Could someone verify how this simplifies?

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In summary, the conversation revolved around the simplification of the expression $$\sqrt{\frac{1}{2}(1+\sqrt{5})+1}$$ into $$\frac{1}{2}(\sqrt{5}+1)$$ and the confusion surrounding it. The participants discussed the use of the golden ratio and nested radicals to denest the expression. Despite some initial doubt, it was eventually proven that the two expressions are equivalent. The conversation ended with the departure of the original poster, who was unable to provide proof of their conjecture.
  • #1
TheGreatEscapegoat
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I don't know why but I'm having this mental block where I can't see how this simplifies.

I have the expression $$ \frac{1}{2}(1+ \sqrt{5}) $$. Now, I substitute for x into the expression $$ \sqrt{x+1}. $$ to make $$ \sqrt{\frac{1}{2}(1+ \sqrt{5})+1}. $$
The result should be $$\frac{1}{2}( \sqrt{5}+1)$$.
However, when I simplify it, I obtain $$ \sqrt{\frac{1}{2}( \sqrt{5}+3)} $$ and for some reason I am not seeing how to simplify that into $$ \frac{1}{2}( \sqrt{5}+1). $$
 
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  • #3
lurflurf said:
notice that

$$x^2=x+1$$

verify this

that is what makes the golden ratio a special number

https://en.wikipedia.org/wiki/Golden_ratio
I don't see how that proves the simplification. If you don't know how to prove it that's fine I have nothing against you for it, and this isn't meant to be insulting, it's just very commonly held that it is rude to interject random comments for your own personal sake that don't actually address the topic.
 
  • #4
TheGreatEscapegoat said:
when I simplify it, I obtain $$ \sqrt{\frac{1}{2}( \sqrt{5}+3)} $$ and for some reason I am not seeing how to simplify that into $$ \frac{1}{2}( \sqrt{5}+1). $$
It would be astounding if your COULD see it since it is not true

EDIT: HM ... seems I was mistaken
 
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  • #6
phinds said:
It would be astounding if your COULD see it since it is not true
I thought it was wrong too, so I checked it with my calculator. Seems it is true, but I'm not seeing the algebra to make it true yet...
 
  • #7
berkeman said:
I thought it was wrong too, so I checked it with my calculator. Seems it is true, but I'm not seeing the algebra to make it true yet...
Yep. Me too.
 
  • #8
Ha. Got it. It does simplify. I think lurflurf knew that all along :smile:
 
  • #9
phinds said:
Ha. Got it. It does simplify. I think lurflurf knew that all along :smile:
That's a fine conjecture, but you're missing proof, so for all I know you are wrong. Remember that the goal is to prove the simplification, not that it is a fixed point of $$ f(x) = \sqrt{x+1}$$
 
  • #10
TheGreatEscapegoat said:
That's a fine conjecture, but you're missing proof, so for all I know you are wrong.
You may consider it to be so if you wish. lurflurf gave you the only hint you need
 
  • #11
phinds said:
You may consider it to be so if you wish. lurflurf gave you the only hint you need
Again, that comment doesn't mean anything without proof. Furthermore, it is rude and counter-productive of you to assume you have the right to arbitrarily dictate what others need and don't need, you only determine what you need.
 
  • #12
TheGreatEscapegoat said:
Again, that comment does mean anything without proof. Furthermore, you don't determine what others need and don't need, you only determine what you need.
I think you misunderstand the methodology of PF. We don't give answers, we help people figure out how to GET the answer. You've been given a sufficient clue. Ignore it if you so choose.
 
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  • #13
Thread closed temporarily for Moderation...
 
  • #14
You can verify that they are equivalent by squaring both values.

The first gives 1/2 ( sqrt 5 + 3 )

And the second gives you 1/4 ( sqrt 5 + 1 ) * ( sqrt 5 + 1 )

Which becomes 1/4 ( 5 + 2*sqrt5 + 1 ) which becomes 1/4 ( 6 + 2*sqrt5)
Which reduces to 2/4 ( sqrt5 + 3 ) or more simply 1/2 ( sqrt 5 + 3 )

So they are indeed equal.

By reversing the verification you could derive the first expression from the second.
 
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  • #15
TheGreatEscapegoat said:
That's a fine conjecture, but you're missing proof, so for all I know you are wrong.

phinds said:
You may consider it to be so if you wish.

TheGreatEscapegoat said:
Again, that comment doesn't mean anything without proof.

phinds said:
I think you misunderstand the methodology of PF. We don't give answers, we help people figure out how to GET the answer.
Since this is your (@TheGreatEscapegoat) problem, the burden of proof is on you, not us.
@phinds is correct in what he says about our principles here at PF.
 
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  • #16
berkeman said:
Thread closed temporarily for Moderation...
After a long PM conversation, the OP has left the building. Thank you to everyone who tried to help him in this thread. Interesting links, BTW. :smile:
 
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Related to Could someone verify how this simplifies?

1. How do you verify simplification in scientific research?

In order to verify simplification in scientific research, one must carefully examine the original data and the simplified version to ensure that the simplification process did not omit any important information or introduce any errors. This can be done through various methods such as peer review, statistical analysis, and replication of the original experiment.

2. Why is it important to verify simplification in scientific studies?

It is important to verify simplification in scientific studies because it ensures the accuracy and validity of the research findings. Simplification is often necessary to make complex data more understandable, but it must be done carefully to avoid misleading or incorrect conclusions.

3. How can one determine if a simplification is accurate?

The accuracy of a simplification can be determined by comparing the simplified version to the original data and evaluating the level of similarity. If the simplified version is sufficiently similar to the original data, it can be considered accurate. Additionally, seeking input and feedback from other experts in the field can also help verify the accuracy of a simplification.

4. What are some potential challenges in verifying simplification?

One potential challenge in verifying simplification is that it can be subjective. Different individuals may have different opinions on what is considered a simplification and whether it is accurate. Another challenge is that simplification may lead to loss of important details, making it difficult to determine the accuracy of the simplified version.

5. How can one improve the process of verifying simplification?

To improve the process of verifying simplification, it is important for researchers to be transparent about their simplification methods and to provide detailed explanations for why certain information was omitted or simplified. Additionally, seeking feedback and input from other experts and conducting further experiments or analyses can help strengthen the verification process.

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