Mathematics 'sans frontiers': An experimental notation to
teach mathematical operations.
David Womack
Ex. Department of Education. University of the South Pacific.
uspdave123@hotmail.co.uk
Abstract.
This paper takes a 'mathematics sans frontiers' approach which encourages
students to experiment freely with mathematics without borders and so
perhaps discover fresh insights into the nature of numbers and arithmetic. The
paper therefore proposes a new notation, to express all operations and their
inverses. This is intended to remove some of the pedagogical barriers raised
(for example) by the bizarre notation for roots and logarithms. The notation
involves the novel (if enigmatic) concept of a 'non-interchangeable operation' defined as a binary operation in which terms cannot be switched without
changing the meaning of the expression, though not necessarily the numerical
result. Through the proposed 'dot' notation we can then extend current
operations to repeated powers (i.e. tetration).
Keywords: exponentiation, signs, logarithms.
Exploring the labyrinth of formal mathematics.
How can students possibly discover new facts and ideas for themselves in
such a complex subject as mathematics? Let's take a caving analogy. Looking
for new mathematical insights is like looking for entrances to new caves,
where discovering a new cave for oneself is infinitely more rewarding than
being taken on a guided tour of a show cave. There is a further analogy. When
living in Kenya, looking for the entrances to lava tube caves in the Rift Valley, it
was usually necessary to ignore any signposts, leave the Maasai cattle tracks
and rely solely on intuition. And so to discover entirely new ideas, we must
sometimes ignore the well-trodden paths of orthodox mathematics and venture
into areas dealing with concepts for which mathematics currently has no
conventional symbols or signs.
But then how can others follow us into a previously unknown mathematical
terrain? In the African bush, tracks are soon obscured with vegetation and to
find them again, we cut notches in trees. So it is with 'off-road' mathematical
ideas; we need new symbols to 'sign-post' these new routes. And just as it is
difficult to drive a Land Rover out of deep ruts, so it is equally difficult to leave
the mathematical paths marked by familiar symbols. However, for students,
whose mathematical ruts are not so well-worn, 'off-road' thinking is often
easier than for professors of mathematics - and so paradoxically, they may
understand new ideas more intuitively.
Outline of Sections.
In Section 1, a brief overview is given of the essential role of operator and
operand in the proposed 'action-arithmetic'. However, in order to identify
operator and operand, there is a need for clarification of current operation
signs. Section 2 suggests how this might be done and in particular how the
sign for exponentiation (^) might be clarified, using a 'dot' notation. Section 3
reminds us how current inverse operation signs are used, and Section 4,
proposes new signs for the inverse power operations of 'roots' and 'logs'.
Section 5 shows the use of dot notation with the 'new' inverse operation signs,
whilst Section 6 defines the operation of repeated powers (tetration). Finally,
Section 7 reasons that to explore such unorthodox mathematics intuitively,
without 'boundaries' – that is, without the need to formally prove each step can powerfully motivate many brighter students.
Section 1. 'Action-arithmetic' and the need to identify operator and
operand.
I begin from the premise that numbers can be regarded as symbolic objects, to
which we assign a conventional order and sequence. Addition is then a
transformation from one position to a new ('higher') position; multiplication a
repetition of that transformation, and exponentiation (with sign ^) a repetition of
the multiplication operation. Transformation and repetition operations imply an
operator and operand – and we take this as characteristic of all such
operations in the proposed arithmetic.
Note that mathematically we define an operation in a very specific way;
however, here we use 'operator' in its traditional or 'linguistic' sense of 'acting'
number, and use 'operand' in the sense of 'acted-on' number. This is not the
set-theoretic relational view of operations but was the historical position prior
to the introduction of set theory. It is also a view intuitively held by young
children and unschooled adults (Hughes,1986; Gifford,1990). The paper also
assumes the relations between finding the operator and finding the operand
are understood as the mental strategies of take-away/ compare and sharing/
repeated-subtraction. Since these have been the subject of many previous
articles (Womack, 1997; Womack & Williams,1998; Womack, 2001), they will
be mentioned here only in passing.
The aim is to create a consistent notation for all operations – both
commutative and non-commutative. This requires the rather unusual concept
of a 'non-interchangeable operation' - a binary operation in which the
interchange of terms, changes the meaning of the expression – though not
necessarily the numerical result. I have previously called the resulting
operations 'action-operations' or operactions, since action-verbs in the physical
world involve an Agent (cf. operator) and Recipient (cf. operand).
In most English action-verbs, both the Actor and the Recipient of that action
are identified in a sentence by a device known as the active and passive voice
of the verb. So in, 'A hits B' (active voice), A is the Actor; but in, 'A is hit by B'
(passive voice), B is the actor. In the same way, we discriminate operator and
operand by means of the active and passive voice of the operation verb.
Therefore, in '3 multiplied by 4', 3 is the operand and 4 is the operator,
whereas '3 powers 4' implies 3 is operator and 4 is operand.
Note that the precise meaning of the conventional multiplication sign is
unclear. In any one instance, (or at any one time), it can mean either
'multiplies' or 'multiplied-by' but not (I contend) BOTH. In short, I understand
any operation between two terms A and B, to imply either A acting on B OR B
acting on A – but not both at the same time. Therefore, the OR (used above) is
used in an exclusive sense – which is the normal and intuitive usage. For
example, a café deal which advertises, 'free muffin, and tea OR coffee' does
NOT mean the customer can enjoy both tea AND coffee. Note also that set
theory uses the inclusive OR for the operations of + and x.
The present signs (+, x, ^) do not distinguish between operator and operand
and to identify them has been one of the themes of the author's research for
some time (Womack, 1995).
Section 2. A 'dot' notation for the power sign.
Historically, the power operation was expressed in many different ways
(Womack 2015), but here we will use the sign ^ . Conventionally, the
expression '3 powered by 4' is written as 3 with superscript 4. However, we
can't we replace the superscript convention with a 'normal' sign (written on a
line with numbers) because without an agreed convention, the meaning of the
symmetrical symbol ^ is unclear; that is, does 7^3 mean '7 powered by 3' (the
passive voice of the verb 'to power') or '7 powers 3' (the active voice)? See
Womack, Jan 2013). Therefore, if we could indicate whether an operation sign
was to be read as 'active' ('powers') or passive ('powered-by'), then we could
identify operator and operand in expressions such as 3^4.
Addressing this problem for the past 20 years, working with students in the
South Pacific and the West Indies, I have sometimes used a dot placed
adjacent to the sign, but always placed on the side of the operand (e.g.
Womack, 1998). So 3.x4 means '3 multiplied-by 4', and 3.^4, means '3
powered-by 4' . In each case 3 is the operand and 4 is the operator. With the
same notation, 3x.4 means '3 multiplies 4' and 3^.4 means '3 powers 4'.
Note that any passive form can be put into the active form (and vice versa) by
reflecting the sign-with-dot along the vertical axis. For example, 5 .x4 ≡ 4x. 5
(5 multiplied by 4 is equivalent to 4 multiplies 5 (Womack, 2011.)
Section 3. How conventional inverse operation signs are used:
Typically in arithmetic (and algebra) we are given an equation with the problem
of finding a missing term. For example:
Finding the missing operand (d), in the equation d.^ 4 = 81. To solve, we find
'the 4th root of 81'. We write '4 √ 81' but note that we do not (normally) read
the expression as ‘81 rooted by 4' (passive voice).To find the missing operand
for any operation we split the resulting number into 4 constituent 'parts'.
Finding the missing operator (r), in the equation 3.^ r = 81. To solve, we can
repeatedly divide 81 by 3. We write, 'Log (subscript 3) 81' and read the
expression as 'Log to base 3 of 81'. To find the missing operator for any
operation, we count the number of 'lower' operations required to reach the
resulting number.
Section 4. Proposed signs for 'roots' and 'logs':
This bizarre terminology and notation for 'finding the root' (i.e. the operand)
and 'finding the logarithm' (i.e. the operator) can often be confusing to
students. We could rationalise this terminology (and indeed all inverse
operations), by inventing new signs. However, ideally and more intuitively we
could keep the direct operation sign and amend it. We can then use a similar
notation for all operations.
I have previously (Womack, 2009) appended ‘direct’ operation signs by using
a hyphen (─) and a forward slash ( / ). (Handwritten, they are placed above the
operation sign but in printing, we insert them in front of the sign.) For
readability, we will assume a passive interpretation of the 'power' ^ sign :For ‘finding the root’ we use: ─^
This is the inverse operation which ‘finds’ the operand.
For ‘finding the logarithm’ we use: /^
This is the inverse operation which ‘finds’ the operator.
For example, we can write:81 ─^4 meaning ‘81 rooted-by 4’ (i.e. the 4th root of 81: answer 3)
81 /^3 meaning ‘81 logged-by 3’ (i.e. the log to base 3 of 81: answer 4)
Entirely in symbols, this is:3 ^4 = 81 (3 powered-by 4 equals 81) implies 81 ─^4 = 3 and 81 /^3 = 4
In the general case, I will call the procedure to find the operand, the inveroperation and the procedure to find the operator, the anti-operation.
[Note that these are better thought of as 'inversing strategies' rather than
'operations'.]
So 3 ^ 4 = 81 implies
3 = 81 inver-powered-by 4 and
4 = 81 anti-powered-by 3
We will use this terminology for all subsequent inverse operations.
Section 5. Inverse operations and 'dot' notation:
We can now use dot notation (if preferred) with the proposed inverse notation.
The Inver-operation.
10 .–+7
which means '10 subtract 7'
(Solution: take-away 7 from 10)
18 .–x6
which means '18 divided-by 6' (Solution: share 18 among 6)
81 .-^4
which means 'take 4th root of 81'.
The Anti-operation.
10 ./+7
which means '10 subtract 7'
(Solution: compare 10 with 7)
18 ./x6 meaning '18 divided-by 6' (Solution: repeatedly subtract 6 from 18.)
81 ./^3
which means 'find the log of 81 to base 3'.
Section 6. Extending operations to tetration (repeated powers).
Exponentiation is usually introduced to students as a ‘shorthand’ for repeated
multiplication; however, rarely if ever do we refer to tetration (repeated
powers). The reasons for ignoring this operation are probably twofold; firstly,
the very large numbers generated, and secondly, the confusing notation for
the power operation on which current notation for tetration is based (see
Cajori, 1993). It is this index-number power notation which makes the algebra
of tetration and its inverses so complex that only the most dedicated
mathematician can, (or would even want to) understand it. However, the
operation is certainly recognised by mathematicians and is discussed from
time to time on the Web (see for example, Wikipedia, ‘Tetration’).
Tetration is the operation of repeated powers, so that 'A tetrated by 5' is
defined as a 'power tower' consisting of a hierarchy of five levels of the number
A. In accordance with current convention, we assume a right-bracketing of
these indexed repeated powers. We will continue to use the carat sign (^) for
the power operation but for the sake of readability, here we will assume a
passive interpretation of the sign. That is to say, ^ will mean ‘powered-by’
rather than ‘powers’.
So, A tetrated by 5
is written
A^ ( A^ ( A^ ( A^ A.
(omitting
unnecessary brackets.)
For tetration we will use a single vertical arrow sign (↑) and provisionally
interpret this as ‘tetrated-by’ – that is (again), the passive form of the operation
verb.
[Note that this arrow has no connection with the arrow proposed by Donald
Knuth for powering. (See Wikipedia ‘Tetration’). ]
Therefore, 'A tetrated by 5' is written A ↑5 with meaning A^( A^( A^( A^A
Here are some examples of tetrations with counting numbers (right
bracketing).
3↑2 = 3^3
= 27
(3 tetrated by 2)
2↑3 = 2^(2^2
= 16
(2 tetrated by 3)
2↑4 = 2^(2^(2^2
= 65536
(2 tetrated by 4)
3↑3 = 3^(3^3
= 7,625,597,484,987 (=327) (3 tetrated by 3)
[Inverse tetration is the subject of an earlier article (Womack, 2013). ]
Section 7. Exploring 'off-road' mathematics ….
To recap, I suggest that standardising the notation we use for all operations
will make the operations of powering and tetration entirely comprehensible to
college students. It will also provide students with the tools to investigate an
operation which lies at the cutting edge of mathematical research. Note that
tetration is described in some detail in Wikipedia, using the current notation
and terminology of powers, roots and logs – such as 'iterated powers', 'superexponentiation' etc. However, when we attempt to extend tetration to all real
and complex numbers, the notation becomes extremely complicated and
specialised. Inverse functions are even more difficult to understand since the
various notations are described within the framework of set theory. However,
'missing' non-discrete tetration operands are relatively easy to calculate and
students should be encouraged to do this (see Womack, 2013).
I have not attempted to resolve these complex mathematical issues in any
way; nor would I presume to present a set of axioms and theorems involving
functions, relations and limits etc. Aside from the enormous task this would
present, it is the responsibility of the mathematics community to decide where
the 'new' numbers fit into the present real number system. Therefore, I make
no apologies for these omissions but present these preliminary ideas in the
hope that they will introduce to students the real possibility (even probability!)
of making new discoveries about numbers.
References
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