Why the cube root of two is no existing number on the real number line?
- JimLv 74 weeks ago
Cube root are "closed" (always possible) in Real Numbers.
- sepiaLv 74 weeks ago
Given a number x, the cube root of x is a number a such that a^3 = x.
If x positive a will be positive, if x is negative a will be negative.
- PinkgreenLv 71 month ago
Let x=2^(1/3)=1.2599210... is an irrational number, it must exist in the real
axis theoretically & 1<x<1.26. Not like sqr(2) or sqr(3) which individually can
be represented by a line segment & marked on the real axis. Maybe, x could
be, I am not certain.
- MorningfoxLv 71 month ago
The real cube root of 2 = 1.2599210498948731647.... That's certainly on the real number line.
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- KrishnamurthyLv 71 month ago
In mathematics, the cube root of a number x is a number y such that y^3 = x.
All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots.
- Wayne DeguManLv 71 month ago
2¹/³ => 2¹/³[cos2nπ + isin2nπ]¹/³
so, 2¹/³[cos(2nπ/3) + isin(2nπ/3)]
With n = 0, we have, 2¹/³[cos(0) + isin(0)] = 2¹/³
With n = 1, we have, 2¹/³[cos(2π/3) + isin(2π/3)]
i.e. 2¹/³(-1/2 + i√3/2) => -0.63 + 1.09i
With n = 2, we have, 2¹/³[cos(4π/3) + isin(4π/3)]
i.e. 2¹/³(-1/2 - i√3/2) => -0.63 - 1.09i
So 2¹/³ = 1.26 is indeed a real cube root of 2
The other two cube roots are imaginary.
- 1 month ago
I'm not sure what you mean by that, the real-valued cube root of 2 is definitely a real number. There are also two more which are complex if you'd like?
- no sea naboLv 61 month ago
∛2 e⁰ ≈ 1.25992 (real, principal root)
∛2 e^((2 i π)/3) ≈ -0.6300 + 1.0911 i
∛2 e^(-(2 i π)/3) ≈ -0.6300 -1.0911 i
- PuzzlingLv 71 month ago
There are 3 cube roots of 2. One is real (but irrational) and the other two are complex.
The principal cube root is real. Remember just because a number is irrational (can't be represented as the ratio of two integers) doesn't mean it isn't real. It is real and on the number line. It's between 1 and 2 which can be shown as follows:
1 < 2 < 8
∛1 < ∛2 < ∛8
1 < ∛2 < 2
Btw, the link below contains a proof that the (principal) cube root of 2 is irrational, if you need that.
- BaalLv 61 month ago
My ancient Scientific Calculator says the cubed root of 2 is 1.25992105 and so does this online calculator.