)Complex numbers are numbers that have a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part and bi is the imaginary part. Real numbers only have a single value on the number line, while complex numbers have a two-dimensional representation on the complex plane.
To add or subtract complex numbers, simply combine the real parts and the imaginary parts separately. To multiply complex numbers, use the FOIL method just as you would with binomials. To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator, which is the same expression with the sign of the imaginary part flipped.
The imaginary unit, i, is defined as the square root of -1. It allows us to represent numbers that cannot be represented on the real number line. It also plays a crucial role in the fundamental theorem of algebra and in many other mathematical applications.
Complex numbers have various applications in fields such as engineering, physics, and economics. They are used to describe oscillatory motion, electrical circuits, and control systems. They are also used in signal processing and image processing.
Yes, complex numbers can have real world values. For example, in electrical engineering, the impedance of a circuit can be a complex number with both a real and imaginary component. In physics, the wave function in quantum mechanics can also be represented by a complex number. Complex numbers are often used to model and solve real world problems.