Countless Numbers

Countless numbers

There are numbers and numbers and they are even in great numbers.

Let us start with what we can consider the principle, using a short story.

José, a young man, decided to be a pastor. He gathered several sheep and a moment came when he felt the need to count them: one, two, three,… He did it using the natural numbers (N). Let's say you used that kind of numbers that allow us to make counts:

Natural Numbers

Note that José just felt the need to tell, because he had something to tell, that is, zero does not belong to natural numbers, if there is 'nothing' to tell, there is no need to tell.

Let us continue then, beyond the natural numbers.

Meanwhile, our shepherd, who only had 20 sheep, was involved in disastrous business with another fellow shepherd, having owed 30 sheep, that is, 10 sheep were missing (he started to have less ten sheep) for José to be able to fulfill his obligations. As he was very careful with his accounts, he intended to record this debt, but with the numbers available, the natural ones, he felt unable to do so.
He then created the negative integers, which, combined with natural numbers, gave rise to the set of so-called whole numbers (Z):

Integers

Well, I'm sure you are finding something strange in this sequence ... That's it, the zero is missing!

Its absence is momentary and is purposeful. Because zero is a much more recent number in the history of the evolution of the concept of number (I promise we will talk about it later), but it is in fact considered an integer, like this:

Integers; Natural Numbers

Note: the symbol it reads and means “reunited with”.

So we actually have:

Eq 4

Well, José meanwhile married Maria and they started to think about how they would divide the flock of eighty-four sheep equally when they had children. As a first hypothesis, they started by imagining that they would have four children and easily concluded: there would be twenty-one sheep for each one (the result, twenty-one, is a whole number already known).

Then they hypothesized that they would have eight children and ... oops, a new problem to solve: the result is not available in the set of whole numbers.
Thus, the fractional numbers which, combined with the whole numbers, constitute the set of rational numbers (Q), that is, those numbers that can be represented by a reason or fraction:

Rational Numbers; Fractional Numbers

José and Maria also remembered to consider the possibility of not having children and concluded that, in this case, could not perform the division because there was no one to distribute the sheep to.

This means that in mathematics, as in real life, it is not possible to divide by zero!

Well, let it be clear, it is neither possible nor necessary, so it is not a matter of mathematical incapacity, but only the realization of real life through mathematics, nobody needs to divide 10 candies by zero people ...

The rational numbers are thus well understood,

At a certain point, the couple José and Maria decided to build, on separate lands, two fences for the sheep not to get too far away for what they needed to do calculations to determine the length of the net they had to buy. As both fences were shaped like a right triangle, José remembered the theorem Pythagoras (it is true, José was curious, very fond of learning new things and had been reading a book about the life of Pythagoras).

To save work, he decided to measure the length of the two sides and apply the Pythagorean theorem to calculate the length of the hypotenuse (I am assuming that my readers know this theorem: the square of the length of the hypotenuse is equal to the sum of the squares of the collet lengths).

One of the fences had the legs with the following lengths (in kilometers): 3 and 4, reason why they easily concluded that the hypotenuse (h) would be:

Eq 6

(the result, five, is one of the rational numbers, which they already knew very well).

However, for the second fence, whose legs, also in kilometers, measured 2 and 3, made the same calculations applying the Pythagorean theorem:

Eq 7

and, again, another problem: there is no rational number, that is, a fraction, capable of representing this quantity.
Thus, the irrational numbers which, together with the rational numbers, constitute the set of real numbers (R):

Number of Reais; Irrational Numbers

After the real numbers, what José and Maria never got to know is that square roots of negative numbers can also be calculated, provided that the real numbers imaginary numbers, in order to obtain the set of complex numbers (C),

Imaginary Numbers, Complex Numbers

Real numbers and imaginary numbers make up the complex set, but this is part of the history of the evolution of the concept of number that let's leave it for later.

Finally, it remains to be noted that José and Maria were very happy all their lives and that they had only one child!

A son, in real numbers!

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4 thoughts on “Countless numbers”

  1. Pingback: The Life and Work of Pythagoras, the Theorem of Pythagoras and the Symbology of Numbers -

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