Complexzahlen

Complexzahlen is a mathematical concept that extends the real number system by introducing the imaginary unit, denoted by 'i'. The imaginary unit is defined as the square root of negative one (i² = -1). A complex number is generally expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. Both 'a' and 'b' are real numbers.

The set of complex numbers is denoted by the symbol C. Every real number can be considered a complex number with an imaginary part of zero (e.g., 5 can be written as 5 + 0i). Complex numbers are used extensively in various fields of mathematics, physics, and engineering. They are particularly useful for solving algebraic equations that have no real solutions, such as x² + 1 = 0.

Arithmetic operations like addition, subtraction, multiplication, and division can be performed with complex numbers. For addition and subtraction, the real parts are combined separately, and the imaginary parts are combined separately. Multiplication involves distributing the terms, remembering that i² = -1. Division typically involves multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi.

Geometrically, complex numbers can be represented on a complex plane, also known as the Argand diagram. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. This geometric interpretation provides a visual understanding of complex number operations and their properties. The modulus of a complex number represents its distance from the origin in the complex plane, and the argument represents the angle it makes with the positive real axis.