In mathematics, complex numbers are usually represented by the letter Z, where Z is a complex number, and the real and imaginary parts of Z are represented by "Re(z)" and "Im(z)". Real numbers can be regarded as a special case of complex numbers, that is, complex numbers with imaginary numbers of 0.
The complex form is a+bi, where a and b are real numbers, and I is an imaginary unit, satisfying I 2 =- 1. Both the real part and imaginary part of a complex number are real numbers, and four operations can be performed respectively. Complex numbers also have many applications, such as signal processing, power engineering, physics and economics.
Numbers in the form of a+bi (both a and b are real numbers) are complex numbers, where a is called the real part, b is called the imaginary part, and I is the imaginary part. Complex numbers are usually represented by z, that is, z=a+bi. When the imaginary part b of z = 0, z is a real number. When the imaginary part b≠0 and the real part a = 0 of z, z is often called pure imaginary number.
Complex number field is an algebraic closure of real number field, that is, any polynomial with complex coefficients always has roots in complex number field.
The concept of complex number originates from the solution of quadratic equation, which can be expressed as a+bi, where a and b are real numbers. By introducing imaginary unit I, we can express the solution of quadratic equation as the sum of real number and imaginary number.
Complex numbers have many applications in mathematics. For example, when solving some problems involving square roots, we need to use complex numbers to represent the square roots of negative numbers. In addition, complex numbers are widely used in signal processing, power engineering, physics and economics.
Besides the real and imaginary unit I, complex numbers can be expressed in other forms. For example, we can use polar coordinates to represent complex numbers, that is, z = r(cosθ+isinθ), where r is the mode length and θ is the radiation angle. This representation method can conveniently deal with some problems involving rotation and translation.
The four operations of complex numbers include addition, subtraction, multiplication and division. In addition and subtraction of complex numbers, we need to add and subtract the real part and imaginary part respectively. In the multiplication and division of complex numbers, we can use distribution law and associative law to calculate.