Necessary and sufficient x^3=1

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In summary, we have learned that the cube root is not a function and therefore taking the cube root of both sides of an equation will result in multiple solutions. In formal logic, this is represented as ##x^3=1 \Leftrightarrow (x=1\vee x=\omega\vee x=\omega^2)##. The complex roots can be retrieved easily by using the formula ##exp(2k \pi i /n)\sqrt[n]{y}##, where ##n## is the number of solutions needed.
  • #1
Mr Davis 97
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So we have ##x^3=1##. I have a really simple question. Why isn't it true that ##x^3 = 1## if and only if ##x = 1##, when we consider that if we take cube root of both sides and we can then take the cube again? Of course there are two other complex roots, but what am I missing in my naive logical argument?
 
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  • #2
The cube root is not a function, as it is a one-to-many mapping, in fact one-to-three. So taking the cube root of both sides gives three separate equations, not one: ##x=1,\ x=\omega## and ##x=\omega^2##, where ##\omega=e^{2\pi/3}##.

Expressed in formal logic, we have
$$x^3=1 \Leftrightarrow (x=1\vee x=\omega\vee x=\omega^2)$$

Given the properties of ##\vee## (OR), this allows us to deduce that
$$x=1\Rightarrow x^3=1$$
but not that
$$x^3=1 \Rightarrow x=1$$
 
  • #3
andrewkirk said:
The cube root is not a function, as it is a one-to-many mapping, in fact one-to-three. So taking the cube root of both sides gives three separate equations, not one: ##x=1,\ x=\omega## and ##x=\omega^2##, where ##\omega=e^{2\pi/3}##.

Expressed in formal logic, we have
$$x^3=1 \Leftrightarrow (x=1\vee x=\omega\vee x=\omega^2)$$

Given the properties of ##\vee## (OR), this allows us to deduce that
$$x=1\Rightarrow x^3=1$$
but not that
$$x^3=1 \Rightarrow x=1$$
I think I see. So with square roots, ##+ \sqrt[2]{x}## is the principal square root which defines a function, while ##\pm \sqrt[2]{x}## is the value of both roots. So analogously ##\sqrt[3]{x}## is the principal cube root, while... What's the analogous way of retrieving the two complex roots easily?
 
  • #4
^the n solutions of $$x^n=y$$ are
##exp(2k \pi i /n)\sqrt[n]{y}##
for
k=0,1,...,n-2,n-1

thus the complex cube roots of a positive real y are
##exp(2 \pi i /n)\sqrt[n]{y}=\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)\sqrt[n]{y}##
and
##exp(4 \pi i /n)\sqrt[n]{y}=\left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)\sqrt[n]{y}##
 
  • #5
Mr Davis 97 said:
What's the analogous way of retrieving the two complex roots easily?
Polynomials split over the complex numbers. So we have to look at complex multiplication. If we represent complex numbers as points in the plane, then multiplication means to multiply the real length of two numbers (= distance to the origin), and add there angles (= towards the real axis). This means in return, that numbers of length one are all on the unit circle, and multiplication of those is adding angles, i.e. in order to solve ##x^n=x \cdot \ldots \cdot x = 1##, we have to divide the unit circle in ##n## equal angles and all possible ##x## are those points on the circle which @lurflurf listed.
 

1. What does "necessary and sufficient x^3=1" mean?

The statement "necessary and sufficient x^3=1" means that for a value of x to satisfy the equation x^3=1, it must be both necessary and sufficient. In other words, the value of x is required (necessary) and enough (sufficient) to make the equation true.

2. How do you solve for x in "necessary and sufficient x^3=1"?

To solve for x in "necessary and sufficient x^3=1", you can take the cube root of both sides of the equation. This will give you the value of x that satisfies the equation.

3. What is the significance of "necessary and sufficient x^3=1" in mathematics?

The statement "necessary and sufficient x^3=1" is important in mathematics because it helps to define the concept of necessary and sufficient conditions. It is also used in various mathematical proofs and reasoning.

4. Can there be more than one value of x that satisfies "necessary and sufficient x^3=1"?

Yes, there can be more than one value of x that satisfies "necessary and sufficient x^3=1". In fact, there are three possible values of x that satisfy this equation, which are 1, -1/2 + i√3/2, and -1/2 - i√3/2. These values can be found by using the cube root function.

5. How is "necessary and sufficient x^3=1" related to the concept of equality?

The statement "necessary and sufficient x^3=1" is a way of expressing equality in mathematics. It means that the value of x is equal to 1, but it also implies that any other value of x would not satisfy the equation. In other words, it is a unique solution to the equation x^3=1, making it a strong form of equality.

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