zhermes said:Yeah! that's exactly right.
If you imagine that complex numbers are a position in the x-y plane, where the x-axis is real numbers and the y-axis is imaginary numbers; then a+ib is just a standard rectilinear (Cartesian) way of describing a point [e.g. 5+2i = 5x + 2y = (5,2) ]; when you use r*e^{i\theta}, its like writing it in polar coordinates, r is the magnitude, and theta the angle with the x-axis.
"Complex functions" are the general term for functions which operate on (or yield) complex numbers. But note, you have to input two scalars (the equivalent of a complex number)madah12 said:I mean like f(x)=2x+2ix and you input (2-i) and get 6+2i or does no such thing exist in mathematics?
The polar form of a complex number is a way of representing a complex number in terms of its magnitude (r) and angle (θ) from the positive real axis. It is written in the form r∠θ, where r is the modulus (distance from the origin to the complex number) and θ is the argument (angle between the positive real axis and the complex number).
To convert a complex number from rectangular form (a + bi) to polar form (r∠θ), you can use the following formula: r = √(a² + b²), θ = tan⁻¹(b/a). This formula calculates the modulus and argument of the complex number, which can then be used to write it in polar form.
The polar and rectangular forms of a complex number are equivalent, meaning they represent the same complex number. The rectangular form is written as a + bi, where a and b are real numbers, while the polar form is written as r∠θ, where r is the modulus and θ is the argument. Both forms contain the same information about the complex number, just in different representations.
The advantage of using polar form is that it makes it easier to perform certain operations on complex numbers, such as multiplication, division, and exponentiation. In polar form, these operations involve simply adding or subtracting angles and multiplying or dividing moduli, which is often simpler than using the algebraic rules for rectangular form.
In polar form, the complex number r∠θ can be plotted on a polar coordinate plane, with r as the distance from the origin and θ as the angle from the positive real axis. This allows for a visual representation of the complex number and can be useful in understanding the properties of complex numbers, such as conjugates and roots.