A complex number is a number that has both a real and an imaginary part. It is 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.
The basic operations on complex numbers are addition, subtraction, multiplication, and division. Addition and subtraction are carried out by adding or subtracting the real parts and the imaginary parts separately. Multiplication is done by using the distributive property and the fact that i squared is equal to -1. Division is done by multiplying the numerator and denominator by the complex conjugate of the denominator.
To graph a complex number, you can use the real axis as the x-axis and the imaginary axis as the y-axis. The complex number a + bi can be plotted as the point (a, b) on the complex plane.
The conjugate of a complex number a + bi is the complex number a - bi. It is obtained by changing the sign of the imaginary part. Conjugates are useful in simplifying complex expressions and finding the modulus of a complex number.
Complex numbers are used in various fields of science, such as physics, engineering, and mathematics. They are particularly useful in representing and solving problems involving alternating currents, quantum mechanics, and signal processing. They also have applications in data analysis, image processing, and cryptography.