Where would I start in finding the real and complex solutions to the equation x^3+1=0?
let's solve this normally
x^3 + 1 = 0
=> x^3 = -1
=> x = (-1)^(1/3)
the real solution is of course, -1. since (-1)^3 = -1
there are no complex solutions to this as far as i can tell, unless we want to make things complicated and do something like i^(2/3), that way, (i^(2/3))^3 = i^2 = -1, and we could try to come up with a formula for all fractional powers of i that causes this to happen, but that's just overkill as far as i'm concerned
Hello, Gretchen!
Where would I start in finding the real and complex solutions to the equation: x³ + 1 = 0?
Factor: .(x + 1)(x² - x + 1) .= .0
Set each factor equal to 0 and solve.
. . x + 1 .= .0 . → . x = -1
. . x² - x + 1 .= .0 . . . This requires the Quadratic Formula.
. . . . . . . . . . .__________ . . . . . . . . _
. . . . . . .1 ± √1² - 4(1)(1) . . . . 1 ± i√3
. . x .= . --------------------- . = . ----------
. . . . . . . . . . 2(1) . . . . . . . . . . . .2
I want to spend some time explaining why this approach does not work.
Say we have,
x^2-1 = 0
We write
x^2 = 1
And take square roots,
x = 1.
But that is not true.
Because the statement sqrt(x^2)=x is false.
It should have been |x|=1
And then solve for "x".
Say we have,
x^4-1=0
Same thing,
x^4=1
|x|=1
But there is a problem.
|x| represens all complex numbers on the unit circle (have any idea what I am talking about?)
Thus, we want to find all complex numbers 1 unit away from the origin. The problem is there is an infinitude of such solutions. And we cannot check all of them. This is why this approach does not work.
Hello, ThePerfectHacker!
I want to spend some time explaining why this approach does not work.
Say we have: .x²-1 .= .0
We write: .x² = 1
And take square roots: .x = 1.
But that is not true.
Because the statement: sqrt(x²) = x is false.
It should have been: |x| = 1
And then solve for x.
Say, we have: .x^4 - 1 .= .0
Same thing: .x^4 = 1 . → . |x| = 1
But there is a problem.
|x| represents all complex numbers on the unit circle. .I disagree.
Thus, we want to find all complex numbers 1 unit away from the origin.
The problem is there is an infinitude of such solutions.
And we cannot check all of them.
This is why this approach does not work.
You are mixing absolute value of a real number
. . with the magnitude of a complex number.
If we solve: .x² = 9, we have: .|x| = 3
. . Then: .x = ±3 . . . and that's it.
But if we solve, say: .z^4 = 1
. . there are four fourth roots: .z .= .±1, ±i
These happen to be on a unit circle, but they are only four points.
. . They do not represent the entire circle.
You can write: .|z| = 1 . . . and this is true.
. . Each of the four roots is exactly one unit from the origin.
We take the nth root of 1, we get n roots . . . some real, some complex.
. . And all n of them lie on the unit circle.
They are the vertices of a regular n-gon.
It is certainly true that there are unaccountably many points on the unit circle. However, viewing those points as complex numbers there are exactly four of them the fourth power of which is -1.
That is impossible to do, because “absolute value of a real number and the magnitude of a complex number” is one and the same concept. It absolute value is simply distance. |w| is the distance from zero for any complex number,w, real or not.
In my defense, I will quote Plato. Who said that "absolute value" is the same concept for both complex and real numbers.
Furthermore, I was showing necessary but not sufficient condtions. In order to solve an equation we must find the necessary conditions and show that they are sufficient.