We will in this presentation introduce a new Accounting concept: Hard Cash.   Hard Cash is to Cash what Cash is to other types of Assets. The Ultimate form of Wealth.   To define these notions properly we will spend some time with modern mathematical ideas such as non-commutativity coming from Group theory and show how they apply in the Financial realm.   This will naturally lead us to our key concepts.

1. HARD
CASH
ACCOUNTING
©
North
Delta
College
2015
Mathema'cs
applied
to
Business
Theory
1
NON-­‐COMMUTATIVE
STRUCTURES
IN
MODERN
FINANCE

2. INTRODUCTION
We
will
in
this
presentaAon
introduce
a
new
AccounAng
concept:
Hard
Cash.
Hard
Cash
is
to
Cash
what
Cash
is
to
other
types
of
Assets.
The
UlAmate
form
of
Wealth.
To
define
these
noAons
properly
we
will
spend
some
Ame
with
modern
mathemaAcal
ideas
such
as
non-­‐commutaAvity
coming
from
Group
theory
and
show
how
they
apply
in
the
Financial
realm.
This
will
naturally
lead
us
to
our
key
concepts.
Mathema'cs
applied
to
Business
Theory
2

3. SUMMARY
Mathema'cs
applied
to
Business
Theory
3
PART
1:
THE
EXISTING
PARADIGM
A)
MathemaAcians
versus
Society
at
large
B)
US
Dollars:
The
UlAmate
commutaAve
currency
C)
The
Paradigm
D)
Why
we
disagree
with
the
Paradigm
PART
2:
WHAT
DO
WE
MEAN
BY
NON-­‐COMMUTATIVE
STRUCTURE?
A)
Examples
of
non-­‐commutaAvity
B)
Formal
definiAon
C)
Everyday
life
is
non-­‐commutaAve
PART
3:
INTRODUCTION
TO
GROUP
THEORY
A)
Non-­‐commutaAvity
and
Group
theory
B)
Group
theory
in
MathemaAcs
C)
What
is
Group
theory?
PART
4:
HARD
CASH
ACCOUNTING
A)
What
is
Hard
Cash?
B)
Why
is
Cash
non-­‐commutaAve?
C)
Hard
Cash
living
FINAL
STATEMENT

4. PART
1:
THE
EXISTING
PARADIGM
The
web
of
structures
behind
modern
finance
is
sAll
at
its
very
core
nothing
more
than
playground
mathemaAcs.
On
the
other
hand,
modern
mathemaAcs
uses
very
subtle
formalism
that
would
leave
the
financier
or
the
layman
in
total
bewilderment
if
they
were
aware
of
them.
We
will
show
in
this
presentaAon
why
the
Society
at
large
has
to
bridge
some
of
the
gaps
they
have
with
mathemaAcians
and
how
modern
Maths
can
help
us
understand
new
financial
concepts.
Mathema'cs
applied
to
Business
Theory
4
MathemaAcians
versus
Society
at
large
MathemaAcs
Finance
Layman
Abstract
Structures
Elementary
Algebra
Common
Sense
Figure
1:
MathemaAcs
versus
Society

5. US
Dollars:
The
UlAmate
commutaAve
currency
The
illusion
that
Cash
and
accounAng
are
commutaAve
might
come
from
the
fact
that
the
green
currency,
the
US
Dollar
has
become
universally
the
unique
reference
for
monetary
trade.
The
fact
that:
1)
It
can
be
used
anywhere
2)
Be
used
interchangeably
whether
under
its
physical
bank
notes
aspects
or
its
more
digital
avatars
3)
And
nothing
in
the
physical
aspect
of
the
currency
can
hint
at
disconAnuiAes,
ruptures
or
upheavals
between
the
currency
and
what
can
be
purchased
through
it.
Mathema'cs
applied
to
Business
Theory
5
PART
1:
THE
EXISTING
PARADIGM
Figure
2:
US
Dollar
–
The
Universal
Currency

6.
Mathema'cs
applied
to
Business
Theory
6
The
Paradigm
The
most
fundamental
assumpAon
of
the
current
financial
world
is
that
the
inner-­‐
structures
of
cash
transacAons
and
the
banking
system
at
large
follow
very
smooth,
easy
to
understand,
commutaAve
pa`erns.
That
no
financier
or
accountant
needs
to
study
higher
mathemaAcs
in
order
to
understand
financial
flows.
That
one
Ames
ten
pounds
is
equal
to
ten
Ames
one
pound.
In
other
words,
1x10
=
10x1.
(MathemaAcal
definiAon
of
CommutaAvity).
PART
1:
THE
EXISTING
PARADIGM
1
x
10
=
10
x
1
Figure
3:
CommutaAvity
of
Cash
–
the
ExisAng
Paradigm

7. Why
do
we
challenge
such
a
basic
assumpAon?
Our
firm
belief
is
that
the
workings
of
currencies
in
any
country
when
dealing
with
cash
is
not
commutaAve.
That
for
accounAng
transacAons:
1x10
is
not
necessarily
equal
to
10x1.
This
shocking
truth
will
be
demonstrated
in
part
4
of
this
presentaAon.
The
Algebra
behind
Cash
is
far
more
complex
than
the
one
dictated
by
using
real
numbers.
Mathema'cs
applied
to
Business
Theory
7
Why
we
disagree
with
the
Paradigm
PART
1:
THE
EXISTING
PARADIGM
Cash
Management
ExisAng
Paradigm
Our
Model
Real
Numbers
Non
commutaAve
Structures
Figure
4:
Our
model

8. Examples
of
non-­‐commutaAvity
What
do
we
mean
by
non-­‐commutaAve?
Let
us
take
the
example
of
heaAng
some
baked
beans.
In
order
to
accomplish
this
task
one
needs
first
to
open
the
An
of
baked
beans
and
then
pour
it
in
a
pan.
You
cannot
do
it
the
reverse
way.
Try
pouring
the
beans
in
a
pan
before
opening
the
An
and
you
will
see
where
you
will
reach.
The
2
operaAons,
opening
the
An
and
pouring
in
a
pan
have
to
be
performed
in
a
definite
order.
Hence
they
do
not
commute.
Mathema'cs
applied
to
Business
Theory
8
PART
2:
WHAT
DO
WE
MEAN
BY
NON-­‐COMMUTATIVE
STRUCTURE?
Figure
5:
An
Example
–
HeaAng
Baked
Beans

9. Formal
definiAon
Now
the
formal
definiAon
of
non-­‐commutaAvity.
A
process
is
commutaAve
if
when
one
splits
its
component
tasks
into
operaAon
A
and
operaAon
B,
it
doesn’t
ma`er
if
operaAon
A
is
done
before
or
aeer
operaAon
B.
Non
CommutaAvity
on
the
contrary
says
it
does
ma`er:
Performing
A
before
B
is
not
the
same
as
performing
B
before
A.
As
we
will
see
daily
life
is
non-­‐commutaAve.
Mathema'cs
applied
to
Business
Theory
9
PART
2:
WHAT
DO
WE
MEAN
BY
NON-­‐COMMUTATIVE
STRUCTURE?
Time
OperaAon
A
OperaAon
B
OperaAon
B
OperaAon
A
Outcome
X
Outcome
Y
X
≠
Y
Figure
6:
Non-­‐CommutaAvity
DefiniAon

10. Everyday
life
is
non-­‐commutaAve
The
processes
of
daily
life
are
most
of
the
Ame
non-­‐commutaAve.
The
order
in
which
ones
performs
tasks
does
ma`er.
Breaking
a
bo`le
of
wine
and
drinking
it
has
to
be
performed
in
that
order
only.
Washing
your
hands
and
opening
the
tap
has
to
be
performed
in
the
reverse
way
in
order
to
be
effecAve.
These
are
Non-­‐CommutaAve
processes.
On
the
other
hand,
switching
the
light
on
and
entering
the
room
can
in
theory
be
done
in
any
order
and
is
therefore
part
of
a
commutaAve
process.
Mathema'cs
applied
to
Business
Theory
10
PART
2:
WHAT
DO
WE
MEAN
BY
NON-­‐COMMUTATIVE
STRUCTURE?

11. Non-­‐commutaAvity
and
Group
theory
Non-­‐commutaAvity
was
first
introduced
into
MathemaAcs
by
Evariste
Galois
in
the
early
19th
century
when
he
discovered
Group
theory
for
the
very
first
Ame.
Groups
were
the
first
mathemaAcal
objects
where
commutaAvity
were
not
assumed.
Groups
exhibit
the
very
first
example
of
structures
where
someAmes
operands
do
not
commute.
Mathema'cs
applied
to
Business
Theory
11
PART
3:
INTRODUCTION
TO
GROUP
THEORY
1800
1900
2000
2100
Now:
2015
ApplicaAons
outside
Maths
Development
of
the
Theory
Discovery
of
Group
Theory
Figure
7:
History
of
MathemaAcs

12. Group
theory
in
MathemaAcs
Group
theory
occupies
a
very
special
place
in
the
mathemaAcal
landscape.
They
were
the
first
algebraic
structure
to
be
discovered
and
are
the
simplest
example
of
such
structures.
Their
applicability
is
almost
universal,
mostly
in
Physics.
They
exemplify
the
paradigm
shie
in
MathemaAcs
from
numbers
and
computaAons
to
structures,
pa`erns
and
conceptualisaAons.
As
such,
they
are
a
model
of
where
every
scienAfic
endeavor
is
heading
to.
Mathema'cs
applied
to
Business
Theory
12
CEO
PART
3:
INTRODUCTION
TO
GROUP
THEORY
ComputaAons
Concepts
MathemaAcal
Research
Figure
8:
EvoluAon
of
MathemaAcs

13. What
is
Group
theory?
More
pracAcally,
what
is
Group
theory?
Groups
are
mathemaAcal
objects
where
the
consAtuent
elements
of
the
object
are
interlinked
through
an
operaAon
which
obeys
3
basic
laws.
CommutaAvity
is
not
pre-­‐supposed
in
these
3
laws.
If
the
operaAon
is
noted
*
and
a
and
b
are
2
consAtuent
elements
of
the
Group
a*b
is
not
necessarily
equal
to
b*a.
We
will
show
in
part
4
why
Financial
models
can
benefit
from
integraAng
in
their
logic
the
facts
that
there
exists
processes
where
operands
do
not
commute.
Mathema'cs
applied
to
Business
Theory
13
PART
3:
INTRODUCTION
TO
GROUP
THEORY

14. What
is
Hard
Cash?
Mathema'cs
applied
to
Business
Theory
14
PART
4:
HARD
CASH
ACCOUNTING
Let
us
now
switch
to
Hard
Cash.
What
do
we
exactly
mean
by
such
a
concept?
Hard
Cash
are
those
elements
in
the
currency
with
the
highest
purchasing
power.
If
Cash
was
commutaAve,
This
definiAon
would
be
void.
But
we
will
show
that
Cash
is
not
CommutaAve,
and
therefore
this
definiAon
makes
sense.
More
precisely,
we
have
Hard
Cash
when
we
can
maximise
the
following
funcAon:
Purchasing
power
of
the
denominaAon
/
Value
of
the
denominaAon
Figure
9:
Cash
and
Hard
Cash

15. What
is
Hard
Cash?
What
do
we
mean
by
purchasing
power
of
the
denominaAon?
If
you
could
only
buy
a
cup
of
coffee
with
a
unit
value
of
the
denominaAon
you
wouldn’t
go
very
far.
Therefore
purchasing
power
comes
with
purchase
of
really
valuable
items
to
the
eyes
of
the
populaAon
with
regards
to
their
income
capability.
Thus,
a
middle
income
individual
in
an
emerging
market
would
consider
having
Hard
Cash
in
hand
if
with
no
more
than
3
units
of
that
denominaAon
he
could
buy
a
TV
or
a
mobile.
Something
he
would
really
value
intrinsically.
Mathema'cs
applied
to
Business
Theory
15
PART
4:
HARD
CASH
ACCOUNTING
Figure
10:
Buying
items
which
are
intrinsically
worth

16. Why
is
Cash
non-­‐commutaAve?
Mathema'cs
applied
to
Business
Theory
16
Why
does
Hard
Cash
make
sense?
Because
Cash
is
indeed
non-­‐commutaAve.
One
Ames
10
pounds
is
not
equal
to
ten
Ames
1
pound.
Even
if
in
theory
it
is
true,
in
pracAce
it
is
not.
It
is
easier
and
more
likely
you
will
carry
a
ten
pounder
rather
then
10
coins
of
1
pound
in
your
wallet.
Moreover,
the
shopkeeper
could
in
some
instances
refuse
too
much
pe`y
cash
and
ask
for
hard
cash
without
naming
it.
Therefore,
depending
on
what
you
buy,
denominaAons
do
not
necessarily
follow
laws
of
equal
relevance.
PART
4:
HARD
CASH
ACCOUNTING

17. Why
is
there
a
Max
for
Hard
Cash?
An
easy
mistake
at
this
stage
would
be
to
think
the
bigger
the
denominaAon,
the
harder
it
is.
This
is
clearly
wrong
as
there
exists
opAmum
denominaAons
depending
on
your
monthly
income.
For
example,
a
denominaAon
could
be
too
big
for
the
given
expense.
Moreover,
for
a
certain
monthly
income,
there
are
targeted
valuable
purchases
to
be
made
which
clearly
restrict
the
Hard
cash
to
a
certain
range
of
ideal
denominaAons.
(not
too
big,
not
too
small).
Mathema'cs
applied
to
Business
Theory
17
PART
4:
HARD
CASH
ACCOUNTING
DenominaAon
Purchasing
Power
/
Value
of
DenominaAon
Hard
Cash
Figure
11:
Why
Purchasing
Power
reaches
a
Max

18. Mathema'cs
applied
to
Business
Theory
18
Examples
of
Hard
Cash
Let
us
give
some
examples.
In
the
UK,
for
an
average
income
of
£2,500
per
month
(a
standard
salary
in
2015),
a
typical
Hard
Cash
denominaAon
would
be
the
blue
£20
note
or
the
red
£50
note.
In
Dubai,
for
an
average
income
of
AED
8,000
(
a
reasonable
income
by
2015
standards),
a
typical
Hard
Cash
denominaAon
would
be
the
red
AED
100
note.
In
both
cases,
few
units
of
Hard
Cash
will
give
access
to
the
middle
class
dream
lifestyle,
this
middle
class
is
precisely
aspiring
to.
è
Good
restaurants,
technological
gadgets,
branded
clothes,
exciAng
accessories
(watches,
shoes
etc…)
PART
4:
HARD
CASH
ACCOUNTING

19. Hard
Cash
Living
Hard
cash
generates
its
own
lifestyle
which
is
the
lifestyle
of
the
future.
Taking
its
source
in
the
American
dream,
but
exemplified
more
accurately
in
place
like
Dubai.
A
heavy
spending
middle
class
lifestyle,
with
branded
products,
shopping
malls
and
no
regrets.
The
whole
Hard
Cash
concept
and
its
subsequent
lifestyle
might
seem
shallow
to
an
European
intellectual,
nevertheless
it
is
THE
lifestyle
where
all
the
emerging
markets
are
converging
to
and
in
20
years
Ame,
if
the
world
passes
through
the
economic
storm
there
will
be
nothing
else
lee.
Mathema'cs
applied
to
Business
Theory
19
PART
4:
HARD
CASH
ACCOUNTING
Figure
12:
Dubai:
The
laboratory
of
the
world
to
come

20. Final
Statement
We
have
therefore
shown
in
this
presentaAon
how
proper
usage
of
modern
mathemaAcal
concepts
can
enlighten
very
down
to
earth
subjects.
In
parAcular,
we
extracted
the
noAon
of
non-­‐commutaAvity
from
Group
theory
and
applied
it
to
a
well
known
accounAng
concept
such
as
Cash.
This
led
us
to
define
Hard
Cash.
Hard
Cash
is
the
cash
you
have
in
hand
that
allows
you
to
chase
your
dream
material
lifestyle.
Although,
theoreAcally
a
void
concept,
in
pracAcal
terms
it
is
not.
Once
one
becomes
aware
of
its
existence,
one
will
soon
realise
that
regarding
Money
and
spending,
sole
Hard
Cash
gives
you
access
to
the
Holy
Grail.
Mathema'cs
applied
to
Business
Theory
20
FINAL
STATEMENT