Learn to use the number i in mathematics

This is another interesting mathematical theorem to explore:  the imaginary number i .

If we know that  the square of any relative number is positive  (-4² = 16, for example), we know that  the square root of x is the number that, squared, is equal to x  (the square root of 16 gives 4).

However, we cannot extract a square root of negative numbers, since the square of a number, whatever its sign, produces a positive result.

To solve this great mathematical problem, whose history spans centuries,  a pure imaginary number was invented. The number called i allows to contemplate the extraction of the square root of a real number:  root of -4 = 2i.

According to the rules of signs (the product of two negative numbers is positive), the square of -1 is positive since -1² = (-1) x (-1) = 1. The square root of -1 would be a number that squared would equal -1, so it wouldn’t exist!

We are in a dead end? No, because  mathematical scholars have shown their imagination , adding i to the square root of the number -1.

 Thus, i is the number whose square is -1, and its algebraic notation is i² = -1.

The history of this imaginary number dates back to the 16th century, when Gerolamo Cardano (1501-1576) sought to extract   to solve a third degree equation: this is how complex numbers arise in mathematical language.

Mathematical research, at that time, tries to give non-real solutions to impossible equations. L. Euler created the notation i in 1777, to qualify supposedly impossible or imaginary numbers.

The mathematicians CF Gauss (1777-1855) and Augustin Louis Cauchy (1789-1857) deepened their work on pure imaginary numbers, allowing them to be incorporated between real numbers in calculations.

Since the number i allows solving equations that have no solution in a real set, the field of possibilities in Mathematics is greatly expanded.

In fact, any equation that results in negative has no solution in its set of natural numbers (for example, the equation x – 10 = -20 = -10), but has a solution in the set of relative numbers.

The number i has allowed progress in physical research and electricity, especially for the development of the printed circuit for computers during the computer revolution.

Discover our offer of private math classes at Superprof. Learn to use the number i in mathematics