# Why the cube root of two is no existing number on the real number line?

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• Jim
Lv 7
4 weeks ago

Cube root are "closed" (always possible) in Real Numbers.

• sepia
Lv 7
4 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.

• 1 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.

• 1 month ago

The real cube root of 2 = 1.2599210498948731647.... That's certainly on the real number line.

• 1 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.

• 1 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?

• 1 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

• 1 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.

• Baal
Lv 6
1 month ago

My ancient Scientific Calculator says the cubed root of 2 is 1.25992105 and so does this online calculator.