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 (√-1). Complex numbers are typically used in mathematics and engineering to represent quantities that cannot be expressed using only real numbers.
To add or subtract complex numbers, simply combine the real parts and the imaginary parts separately. For example, to add (3 + 4i) and (2 + 5i), we add 3 and 2 to get the real part 5, and add 4i and 5i to get the imaginary part 9i. So the sum is (5 + 9i). To subtract, we follow the same process but with subtraction instead of addition.
The conjugate of a complex number a + bi is the number a - bi. In other words, it is the complex number with the same real part but with the sign of the imaginary part flipped. For example, the conjugate of 3 + 4i is 3 - 4i.
To multiply complex numbers, we use the FOIL method (First, Outside, Inside, Last). For example, to multiply (3 + 4i) and (2 + 5i), we multiply 3 by 2 to get 6, then 3 by 5i to get 15i, then 4i by 2 to get 8i, and finally 4i by 5i to get 20i². Simplifying, we get (6 + 23i - 20) which becomes (-14 + 23i). To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator and then simplify.
Complex numbers have a wide range of applications in various fields such as physics, engineering, economics, and even in the study of fractals. They are used to represent alternating currents in electrical engineering, to solve differential equations in physics, and to model financial markets in economics. They also have applications in signal processing, control systems, and quantum mechanics.