
3. n determines flatness at x = 0
4. s determines tail at large x
4. Fit to data using the 4 parameter with programs like
MATLAB”, “IGOR” or
“Excel” (insert , name , define) + (extras , solver)
5. (A , B) – diagram allows rough estimates
of parameter n and s
binomials.doc
18
W.Joho
typical profiles y(u) in (A,B)-plot
1.0
13 =square
8
7
12
4
B
6
21
5
Gaussian
3
0.5
11
2
1
9
0.0
14
B=A4
16
0
0.5
1 y  1 u
13
2
y  (1  u ) 2
parabola, convex
3
y  1 u
triangle
4
y  1 u
5
y  (1  u 2 ) 2 biquadrati c
6
y  1 u2
parabola, concave
7
y  1 u2
quarter circle
8
y  1 u4
9
10
y  e
11 y  e u
12
2
y  e u
A
1
y 1
square
1
1 u
1
15 y 
(1  u ) 2
1
16 y 
1 u2
1
17 y 
Lorentzian
1 u2
1
18 y 
bi - Lorentzian
(1  u 2 ) 2
1
19 y 
1 u3
1
20 y 
magnetic fringe field
(1  u 7 )1/ 3
1
21 y 
1  u12
14
u
y  e u
20
17
10
15
19
18
exponentia l decay
Gaussian
6
19
y
W.Joho
representations
beam
profiles
binomials
Representation ofofbeam
profiles
withwith
binomials
1) Gaussian:
y  e1/ 2 ( x / s )
2
2) Binomials :
x
y  [(1  ( ) 2 ]m 1 / 2 , x  xL , xL  2(m  1) s
xL
clipped tails at xL (e.g. m=3.5 => xL= 3s ; m=7 => xL= 4s)
1.0
y
binom m=7
0.8
Gauss
binom m=3.5
0.6
Profiles
0.4
0.2
0.0
-4
-3
-2
-1
0
1
2
x/s
3
4
0.20
y
binom m=7
Tails of Profiles
Gauss
0.15
binom m=3.5
(full width at 10% level :
≈ 4.4 σ for large range of m)
0.10
0.05
0.00
1.5
2
2.5
3
3.5
x/s
20
W.Joho
clipped binomial phase space densities
 ( x, x ' )  (1  a 2 ) m 1
(a  u  v  1 ,
2
2
2
projected profile :
x
u
,
xL
x'
v ' )
xL
y ( x)  (1  u 2 ) m 1/ 2
we get again a binomial with the exponent reduced by 1/2
big trick:
plot fraction
which is outside
of ellipse!
Sacherer,
Lapostolle
εp/ε
for m1.5 the curves have a crossing point at εp ≈ ε and p
≈ 13%; i.e. ca. 87% of all particles are inside an ellipse
with emittance ε=(2σ)·(2σ‘) , independent of m.
For a Gaussian distribution we have p=exp(-2εp/ε),
which gives a straight line in this diagram (m=).
21
W.Joho, 1980
PSI report TM 11-4
W.Joho
correlations x  y
example:
income and research for 50 US companies in 1976
(from journal „Physics Today“, march and september 1978 )
x = income / sales
y = research budget / sales
There are 3 possibilities to show a correlation:
1. linear fit of y(x) : income stimulates research !
2. linear fit of x(y) : research stimulates income !
3. correlation ellipse from <x y> : high income  strong research
Y=
fit: y(x)
fit: x(y)
X=
22
W.Joho
Representationofofrms-beam
rms beam
ellipse
Representation
ellipse
in phase space
in
space (x,
(x, x)
x‘ )
For an arbitrary distribution of particles define the statistical values:
=> shift first origin such that <x> = <x> = 0
<x2>  s2 , <x2>  s2 , <x x>
the rms-emittance  is then defined as:
  <x2><x2> - <x x>2
the traditional representation of the rms-ellipse is given with
the Courant-Snyder invariant :
 x2 + 2 x x +  x2 = 
the parameter    are defined by
 = s2/ ,  = s2/ ,  = - <x x>/
     
with the relation
the graph representing this ellipse is akward to remember
and not easy to plot!
23
W.Joho
The parametric representation of the
rms beam ellipse in phase space (x, x‘)
 use parametric representation of the rms-ellipse
x = s cos 
(0
x = s sin ( +)



2p =
"running parameter" )
but what is phase shift  ?
sin   r12   xx' 
ss '
= correlation parameter
(in a drift,  is just the phase advance  from the waist position)
this representation of the rms-ellipse is easy to remember
and easy to plot! ( = 0 gives a circle)
s sin
x
s
scos
ssin
3
2
4
x
1
s
s cos
24
W.Joho
Dictionary for Beam Parameters
Some useful quantities are easy to guess from the factors
sin or cos ( is 0 at a waist!) and dimensional arguments
(using m, mm and mrad)
emittance:
 = ss cos [mm mrad]
slope of envellope:
ds/ds = s sin
[mrad]
virtual waist size:
xw = s cos
-function at virtual waist:
min = (s/s) cos
[mm]
distance from virtual waist: Lw = (s/s) sin
[m]
[m]
= min tan
phase advance from virtual waist:  = 
the dictionary between the 2 representations is:
 = - tan  = -  xx' 

( = - x3/x1 = -x2/x4 )
2
s
=s =

s ' cos 
( = x2/x4 )
'2
s'
=s =

s cos 
( = x3/x1 )
[1]
[m]
[m-1]
(as a check :  = 1/cos2 = 1 + tan2 = 1 + 2 )
25
W.Joho
Convolution of two ellipses
Example: convolution of the electron beam ellipse (x1, x1),
with parameter σ1,σ1’,1 and the diffraction limited photon
beam (x2, x2), with parameter σ2,σ2’,2 from an undulator.
Simple recipe:
add variances and correlations linearly
to form the combined ellipse (X,X’) with parameter , ’, 
<X2> = <x12> + <x22>
<X2> = <x12> + <x22>
<XX> = <x1x1> + <x2x2>
or
2 = s12 + s22
2 = s12 + s22
  sin = s1s1 sin1 + s2s2 sin2
the convoluted emittance is
 =  cos 
(  1 + 2)
with the dictionary one can, if necessary, transform these
values back to the Courand-Snyder values , ,  .
26
W.Joho
e
ip
 1
This formula from Euler combines beautifully
3 fundamental numbers in mathematics
another „gem“ from Euler is:
3 4 5 6
3
3
3
3
or from Ramanujan comes:
9  10  1  12
3
3
3
3
in the the same spirit one can write:
13  23  33  ... n3  (1  2  3  ... n)2
(n  1) 2 n 2
4
[
(n  1)n 2
]
2
(I figured this out myself, but I am sure it exists somewhere in the
literature, but I could not find yet the proper reference)
27
W.Joho
treacherous predictions !
1) If you see a series of numbers: 2, 4, 6, 8, 10, 12, …
created by a formula F(n), for n=1, 2, 3, …6
you probably guess, that the next term is 14 !?
Now give me the number Y, your year of birth.
I give you below a formula F(n), where the next term
in the series (for n=7) is not 14, but exactly Y !
F(n)=2n+ (Y-14)(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)/6!
-----------------------The above example was easy to construct.
But what about the next treacherous example?
2) The following formula was constructed by the Swiss
Physicist Leonard Euler:
P(n)=n(n-1)+41
Believe it or not, but for n=1, 2, 3….up to 40
this formula gives a prime number !
It fails the first time at n=41, where
P(41)=41*41=1‘681
(It then fails further at n=42, 45, 50, 57, 66 etc.)
28
W.Joho
winning money with statistics !
throw simultaneously 6 dices:
If all 6 dices show different
numbers (1, 2, 3, 4, 5, 6)
 I give you 20 times your
betting sum
…..but chance is only 1.5% !!
to be fair, I should offer you
65 times your betting sum!
5 4 3 2 1
5!
5
(p=     = 5 =
= 1.54% )
6 6 6 6 6
6
324
29
W.Joho
winning money with statistics !
what is the chance, that in a group of n persons, 2
people have the same birthday (disregarding the year
of birth and the 29th february)?
364 363 362
(366 n)
probability w  1 



365 365 365
365
with 23 people the chance is already 50%,
with 40 people it is 87% and with 80 people a
double coincidence is a „sure bet“ and a
triple coincidence has a 42% chance!
Chance for double or triple birthday
1.0
0.9
0.8
double or more
Chance [%]
0.7
0.6
triple or more
0.5
0.4
0.3
0.2
0.1
0.0
0
10
20
30
40
50
60
70
80
Persons
30
W.Joho
exponential growth
with compound interest
With an interest rate of 2% it takes 35 years to double
the income (50 years without compound interest)
How can we get this result very quickly?
For a quick estimate of exponential growth one can use:
e7 ≈ 210 ≈ 103
example:
With an interest rate of p(%) it takes T2 years
to double an initial capital investment C0.
T2 = 70 years/p(%)
(70≈100 ln2)
To have an increase by a factor of 1‘000 (≈210)
it takes T1000 years:
T1000 = 10 T2 = 700 years/p(%)
31
W.Joho
Growth of Capital
William Tell deposited 1 Fr. in a bank account,
700 years ago!
=> assume he gets 3% interest before taxes,
and 2% netto after taxes
The difference goes to the government
(which does not pay taxes)!!
 after 700 years the government has 109 Fr.
the descendents of William Tell „only“ 106 Fr. !?
Growth of Capital
1,000,000,000
100,000,000
2% interest
10,000,000
3% interest
1,000,000
difference
100,000
value
10,000
1,000
100
10
1
0
0
100
200
300
400
500
600
700
year
32
W.Joho
„Hamiltonian Table“
5
You want to construct a nice table for
your living room with 4this shape?
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
-4
-5
33
W.Joho
„Hamiltonian“ Plots
Try to plot a curve F(x,y)=const.=c ,
where F(x,y) can be quite complex like
x
y
F ( x, y )  ( ) n  ( )n  S x  S y
x
y
a
b
x
y
 G exp[ ( ) 2 ]  G exp[ ( ) 2 ]
x
y
L
L
x
y
One method is to choosea certain x and then solve
the corresponding nonlinear equation for y with iterative methods.
Then repeat thi s procedure for the next choice of x.
A more elegant method is the following :
1) Pretend that the function f(x, y) is a Hamilton ian H(x(t), y(t))
2) Solve the equations of motion
(e.g. with the Euler - Cauchy integratio n or the Runge - Kutta method .
x 
H
,
y
y  
H
,
x
To get some initial conditions one can choosee.g. y0  0
and then solve f(x ,0)  c to get the corresponding x0 .
0
The beauty of this method :
For a given integratio n step dt the points x(t), y(t) move smoothly
along the desired function F(x, y).
34
W.Joho
Example of „Hamiltonian“ Plots
with F(x,y)=const.
x
y
x
y
F ( x, y)  ( )n  ( )n  S x x  S y y  Gx exp[( )2 ]  G y exp[( ) 2 ]
a
b
Lx
Ly
5
5
4
4
3
3
2
2
1
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-1
-2
-2
5
-3
4
-4
3
-5
-3
-4
-5
2
1
1
0
-4
-3
-2
-1
4
3
2
-5
5
0
1
2
3
4
5
0
-1
-5
-4
-3
-2
-1
0
-2
-1
-3
-2
-4
-3
-5
-4
1
2
3
4
5
You want to construct a nice table for your living room with one
of these shapes?
I give you the corresponding parameters for a modest royalty!
-5
35
W.Joho
„real fun“ with formulas !
two very famous equations are:
a2 + b2 = c2
Pythagoras (500 BC)
E = mc2
Einstein (1905)
with some easy Algebra we get:
a2 + b2 = E/m
Pythagoras–Einstein-Joho (2008)
The Swiss physicist Paul Scherrer gave a beautiful
analogy for the famous Einstein equation:
„This energy E (=mc2) is deposited on a blocked bank
account“!
(this analogy is mentioned by the author Max Frisch in his
book „Stiller“)
36
W.Joho
Varia
37
W.Joho
photon energy   wavelength 
use of magic numbers
to memorize the relation
  = 1240 eV nm (=hc)
trick => take square root !
VUV-region
35 eV  35 nm
soft X-rays
1.1 keV  1.1 nm
„old fashioned“
3.5 keV  3.5 Å
„infrared-people“ use wavenumber k in [cm-1]
100 cm  100 mm
-1
(correlations for arbitrary numbers are then quickly estimated
by multiplication resp. division)
38
W.Joho
Graphical solution of the lens equation
the lens equation (Newton) solved for a
thin lens with focal length f
1 1 1
= +
f u v
symmetric case:
u = v = 2f
same graph for resistances in parallel,
capacitances in series etc.!
39
W.Joho
Brightness of Synchrotron Radiation
from Bending Magnet with Field B
x
H 2 ( x)  x K ( ) ,
2
2
2
2/3
h( x )  exp[ (
Fit with H2(x) = A x2/3 h(x) ,
x N
) ]
xL
fit of binomial h(x) with 8 data points to ±2%:
A = 2.95 ,
N = 1.11 ,
xL = 1.336
H2 = normalized Brightness
10.000
H2
1.000
0.100
0.010
0.001
0.001
0.01
0.1
1
10
x
40
W.Joho
binomials
short range binomials (s>0)
N 1/S
Y = (1-u )
Y
short range binomials (s>0)
, u = x/xL
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Y = (1-uN)1/S , u = x/xL
Y
case 1, N=0.5, S=1
case 2, N=1, S=0.5
case 3, N=1, S=1
case 4, N=1, S=2
0
0.5
1
1.5
2
2.5
3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
case 5, N=2, S=0.5
case 6, N=2, S=1
case 7, N=2, S=2
case 8, N=4, S=2
0
3.5 x 4
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
X
2
exponentials (s=0)
Y
Y =exp (-uN) , u = x/xL
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
case 9, N=0.5
case 10, N=1
case 11, N=2
case 12, N=6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x
Y
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
long range binomials (s<0)
long range binomials (s<0)
N 1/S
Y = (1+uN)1/S , u = x/x L
Y = (1+u ) , u = x/xL
Y
case 14, N=1, S= -1
case 15, N=1, S= -0.5
case 16, N=2, S= -2
case 17, N=2, S= -1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x
41
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
case 18, N=2, S= -0.5
case 19, N=3, S= -1
case 20, N=7, S= -3
case 21, N=12, S= -2
0
0.5
1
1.5
2
2.5
3
3.5
x
4
W.Joho
Heart  Motor
Who is more reliable, your heart or the
motor of your car ?
Assumptions:
• a car makes about 200’000 km with an average
speed of 40km/h
=> runs for about 5’000 h or 300’000 min.
• the motor runs at an average of 2’000 cycles/min
=> the motor makes about 0.6 ·109 cycles,
If you live 80 years, your heart has made about
2.5 ·109 heart beats
=> your heart will make about a factor 4 more cycles
than the motor of a car!!
by the way:
during your life you experience some special dates:
after ≈ 11 years and 41 weeks you lived 100’000 h
after ≈ 27 years and 20 weeks you lived
10’000 days
after ≈ 31 years and 36 weeks you lived 109 s
42
W.Joho