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Complex Numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as i = √(-1). In this expression, a is called the real part, and bi is called the imaginary part of the complex number.

Complex numbers extend the concept of real numbers by introducing the imaginary unit i, which allows us to represent numbers beyond the real number line. The real part a represents the position of the number on the real number line, while the imaginary part bi represents the position on the imaginary number line.

Complex numbers can be added, subtracted, multiplied, and divided using rules similar to those for real numbers. Additionally, they can be represented geometrically on the complex plane, where the real part corresponds to the horizontal axis (often called the real axis) and the imaginary part corresponds to the vertical axis (often called the imaginary axis).

Complex numbers find applications in various fields of mathematics, physics, engineering, and other sciences. They are used in solving equations, analyzing alternating currents in electrical engineering, studying wave functions in quantum mechanics, and many other areas where mathematical modeling is required.

Common Formulas Related to Complex Numbers:

  1. Addition and Subtraction:

    [ (a + bi) + (c + di) = (a + c) + (b + d)i ]

    [ (a + bi) - (c + di) = (a - c) + (b - d)i ]

  2. Multiplication:

    [ (a + bi) × (c + di) = (ac - bd) + (ad + bc)i ]

  3. Conjugate:

    If ( z = a + bi ) is a complex number, then its conjugate is denoted by ( z ) and is given by:

    [ ( z ) = a - bi ]

  4. Modulus or Absolute Value:

    If ( z = a + bi ) is a complex number, then its modulus or absolute value is denoted by ( |z| ) and is given by:

    [ |z| = √(a2 + b2 ) ]

  5. Division:

    If ( z1 = a + bi ) and ( z2 = c + di ) are complex numbers where ( z2 ) is not zero, then the division of ( z1 ) by ( z2 ) is given by:

    [ (z1 / z2) = (a + bi) / (c + di) = (a + bi) / (c + di) × (c - di) / (c - di) = ((ac + bd) + (bc - ad)i) / (c2 + d2) ]

These formulas are fundamental in performing operations involving complex numbers.