What is the Cartesian form of a complex number?

The Cartesian form of a complex number is a representation of a complex number in terms of its real and imaginary parts.

In the Cartesian form, a complex number is written as a+bi, where a is the real part and b is the imaginary part. The real part represents the horizontal axis and the imaginary part represents the vertical axis on the complex plane.

To convert a complex number from polar form to Cartesian form, we use the following formula:

a+bi = r(cosθ + i sinθ)

where r is the modulus of the complex number and θ is the argument of the complex number. For more about the relationship between Cartesian and trigonometric forms of complex numbers, see Trigonometric Form of Complex Numbers.

To convert a complex number from Cartesian form to polar form, we use the following formulas:

r = √(a^2 + b^2)

θ = tan^-1(b/a)

where r is the modulus of the complex number and θ is the argument of the complex number.

The Cartesian form of a complex number is useful for performing arithmetic operations such as addition, subtraction, multiplication, and division. To add or subtract complex numbers in Cartesian form, we simply add or subtract the real and imaginary parts separately. To multiply or divide complex numbers in Cartesian form, we use the distributive property and FOIL method. Explore the underlying operations in more detail on our page about Operations with Complex Numbers.

In summary, the Cartesian form of a complex number is a representation of a complex number in terms of its real and imaginary parts, and is useful for performing arithmetic operations. This form is also essential when we delve into more advanced topics such as Differentiation Techniques used in calculus.