The Mystery of Numbers: Discover the Unexpected

In mathematics, the term "irrational number" is well-known. What exactly is an irrational number? Here's a sufficient definition for our purposes:

"An irrational number is a real number that is not a rational number, meaning it cannot be expressed as a ratio of two integers. In the decimal representation of an irrational number, there are infinitely many digits to the right of the decimal point without any repeating pattern, which means we can only write a finite number of these digits (to the desired level of precision), with the continuation indicated by an ellipsis (e.g.,
π = 3.14159…)"
(Wikipedia, "Irrational Number", 5.22)
(The underlining is not in the original text)

In other words, we're dealing with a number whose precise value humans cannot determine exactly. Did you know such numbers exist?

For example: You can buy and sell exactly 1 apple, 2 apples, 1.5 apples, 2.25 apples, and so on. All these numbers are rational. However, you cannot precision-count "the square root of 2" apples. No factory in the world, no matter how computerized, could prepare a package with "the square root of 2" apples for us because the exact number is unknown to humanity—and it will never be known. It's a mathematical fact.

But that's just the beginning. It turns out that most numbers on the number line are irrational numbers:

In other words, nearly all numbers in the world have values that are not precisely defined by humanity. Amazing, isn't it?

By the way – notice the use of the ancient Hebrew letter אַ to describe "magnitude" in modern mathematics. A point to ponder. According to Jewish tradition, the letter "Aleph" represents "The Master of the World" – Hashem (see the book "The Code" by Rabbi Zamir Cohen, letter Aleph).

But even here, the matter doesn’t end. It turns out that even the simplest numbers used by all of us in everyday life – are irrational numbers. These numbers are deeply embedded in the design and implementation of nearly every modern technological system, in aviation, maritime, space exploration, missiles, military systems, physics, chemistry, computer science, and more. Humanity in general, and modern science in particular, are based on numbers whose exact value is essentially unknown (!). Here are a few examples of such numbers. You can find diagrams and illustrations for each example in the pdf attached to this article:

1. The length of the hypotenuse in a right-angled isosceles triangle

2. The ratio of a circle's circumference to its diameter

3. The mathematics of radioactive decay – used as a basic tool in the scientific dating of artifacts from the distant past.

In other words, to summarize – it appears that even those components of nature and science we thought were well understood and controlled by humanity, actually contain vast amounts of information that remain unknown and beyond our control. As the verse says:

"Therefore have I uttered that which I understood not

things too wonderful for me, which I knew not."

(Job 42:3)

The text is from the book "Intertwined Worlds". To download and read the chapter in a PDF file, click here >>

The author holds a B.Sc in Physics with honors from the Technion.