The formula Andiπ + 1 = 0said Euler’s identitytells us about powers and complex numbers and is considered by many to be the most beautiful equation in the world. What makes it so fascinating to mathematicians is its ability to put together five of the most important numbers in mathematics. Let’s see what this formula, discovered by the Swiss mathematician Euler in the 18th century, tells us, what it has to do with powers and complex numbers and why it is also connected to wave functions and exponential growth.
THE’Euler’s identity (see figure above) is a formula that deals with powers And complex numbers putting together some of the most important and famous mathematical constants. And it is precisely thanks to its ability to inextricably link the following numbers that many scholars consider it one of the most beautiful formulas in mathematics:
- there mathematical constant And, also known as Nepero numberThat it’s worth it about 2.71
- πwhich is worth it about 3.14to which a day of the year has even been dedicated, Pi day.
- thea strange number that belongs to the world of complex numbers, which is the square root of -1.
- the number 0theneutral element from the sum
- the number 1theneutral element from the multiplication
The protagonists ofEuler’s identity they are all numbers with well-determined values, even if they are indicated by letters, which is why it is more correct to call it identity rather than equationand if we want we can rewrite it without using even a letter, as in the figure below, but as long as we use the symbol ≅ to specify that we are approximating the values of And And π.
Usually we leave it written in letters so as not to run into approximations, but writing it using numbers helps us understand that, ultimately, this formula tells us nothing other than the result of a particular power whose basic it’s the number And (≅2.718) and whose exponent is the product between π (≅ 3.14) and the Complex number the (the square root of -1). So this identity is essentially the calculation of a power, but it is not just any power, but rather it is a power with a special exponent, a Complex number.
When we talk about complex numbers we are referring to an extension of the Real numbers (the decimal numbers that we all know). The peculiarity of these numbers is that some of them, when they are raised to the secondresult in a Negative real number. This is what happens to the number thefor which it is valid the × the = -1and to all its multiples, which are called Imaginary numbers. Every Complex number is the sum of a Real number, called the Real part, and an imaginary number, called the Imaginary part. For example 3+4i is a complex number with real part 3 And imaginary part 4.
The calculation of powers with i complex numbers it is not simple and can give both Real and Imaginary results, or a combination of the two. However, Euler himself enunciated another very important formula (see image below), thanks to which he managed to derive precisely theEuler’s identity which we can use to calculate some powers, including that of our identity.
As we can see, And elevated to theπ/4 results in a complex number composed of both a real part (about 0.71) which gives one imaginary part (about 0.71the), While And elevated to iπ/2 is equal to imaginary number thewithout Royal side. Finally, it may happen that the result is a purely number Realas in the case of And elevated to iπ which is equal to -1as established by theEuler’s identity.
Finally, this formula hides within itself another little jewel that gives a further touch of charm to theEuler’s identity. As you can see from the figure above, the two also appear in the formula trigonometric functions breast And cosine (indicate how sen And so), which serve to describe the waves and which in this formula are associated with the concept of power in its most generalized form which takes the name of exponential function. Euler’s Identity, therefore, is also the result of a formula capable of putting together wave phenomena And exponential growtha little extra touch of charm that we can visualize with a three-dimensional representation of the Real and Imaginary parts of the powers of And.
