After a few years, I am re-opening this Blog and, curiously enough, with
a maths subject.
I have an interest in Physics and I have decided that, to make progress,
I must improve my mathematical skills. Unfortunately, I am liking maths too
much, so it is hard to move ahead, because interesting ideas flood into and
grab my mind at each step.
I will share what I have thought about imaginary number and powers. A
good example of the benefits of my "Cinderella method" as an
epistemological tool. I can advance that this provides an insight for solving
the “zero to the zero-th power” issue,
but let us go step by step, by taking a twisted
… road through imaginary numbers
An imaginary number is the square root of a negative number. That doesn't
exist, it is true, that is why people coined this adjective,
"imaginary". But, if you think of it, the denomination is not
fortunate, because actually all numbers are "imaginary",
just like any other concept: they are ideas that our mind invents, intellectual
artifacts, although -like the Cinderella method reminds us of- they are islands
surrounded by empirical reality from all sides: they are mental
"slippers", but made following the model of a real "foot"
(Cinderella's) and aiming at solving a practical problem (providing the Prince
with an "apt wife", one who is able to dance in unison with his
political and personal troubles...).
For example, the most elementary category, natural numbers = positive integers (0, 1, 2 , 3 ...):
do they exist? No, if you look around, you may find one husband or two houses,
but not any entity called "one" or "two" and being only that. However, this creation turns
out to be most helpful, because it allows you to count those very real items
and draw the very pragmatic conclusion that having more than one house is a
nuisance, not to talk about polygamy, which would be suicidal...
As to negative numbers, more of the same. One day your employer gives
you 10 euros. The day after the bank hands over to you 10 euros. No physical
difference between those two operations and objects. Your cash is hence 20
euros. But tomorrow you may have a legal need: you need to determine how much
of that money is really yours and does not to have be returned. Time to invent
a new "slipper", negative
numbers. You should have recorded the second delivery as a loan, with a
negative sign. Thanks to this, you will be able to add up your total cash with
your (negative) loans and thus infer your net wealth.
Let us jump now, skipping some other categories, to imaginary numbers.
Your son is standing 1 km away from your house to the East. But he should have
moved to the West instead, to point -1 km. Two possibilities: walking straight
through the road or taking the scenic route and rotating in a 180o arc.
Mathematically these two options correspond
to a subtraction (1 - 2 = -1) or a multiplication [1 x ( -1) = -1],
respectively. But what if your son wants
to go scenic but stop after 90o to spend the night with his
girlfriend? For that noble purpose, mathematically, he should half-multiply by
-1… But… what is that, “half-multiplying…”? Well, that amounts to multiplying
by √-1, so why not invent the so called imaginary unit (denominated i), which has that weird value, but beautifully represents the move that your
young son so much desires…?
Now let us compare our story with the standard account of the discovery
of imaginary numbers. The serious version goes like this: mathematicians stumbled
across this puzzling equation
x2 = -1
and found out that it can be solved by equating x to the imaginary unit i = √-1.
Well, our narration also contains an implicit equation, but what is
remarkable is that by wrapping it up in a real-life problem, we have given it
an embellished form, as if we had touched it with a magic wand:
1 ∙ x2 = 1 ∙ x ∙ x = -1 or 1 ∙ i = √-1
I have seen this trick for the first time in a wonderful site for obtaining
insights about maths (betterexplained.com) and also in these other excellent places:
Math
Memoirs and Kahn
Academy. My modest contribution is giving the equation a second touch, so
as to make it apparent why the trick in question works:
1 km ∙ i = √-1
Not much, I admit that the idea is obvious: the magic functions because now
the equation is closer to reality. We said it before: abstract numbers don’t exist, you always use
them to count units of real things (1 km, for instance) and it is by showing
their use in practice that you can ascertain how they operate and what they
serve for. Let us prove it more robustly through an even simpler example, ordinary
powers.
How to explain powers and exponents
The ordinary approach, given for instance in the article of Wikipedia,
is that “exponentiation” involves just
two numbers, the base (b) and the exponent (n), like here:
bn = x
And it means repeated multiplication of b, in
particular the number of times indicated by n.
Then you are told the properties of powers:
·
If n = zero à x = 1.
·
If n = negative à x = 1/b.
·
If n = a
fraction = 1/m à x = √b, with m being the index of the square.
All that is true of course, but very hard to remember, because there is
no logical thread joining all elements. In fact, one (mistakenly but excusably)
tends to think that exponentiation is something done over the base, but then if
you have exponent 2, how is it that you do it only once… and if exponent is 0,
how come that the base turns into 1?
Let us now wave our magic hand in order to embellish the expression by
adding a real-life unit, over which our controverted operation is acting:
1 thing [operation] base = x
And where shall we place the exponent? Well, the advantage of the new form is that now we
don’t need to put such exponent over the base, we dispose of an alternative
solution: we can place it… over the
operation itself! Like here:
1 thing [operation n] base = x
And, yes, that is the right thing to do, because this way all the
qualifications represented by the exponent naturally apply to the operation in
question and everything fits in:
·
Basic meaning of
the operation: MULTIPLICATION of 1 thing by b.
·
Possible
qualifications of the operation:
o
If n = zero à you do NOT MULTIPLY the thing by b (which, yes, is
equivalent to multiplying it by 1).
o
If n > 0 à you MULTIPLY the thing by b n TIMES.
o
If n < 0 à you
ANTI-MULTIPLY the thing by b (i.e., DIVIDE it by b n TIMES).
o
If n = ½ à you HALF-MULTIPLY the thing by b (which is equivalent
to multiplying it by square root of b).
o
If n = 1/m à you FRACTION-MULTIPLY the thing by b (i.e. you
multiply it by root of b with m index).
What is the then the insight about zero to
the zero-th power?
It is usually said that one has mixed feelings with regard to this
expression. On the one hand, 0 raised to any power is 0. On the other hand, any
base raised to 0 is 1. So the authors discuss whether the correct solution for
00 is 0, 1 or should be left as indeterminate. You can google for multiple
discussions and sophisticated demonstrations about one or another solution.
Obviously, our argument points at 1 as the answer: what the expression
is telling us is simply that we should not apply any operation to the object in
question, that we must leave it alone. That is what is called, by the defenders
of this solution an “empty product” or “multiplication by the unit identity.
Well, I am not going to affirm that this is the only possible solution.
I am not qualified to discard that other solutions may make sense in other
contexts. But what I am sure of is that in contexts like ours, the solution is
100% certain: if you are faced with the decision of running 1 km East (or √-1
km North), marrying a husband or wife or buying a house… and someone comes to
you shouting, “but please do it 0 times raised to 0”, don’t worry: she or he is
just confirming good sense, that you should do it once… The concept has been
invented as a slipper to fit around one foot and lead you to one Cinderella, so
it can ONLY have this meaning. If it should ever have another meaning, it will
be because it is a different concept meant to catch another lady with another
foot…
