Real and imaginary numbers

Real and Imaginary Numbers

Basics of Real and Imaginary Numbers

  • Real numbers include all the numbers that can be found on the number line. This includes both rational and irrational numbers.
  • Imaginary numbers are numbers that cannot be found on the number line. They’re based on the imaginary unit, defined as i, where i² = -1.
  • The set of real numbers is denoted by R and the set of imaginary numbers by I.

Complex Numbers

  • Complex numbers are numbers that combine real and imaginary parts. They have the form a + bi, where a is the real part and bi is the imaginary part.
  • A complex number without an imaginary part is a real number. A complex number without a real part is an imaginary number.
  • The set of complex numbers is denoted by C. It brings together all real and imaginary numbers.

Modulus and Argument of Complex Numbers

  • The modulus of a complex number is the distance from the origin to the point represented by the complex number in the complex plane. It’s calculated using Pythagoras’ theorem in the form ** a + bi = √(a² + b²)**.
  • The argument of a complex number is the angle the line drawn from the origin to the point makes with the positive real axis. It can be calculated using trigonometric equations.

Complex Conjugate

  • The complex conjugate of a complex number is created by changing the sign between the real and imaginary parts of the original number.
  • For any complex number a + bi, its complex conjugate is a - bi.
  • Multiplication of a complex number by its conjugate results in a real number.

Polar Form of Complex Numbers

  • Complex numbers can also be represented in Polar form, which is z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.
  • This representation is particularly useful when multiplying or dividing complex numbers.

The Fundamental Theorem of Algebra

  • The Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root. This includes both real and imaginary roots.

De Moivre’s Theorem

  • De Moivre’s theorem is a formula that relates the powers of complex numbers to the product of their modulus and the sum of their argument and is particularly useful when dealing with high powers or roots of complex numbers.
  • The theorem states that ((r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ)) for any real number n.