"Die ganzen Zahlen hat Gott gemacht, alles andere ist
Menschenwerk"
- Leopold Kronecker
At one point numbers were only used to count.
The number system was created by men based on their own observations of
reality, for instance: 1 sheep, 2 cows, 3 rocks, etc. Later on, the definition
Number was extended, not just to count things, but also measure their
properties. Therefore, the numbers used for counting were grouped as “Natural
Numbers” and the new extended system became known as “Real Numbers” to include
such numbers as zero, negative numbers, rational numbers, irrational numbers,
and finally complex numbers.
Zero
What is zero? Is it a number? How can it
be a number if it does not count anything? Does it even behave like a number? The
definition of a mathematical operations states that they are certain procedures
that take one or more numbers as
input and produce a number as
output. This how normal numbers act, therefore, the “zero” should behave as
this.
Test 1:
0 + 1 = 1
Test 2: 3 – 0 = 3
Test 3: 1 × 0 = 0
Test 4: 2 ÷ 0 = #¡DIV/0!
Thus, we cannot divide a number by 0,
because there is no solution to the equation “X × 0 = 3”
Negative Numbers
Afterward, mathematicians were forced to fill
a void when they had 4 sheep and wanted to subtract 5 of them. It was like
playing god, creating a sheep out of the blue (Actually they were trying to
solve the equation X + 1 = 0, not playing god) But this meant that numbers no
longer represented reality. Therefore, negative numbers were the first
imaginary numbers.
Rational and Irrational Numbers
Additionally, this sheep has also 2 ears, 4
legs, 231 bones, millions and millions of cells and so on. So, how exactly do
these numbers explain reality? Is 1 leg the same as 1/24 of a sheep? As I was told in school: "Es
como comparar papas con camotes" ("It´s like trying to compare apples
and oranges")
Complex Numbers
Mathematicians love to solve equations, so
let us consider this equation: X – 1 = 0
This equation has only one “real”
solution… x = 1
Another would be: x2 –
1 = 0
This equation has two “real”
solutions… x = 1, -1
Nevertheless, what if we have the
equation: x2 + 1 = 0
Actually, this equation has no solution
because there is no number that when squared the result is -1. Thus, what
should we do? As we need to have an answer for everything we cannot have this
unsolved equation over our heads, consequently, why not extend the number
system one more time to include solutions to equations of this type? So
everyone settled and said: “OK, as there is no solution for √-1, let us call it
“i” and save us the trouble”
Subsequently, this equation has two
“imaginary” solutions… x = i, -i
However, why does this number appear? Why is
there a mathematical operation that we cannot do?
If mathematicians have created a symbol for an operation we cannot do, why don't they create another for the division by zero (“X × 0 = 3”)?
Reengineering
What if we don’t need to settle? What if the
problem is our numbers system?
As stated before, our number model was
developed by people and then patched up as new discoveries appeared. As it is a
system developed by men, it has the same restrictions and constrains as men.
As I see it, this model can be only applied
until certain level, but if we want to reach the next level (or go back one
level, or go to a level within a level) then it gets complicated and we have to
settle for negative numbers, fractions and even “i”.
We need to start by assessing the reason for
negative numbers existence. Negative numbers are the lack of something, meaning
something being lost. If someone loses something, then another one gains that
something. First example would be debt: If I owe the bank $2,000 in my credit
card, it show as negative in my balance; but it is also positive from the banks
perspective (apart from the interest rate). Another example would be heat
transfer: if we put ice on a glass of water then we need to see it not as the
water loosing temperature, but as the ice gaining it.
It is all a matter of perspective.
It is all a matter of perspective.
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