Complex Numbers

INTRODUCTION TO COMPLEX NUMBERS

Complex numbers are an extension of the set of real numbers and enable the solving of equations that have no solution in the set of real numbers. The basic idea of complex numbers begins with a unit called the imaginary number, denoted by 'i', for which i² = −1. This concept allows for the definition of complex numbers as the sum of a real part and an imaginary part.

STRUCTURE OF A COMPLEX NUMBER

Complex numbers are usually written as a + bi, where 'a' is the real part and 'b' is the imaginary part of the complex number. The real part 'a' represents the coordinate on the real axis, and the imaginary part 'b' represents the coordinate on the imaginary axis in the complex plane. This two-dimensional representation allows for the visualization of complex numbers as points or vectors.

BASIC OPERATIONS WITH COMPLEX NUMBERS

Complex numbers support basic arithmetic operations: addition, subtraction, multiplication, and division.

• ADDITION AND SUBTRACTION are performed by separately adding or subtracting the real and imaginary parts.
  (a + bi) + (c + di) = (a+c) + (b+d)i
  (a + bi) - (c + di) = (a-c) + (b-d)i
• MULTIPLICATION follows the distributive property, remembering that i² = -1.
  (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
• DIVISION often involves multiplying the numerator and denominator by the conjugate of the denominator to make the denominator real.

THE COMPLEX PLANE AND GEOMETRIC INTERPRETATION

The complex plane (also known as the Argand diagram) is a two-dimensional coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number. This allows for a geometric interpretation of operations with complex numbers.

IMPORTANCE

Complex numbers play a crucial role in mathematics and physics. They enable the analysis and solution of various problems related to oscillations, waves, and many other phenomena where real numbers do not suffice. They play a key role in numerous scientific disciplines, including electrical engineering, where they are used to analyze alternating currents; in quantum mechanics for describing the states of particles; and in various branches of engineering. They allow us to solve polynomial equations of all degrees (Fundamental Theorem of Algebra), which is the basis for more advanced mathematical theories and models.

CONCLUSION

These numbers are essential for understanding and applying advanced mathematical concepts. Complex numbers extend the boundaries of mathematical understanding and provide deeper insight into the nature of mathematical, physical, and technical problems. Their practical use demonstrates how necessary they are for progress in science and technology.