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What are complex numbers?

Complex numbers are the combination of real and imaginary numbers. The real part can be expressed by an integer or decimal, while the imaginary part has a square that is negative. Complex numbers arise from the need to express negative numbers’ roots, which real numbers can’t do. This is why they reflect all the roots of polynomials.

Their use extends to different scientific branches, ranging from mathematics to engineering. Complex numbers can also represent electromagnetic waves and electric currents, so they are essential in the field of electronics and telecommunications.

Its mathematical formula is a + b i, where a and b are real numbers, and i is the imaginary number. This expression is known as binomial form because of the two parts that make it up.

What is the origin of complex numbers?

French mathematician René Descartes was the first to emphasize the imaginary nature of numbers, positing that “one can imagine as many (numbers) as already mentioned in each equation, but sometimes, there is no quantity that matches what we imagine.”

However, the conceptualization of complex numbers dates back to the 16th century with the contribution of Italian mathematician Gerolamo Cardano, who proved that having a negative term inside a square root can lead to the solution of an equation. Up until then, it was thought to be impossible to find the square root of a negative number.  

Later, in the 18th century, mathematician Carl Friedrich Gauss consolidated Cardano’s premises, in addition to developing a treatise on complex numbers in a plane and thereby established the modern bases of the term.

What are the main characteristics of complex numbers?

  • The real numbers involved in a complex number formula can be expressed in the form of an ordered pair, a binomial, and a vector.
  • The whole set of imaginary numbers is called i and is the equivalent of 1 in the real numbers. Similarly, the square root of i is -1.
  • Two complex numbers are considered equal when they have the same real and imaginary components.
  • The letter C represents the set of all complex numbers. C also forms a two-dimensional vector space.
  • Unlike real numbers, complex numbers have no natural order.
  • There are pure imaginary numbers, the real part of which is 0; their formula is as follows: 0 + bi = bi.

What is the importance of complex numbers?

While their day-to-day application is not as direct as that of real numbers, their imaginary component makes complex numbers important as they make it possible to work very precisely in specific areas of science and physics. This is the case with measuring electromagnetic fields, which consist of electrical and magnetic components and require pairs of real numbers to describe them. These pairs can be seen as a complex number, hence their importance.

How are complex numbers represented graphically?

Any numerical category (whether natural, integer, or rational) can be represented graphically on a line. In the case of real numbers, they cover the line completely, and every number corresponds to a place on the line (also called the real line).

Complex numbers leave the line to fill a plane called the complex plane. In this case, complex numbers are represented on Cartesian axes, where the X axis is called the real axis and Y the imaginary axis. The formula for complex numbers, a + bi, is represented by the point or end (a,b), called the affix, or by a vector with the origin (0,0).

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