Complex numbers are a form of numbers in mathematics, which consists of two parts, the real part and the imaginary part. The common forms of plural are as follows:
Standard form: The form of a+bi, where a and b are both real numbers, and i is an imaginary unit, satisfying the condition of i = -1.
Exponential form: Shaped like r*(cos +isin), where r is the modulus of a complex number, is the radial angle of a complex number.
Triangular form: Shape like r (cos + isin), where is the amplitude of the complex number.
Polar form: Shape r(cos +isin), where r is the modulus of a complex number, is the radial angle of a complex number.
Cartesian form: A form of the form a+bi, where a and b are both real numbers.
Quaternion form: the form of q=a+bi+cj+dk, where a, b, c, d are real numbers, and i, j, k are quaternion units.
To convert complex numbers into exponential form, Euler's formula can be used. Euler's formula is e (ix) = cos(x) + i*sin(x). where i is an imaginary unit satisfying i 2 = -1.
The specific steps are as follows:
Determine the real and imaginary parts of complex numbers. For example, the complex number z = a + bi, where a is the real part and b is the imaginary part.
Calculate the modulus length r = sqrt(a2 + b2) for complex numbers.
Calculate the radial angle of the complex number theta = atan2(b, a). Note that the atan2 function is needed here, as it can handle the four-quadrant case.
Euler's formula is used to convert complex numbers into exponential form. According to Euler's formula, we have e (ix) = cos(x) + isin(x). Thus, we can convert the complex number z = a + bi into the exponential form re.
Finally, we get a complex number with a modulus length of r and a radial angle of . This is the exponential form of the plural number.