# Diophantus and his equations

When solving many mathematical problems, it is necessary to compose equations and determine from them unknown quantities. A classic problem of this type is described in the tale of Pinocchio, where he had to share imaginary apples. But who first made such an ingenious step in the development of mathematics? For this we need to make a journey into the depths of the centuries, when the great heroes-conquerors and philosophers-scientists lived in parallel. And with them is Diophantus and his equations.

Once the great conqueror Alexander of Macedon wanted to conquer the whole world. But, being a disciple of no less famous Aristotle, he was not indifferent to philosophy and other sciences. Therefore, in the city of Alexandria, founded by him on the Mediterranean coast of Egypt, the scientific center of the whole world was built, where the best minds of mankind flocked. One such scientist was Diophantus, who in his mathematical work used the achievements of the Babylonians, Egyptians and Greeks. And it was he who first introduced the notation for unknown quantities in mathematics into mathematics.

In those days, the language of science was Greek. What is interesting - the numbers in it was not! That is, the numbers were composed of the letters of their alphabet. The first nine letters - alpha, beta, gamma, delta, ... denoted numbers from 1 to 9, the next nine - iota, kappa, ... denoted numbers from 10 to 90, and the next nine - ro, sigma, ... denoted numbers from 100 to 900. In order not to confuse letters and numbers, above the letters denoting the number, a dash was placed on top. In the Greek alphabet of the letters was 28, that is, there was one more letter - the sigma end, which was placed only at the end of the words and did not have a numerical designation. That's it, and decided to take for the designation of the unknown quantity Diophantus, as we now take x.

But what if there are more unknowns, not just the first degree? In his work "Arithmetic" Diophantus designated unknown to the sixth degree with special signs. For example, he denoted the square of the unknown by the sign Δν. In addition, in the symbolic writing of the equation, instead of the words "it is equal," he began to write ισ - the first two letters of the word - ισοζ - equal. How can we not remember isotopes with isobars in physics! Entered Diophantus and a sign of subtraction. To do this, he took the letter Ψ, turned it around and slightly modified it. It turned out something like a square. The sign of addition was not there - the terms simply were recorded next to each other. Diophant wrote the coefficients to the right of the unknown, moreover, in the equations he always put before the free term the symbol Μo - the first letters of the word Μοναζ ("monas" is a unit). Diophantus introduced some more notation into his equations, but there are no analogues of them in our mathematics. Invented by Diophantus and two basic methods of solving equations is the transfer of unknowns to one side of the equation and reduction of such terms.

Once upon a time, Deophant lived around the 3rd century BC, details about his life did not survive, except for the poem-task, according to legend, engraved on his tombstone:

Traveler! Here the ashes are buried by Diophantus,

And the numbers can tell, about a miracle,

How long was the age of his life.

Her sixth part represented a happy childhood.

The twelfth part of his life -

And his chin covered with down.

Seventh in a childless marriage was held by Diophantus.

Five years passed - he was happy

Birth of the beautiful firstborn son,

To whom the rock half is only a life of happy and bright

Gave on earth compared to father.

And in the sorrow of the deep old man of the earthly destiny the end was taken,

After surviving four years since I lost my son.

Tell me how many years have you reached,

Did Diophantus take the death?

Try and solve this puzzling problem.

The ideas of Diophantus and his equations found a response and development in the works of European scholars only in the 17th and 18th centuries. And they got to Europe from the Arab world. As you can see, the path was long, but interesting. But this will be the next story...