Few people found logarithms and integrals useful in their daily lives. Without it, we can accurately compute the interest on a mortgage or the magnitude of a discount at a store.
Recognizing the logarithm and integrals is crucial for figuring out how big discounts should be and how to best use promotion strategies in retailing and advertising.
Maybe you had some instructors in school who were more articulate about the value of science: mental exercise, exposure to new ideas, delving into the workings of the universe, etc. If you are doubtful or have never encountered such instructors, we hope we can change your mind today. However, use caution: perhaps even excessive mathematics will be there. Still, it's not awful.
Useless math
Mathematics' practical value is not always clear. It is not always evident, even to mathematicians. Number theory is a field of mathematics that investigates the mystical interaction of numbers and discovers remarkable patterns. However, it is entirely inappropriate for actual use.
It is unattainable to comprehend the insignificance of mathematics in the absence of mathematics itself. Therefore, you may either accept our word for it or try to determine the action process on your own. Let us conditionally refer to mathematical or non-mathematical mechanisms as "black boxes." Our objective is to provide an accessible explanation for each essential black box included in the article.
Other mathematicians had the task of proving the theorems that Pierre Fermat, a crafty individual, had left behind. As an illustration, the following assertion is contained in the black box of Fermat's Little Theorem:
An integer (i.e., not a fraction) is a certain number a. Along these lines, there exists a simple integer p that is exclusively divisible by itself and by 1. If p divides a but not a, then ap-1-1 is a divisor of p. Seven and three are our numbers. The result of dividing seven by 3 is not an integer. To achieve 48, however, you must first increase 7 to the power of 3-1 and then subtract 1. Plus, there is no leftover when you divide 48 by 3.
This is an excellent theorem. However, it seems to have no practical use. This is particularly true in the view of mathematicians, who historically have no idea how to use it. A little spoiler: this theory safeguards your data continuously; you are unaware of it. I haven't guessed it yet, but I'm sure we will.
Cryptography is the subject of this article.
Every man has his own set of secrets. Even if you can easily repeat the secret to a buddy, trying to communicate it across a distance is a whole different ballgame. The circumstances of the war, conspiracies, and the continual court intrigues contributed to the fact that the statesmen had numerous secrets. This means that political reasons were the driving force behind the invention of the earliest cyphers.
Jews made the Atbash cipher. The rule for cryptography was to change the n-th letter of the word to the i-n+1th letter. The initial letter of the alphabet was switched out for the last, the second for the next-to-last, and so on.
This code's name comes from the Hebrew letters "alef," "tav," "bet," and "shin," which are the first, last, second, and next-to-last letters.
It was a little harder to figure out the Spartans' code, Skital. They used unique tools like cylinders with different diameters to secure and retrieve messages. They wrote the message on a stick by winding a thin paper around it in a circle. The line was hidden when the paper was unwound.
You will always hear about the Caesar cipher in a cryptography class. Caesar changed the values of the letters in the alphabet to send hidden messages to his friends. For instance, in Cyrillic, the letter "A" will become encrypted "G" if it is moved three places forward. If only the average knew what this move meant. If not, the Roman ruler would have doomed his people to long nights of falling apart and himself to becoming the salad with the same name.
In the ROT13 method, the Caesar cipher with a change of thirteen is currently used. There is no hidden meaning behind the number thirteen; there are just 26 letters in the Latin language, and the encryption algorithm and the decryption method are the same when you shift them by 13. And Sir Arthur Conan Doyle wrote a story called "The Dancing Men" about it.
It is the name of all ciphers that use letter moving to hide information. In the wars of Caesar, they did well against Obelix and Asterix. Any smartphone can use brute force to figure them out now, but people could also figure them out back then.
Grammar betrayed us all.
The secret will be revealed at some point. And if telling family members that you like "My Little Pony" doesn't upset them, telling everyone about the army's attack plans could be fatal. It is essential to know this. There is also always someone who wants to listen in on them.
For instance, Aristotle was brought in to determine what the Spartan Scytals meant. And pretty well, too. He used a cone and changed the circumference while wrapping a secret note around it. At some point, the words started to make sense, and the cipher itself stopped making sense. But this method can't be used to break other ciphers.
Cryptographic frequency evaluation was unknown to ancient people, although they actively used it without suspicion, thanks to grammar. English articles are the most glaring illustration of a language's shortcomings, but every language has its own. They weaken the text since they appear so frequently. We can figure out that this is an article by highlighting the identical cipher pieces; then, we can determine the shift and apply brute force and guessing to crack the code.
Caesar, you must improve your encryption methods if the secret is to be kept hidden.
Retaliation from Encryption
Then someone came up with the polyalphabetic cipher. It has a group of mono ciphers, not just one change. The Vigenère cipher is a good example of this. It uses a unique table and a secret word to secure the text. This is a representation of the English cipher table.
The word "encrypted" needs to be encrypted. Come up with a secret. Any word or phrase will do, as long as it's not longer than the cipher. The key must have the same number of letters as the cipher. The word "geese" is the key. The letters in the key are used over and over again until there are no more letters left in the message. "Gusigusigu" it turns out. Let us now encrypt. The letter "s" will be where the first letters of our phrase meet the key. The same will happen where the second letters meet the key, then "e," and so on. "yeshchyudefvo" is what the cipher will look like when it's done.
Frequency analysis can also be used on the Vigenère cipher, but it is much harder to find trends because each letter is moved around for no reason. This worked for everyone who sent texts. It took a long time to list everything, and it was hard to figure out what the message meant in time; the information just became outdated.
But the never-ending struggle between encryption and decryption persisted nevertheless. Cryptographic styles evolved at the turn of the twentieth century. For our benefit, electromechanical devices started encrypting and decrypting for us.
We are only beginning the most challenging and fascinating part of cryptography.