My Math Forum Proof of uniqueness of positive cubes
 User Name Remember Me? Password

 Real Analysis Real Analysis Math Forum

 February 2nd, 2018, 07:32 PM #1 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 Proof of uniqueness of positive cubes Prove that there exists exactly one positive $t\in \mathbb{R}$ with $t^3=c$ for $c>0$. I assume that most of my derivation needs to be algebraic and using some simple axioms. So for example, if I define the set: $$T:=\lbrace t\in \mathbb{R}: t> 0, t^3 \leq c \rbrace$$ I must show that the cubed root of $c$ is the supremum of $T$ and then prove its uniqueness. But I'm not sure how to go about this. How can I concisely prove this?
 February 2nd, 2018, 08:42 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,553 Thanks: 1403 suppose there are $t_1\neq t_2$ such that $t_1^3=t_2^3=c$ then $t_1^3 - t_2^3 = 0$ $(t_1 - t_2)(t_1^2 + t_1 t_2 + t_2^2) = 0$ well we've assumed that $t_1 \neq t_2$ so it must be that $t_1^2 + t_1 t_2 + t_2 ^2 = 0$ $\left(t_1 - \dfrac{t_2}{2}\right)^2 + \dfrac{3 t_2^2}{4} = 0$ It should be pretty clear that this has no real solutions. Thus the original assumption is incorrect and $t_1 = t_2$ Thanks from greg1313, Maschke, Country Boy and 1 others

 Tags cubes, positive, proof, uniqueness

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Tau Math 40 December 31st, 2016 06:23 PM annakar Number Theory 1 December 4th, 2012 11:06 AM aeromantang Applied Math 3 December 28th, 2011 10:34 AM jstarks4444 Number Theory 2 November 16th, 2010 12:19 PM johnmath Advanced Statistics 1 July 24th, 2010 06:56 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.