write the potarform of complex number
Answers
Answer:
Step-by-step
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted using the symbol {\displaystyle \mathbb {C} }\mathbb {C} . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation
{\displaystyle (x+1)^{2}=-9}{\displaystyle (x+1)^{2}=-9}
has no real solution, since the square of a real number cannot be negative. Complex numbers, however, provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case, the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1:
{\displaystyle ((-1+3i)+1)^{2}=(3i)^{2}=\left(3^{2}\right)\left(i^{2}\right)=9(-1)=-9,}{\displaystyle ((-1+3i)+1)^{2}=(3i)^{2}=\left(3^{2}\right)\left(i^{2}\right)=9(-1)=-9,}
{\displaystyle ((-1-3i)+1)^{2}=(-3i)^{2}=(-3)^{2}\left(i^{2}\right)=9(-1)=-9.}{\displaystyle ((-1-3i)+1)^{2}=(-3i)^{2}=(-3)^{2}\left(i^{2}\right)=9(-1)=-9.}
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th-century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers—in his attempts to find solutions to cubic equations.
Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i This means that complex numbers can be added, subtracted and multiplied as polynomials in the variable i, under the rule that i2 = −1. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.
plz mark as brilliant