Complex Number Power Formula
Complex Number Power Formula
The Complex Number Power Formula, which is frequently expressed in rectangular or standard form, is one with the formula a + ib, where a and b are real numbers and i (iota) is the imaginary component and stands for (-1). To calculate the value of a complex number raised to the power of “n”, use the Complex Number Power Formula. For instance, 10 + 5i is a complex number, where 10 denotes the real component and 5i denotes the imaginary component. The values of a and b determine whether they are entirely real or entirely fake. When a = 0, the result is a which is a strictly real number. When b = 0, the result is ib, which is a completely imaginary number.
The value of a Complex Number Power Formula raised to the power of “n” is determined using the complex number power formula. Remember that a complex number is represented by the formula x + iy, where x and y are real values and i is an imaginary number. If i is satisfied, then i2 = -1.
To obtain all of the roots that are not present in real numbers, real numbers must be extended. The Complex Number Power Formula is the term for this extension. If someone were to ask you what complex numbers are, it would simply be that they are an extension of real numbers that contain all of a polynomial’s degree n roots.
The set of numbers with the pattern a+ib is referred to as the Complex Number Power Formula, if i is defined as the answer to the equation x2 = -1.
Since the Complex Number Power Formula are frequently represented by the variable z, z=a+ib. Here, the numbers a and b stand in for the real and imaginary halves of the complex integer, respectively.
In this case, since 7 is independent of i it will be the real component, whereas 3 will be the imaginary part. i is referred to as iota in any complex number. And it has a value of 1.
On the Argand plane, commonly referred to as the complex plane or the Argand diagram, we graphically express. It resembles our standard coordinate plane, however whereas there the x-axis is, here the real axis is. The y-axis is located there, while the hypothetical axis is located here. An ordered pair is used to represent the Argand plane point a+ib (a,b).
A real number and an imaginary number are effectively combined to create the Complex Number Power Formula. The complex number is written as a+ib, where a and ib are real and imaginary numbers, respectively. Additionally, i = -1 and both a and b are real numbers.
Consequently, the Complex Number Power Formula is a straightforward illustration of the addition of two integers, specifically a real number and an imaginary number. It consists of two parts: one that is entirely real, the other entirely fantastical.
Combine comparable phrases while executing complicated number arithmetic operations like addition and subtraction. In other words, add imaginary numbers to imaginary numbers and real numbers to real numbers.
Formula to Calculate the Power of a Complex Number
The Complex Number Power Formula must first be converted to its polar form, which includes the modulus and argument as parts, in order to be expanded according to the exponent supplied. DeMoivre’s theorem is then applied, and it states:
According to De Moivre’s Formula, for all real values of a number, let’s say x,
(x cos + x sin)
Any integer n has the formula n = cos(nx) + isin(nx).
Question- Write the polar form of the square root of 5 + 12i.
Answer-The complex number is: 5 + 12i.
Square root of the given complex number = √(5 + 12i) = (5 + 12i)
r = √(25 + 144) = √169 = 13
tan θ = (12/5)
θ = tan-1(12/5) = 67.38
⇒arg z = 67.38
5 + 12i = 13ei67.38
(5 + 12i)
½ = (13)1/2e(i67.38)/2
√(5 + 12i) = √13 ei33.69
= √13 (cos 33.69 + i sin 33.69)