kompleksikonjugaatti

Kompleksikonjugaatti refers to the complex conjugate of a complex number. 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, satisfying the equation i² = -1. The real part of the complex number is 'a', and the imaginary part is 'b'.

The complex conjugate of a complex number a + bi is denoted by a bar over the complex number, like $\overline{a+bi}$, or sometimes as a* or a'. It is obtained by changing the sign of the imaginary part of the complex number. Therefore, the complex conjugate of a + bi is a - bi. For example, the complex conjugate of 3 + 4i is 3 - 4i, and the complex conjugate of 2 - 5i is 2 + 5i. If a complex number is purely real (i.e., its imaginary part is zero), its complex conjugate is itself. Similarly, if a complex number is purely imaginary (i.e., its real part is zero), its complex conjugate is its negative.

Complex conjugation is an important operation in complex analysis and has many useful properties. For instance, the product of a complex number and its conjugate is always a non-negative real number: (a + bi)(a - bi) = a² - (bi)² = a² - b²i² = a² + b². This value is also the square of the modulus (or absolute value) of the complex number. Another property is that the conjugate of a sum or difference of complex numbers is the sum or difference of their conjugates, and similarly for multiplication and division. The complex conjugate is also used in defining the inverse of a complex number and in solving certain types of equations.