Complex numbers are numbers that consist of a real part and an imaginary part. They are written in the form a + bi, where a and b are real numbers and i is the imaginary unit equal to the square root of -1.
To add or subtract complex numbers, you simply combine the real parts and the imaginary parts separately. For example, (3 + 4i) + (2 + 5i) = (3 + 2) + (4 + 5)i = 5 + 9i.
To multiply complex numbers, you use the distributive property and the fact that i^2 = -1. For example, (3 + 4i)(2 + 5i) = 6 + 15i + 8i + 20i^2 = (6 - 20) + (15 + 8)i = -14 + 23i. To divide complex numbers, you use the conjugate of the denominator to rationalize the denominator and then simplify as needed.
The complex conjugate of a complex number a + bi is a - bi. It is used in dividing complex numbers and in finding the absolute value of a complex number.
Complex numbers are used in a variety of fields, including physics, engineering, and finance. They are used to represent alternating current in electronics, solve differential equations in engineering, and model stock market fluctuations in finance.