My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
March 26th, 2014, 10:51 PM   #1
Senior Member
 
shunya's Avatar
 
Joined: Oct 2013
From: Far far away

Posts: 422
Thanks: 18

Prove 1+sqrt(3) is irrational

Question) Prove that 1 + sqrt3 is an irrational number. Show similarly that m + n(sqrt3) is an irrational number for all rationals m and n (n not 0)

In the question, sqrt3 means square root of 3. I will use the same notation everywhere in the question


My attempt: We will assume (correctly) that sqrt3 is irrational

Let 1 + sqrt3 = a/b
implies sqrt3 = (a/b) - 1
implies sqrt3 = (a-b)/b
implies sqrt3 is rational
but sqrt3 is irrational
Therefore by contradiction, 1 + sqrt3 is irrational

Second part of question:

Let m + n(sqrt3) = a/b
implies n(sqrt3) = (a/b) - m
implies n(sqrt3) = (a - bm)/b
implies sqrt3 = (a-bm)/bn
implies sqrt3 is rational
but sqrt3 is irrational
Therefore by contradiction, m + n(sqrt3) is irrational

Are my proofs acceptable? Is there a better way to do this because after looking at similar problems, this has become rather mechanical for me.
Thanks
shunya is offline  
 
March 26th, 2014, 11:51 PM   #2
Senior Member
 
Joined: Jun 2013
From: London, England

Posts: 1,316
Thanks: 116

Re: Prove 1+sqrt(3) is irrational

Those proofs are fine.
Pero is offline  
March 28th, 2014, 05:39 PM   #3
Senior Member
 
Joined: Nov 2013

Posts: 246
Thanks: 2

Ah but you can multiply any square root by another square root by just multiplying what is in the radicals. Sqrt 2 * Sqrt 50. Both of these are irrational but multiplying them gives 10, a rational number. Why? Because you multiply 2 * 50 = 100 and 100 is a perfect square.
caters is offline  
March 28th, 2014, 09:52 PM   #4
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,617
Thanks: 2608

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by shunya View Post
We will assume (correctly) that sqrt3 is irrational
I would prove this rather than assuming it. The best proof I know of is to suppose that m, n, p and q are integers, both and are in their lowest terms and

Now, since p and q share no factors, every factor of q^2 must divide n. In other words

But since m and n are coprime, and so .
In other words, there is no rational number that is a perfect square unless both m and n are perfect squares. In particular, when n=1, an integer cannot be the square of a rational number, unless that number itself is an integer. And is not an integer because so .
v8archie is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
irrational, prove, sqrt3



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Prove cuberoot 2 is irrational shunya Elementary Math 2 March 20th, 2014 04:11 AM
Prove square root 3 is irrational shunya Elementary Math 2 March 20th, 2014 04:06 AM
prove that Z[sqrt(2)] is an integral domain rayman Abstract Algebra 7 March 6th, 2012 09:18 AM
Prove that if x is irrational then thedarjeeling Real Analysis 1 February 4th, 2012 02:21 PM
How to prove that "i-(ioota)" is irrational? muhammadmasood Abstract Algebra 6 December 31st, 1969 04:00 PM





Copyright © 2019 My Math Forum. All rights reserved.