I'm not sure what you mean by "z= a+ bi assumption". Typically that is NOT an "assumption", it is one way of defining complex numbers.Outrageous said:Find (2+6i)^(1/3). Then we need to let z=a+bi, then z^3... Then we will get a=? And b=?
Why we can do z=a+bi assumption? Any prove show what we assume is correct and how do we know the answer come out is correct?
HallsofIvy said:One can also show that [itex](0, 1)^2= -1[/itex]: (0, 1)(0, 1)= (0(0)- 1(1), 0(1)+1(0))= (-1, 0) which, as above, we identify with the real number -1. We define "i= (0, 1)" so that [itex]i^2= -1[/itex]. And, then, we can write (a, b)= (a, 0)+ (0, b)= (a, 0)+ (b, 0)(0, 1)= a+ bi.
HallsofIvy said:Yes, that is write. By any definition of "complex" number, we can write any complex number in the form z= a+ bi for some real numbers "a" and "b". That form is also "unique" that is, if z= a1+ b1i and the same z= a2+ b2i, we must have a1= a2 and b1= b2.
Now, in this problem, attempting to find (2+ 6i)1/3, they are suggesting that you write the answer as "a+ bi", find (a+ bi)3 and set the "real" and "imaginary" parts equal. For example, (a+ bi)2= (a+ bi)(a+ bi)= a2+ 2abi+ b2i2= (a2- b2+ 3abi. If we wanted to find (1+ i)1/2, we could now set s2- b2= 1 and ab= 1, giving two equations to solve for a and b.
In order to prove that the assumption a+bi is correct, we must first understand what it means. In this context, a+bi represents a complex number with a real part (a) and an imaginary part (bi). Therefore, when we raise this complex number to the power of 1/3, we are essentially finding the cube root of the complex number. To prove that the assumption is correct, we can use the binomial theorem to expand (a+bi)^3 and compare it to the original complex number (2+6i). If they are equal, then our assumption is correct.
Proving the assumption a+bi is correct is important because it ensures the accuracy of our calculation and guarantees that we have found the correct solution. Without proving the assumption, there is a possibility of error in our calculation and we cannot be certain that we have found the correct answer.
To find the cube root of (2+6i), we first need to express it in polar form. This can be done by converting the complex number to its magnitude and argument form using the Pythagorean theorem and inverse tangent function. Then, we can use De Moivre's theorem to find the cube root by taking the cube root of the magnitude and dividing the argument by 3. Finally, we can convert the result back to rectangular form to get the final answer.
One common mistake is not fully understanding the concept of a+bi and how it relates to finding the cube root of a complex number. Another mistake is making errors in the calculations, such as not properly applying the binomial theorem or making mistakes in converting from rectangular to polar form. It is also important to check for extraneous solutions when using De Moivre's theorem.
Yes, there are alternative methods such as using the concept of conjugates and the fundamental theorem of algebra. However, these methods may be more complex and may not always yield a simple and straightforward solution. Therefore, using the binomial theorem and De Moivre's theorem is the most commonly used and efficient method for proving the assumption a+bi is correct.