Our “drunken photon” from last week has to travel 700,000 kilometers to get from the center of the sun to its surface, i.e. 70,000 million centimeters; Therefore, according to the formula d = √n, where d is the distance from the starting point and n is the number of unit steps taken at random, the photon must travel 4,900 trillion random steps of one centimeter, which is exactly what will happen to it despite its enormous speed of 300,000 kilometers per second take about 5,000 years. While that may seem like a lot, this is a very low estimate: other estimates suggest that photons take between 100,000 and 150,000 years to leave the Sun.
I discovered the analogy to the drunk taking one-centimeter steps in my childhood in George Gamow’s fascinating book One, Two, Three… Infinite, a book written eighty years ago and not in the least interesting lost (despite the … The fact is that some of the theorems he mentioned, which had already been proven, were still conjectures at the time, such as the topological theorem of four colors. And it was not the only surprise that this miracle treatise had in store for me: Among other things, I also discovered how useful imaginary numbers can be, these impossible square roots of negative numbers, which Leibniz defined as “a kind of amphibian between being and nothingness”, and which Descartes contemptuously christened himself with the name (speaking of Gamow, the same thing happened with the Big Bang: a derisive name eventually became the official name of his correct theory).
The imaginary treasure
One of the instructive stories in Gamow’s book, which has a lot to do with what was said in the previous paragraph, is the following:
Once upon a time, an intrepid young man found a scroll among his great-grandfather’s papers that revealed the location of a buried treasure. After the scroll gave the longitude and latitude of a deserted island, it said:
“On the north coast of the island there is a meadow where you can see a single oak and pine tree. You will also see an old gallows where traitors were once hanged. Walk from the gallows to the oak tree, counting the steps as you go. Under the oak tree you need to turn right at a right angle and take the same number of steps. Drive a stake into the ground. Then he returns to the gallows and from there walks to the pine tree, counting his steps. Under the pine tree you need to turn left at a right angle and take the same number of steps that got you there from the gallows. Drive another stake into the ground. Dig halfway between the two posts and you will find the treasure.”
The instructions were clear and simple, so the intrepid young man chartered a boat and sailed to the deserted island. He found the meadow, the oak and the pine, but to his great regret he discovered that the gallows was missing. Too much time had passed since the instructions were written; The rain, sun and wind had degraded the wood and there was not the slightest trace of where it had been.
Our young adventurer became desperate and began digging indiscriminately in anger. But the meadow was too big and their efforts were in vain. So he returned empty-handed, and the treasure is probably still there.
A sad story. But what makes it even sadder is the fact that the intrepid young man could have found the treasure if he had known a little more about mathematics, and more specifically, if he had known how to use imaginary numbers correctly.
What would you have done in his place?
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