Golden Ratio Version 1


a+ba=abab+b2=a2b2+2(a2)b+(a2)2=a2+(a2)2(b+a2)2=a2+a24b+a2=±5a24b=(1±5)a2\begin{aligned} \frac{a+b}{a}&=\frac{a}{b}\\ ab+b^2&=a^2\\ b^2+2\left(\frac{a}{2}\right)b+\left(\frac{a}{2}\right)^2&=a^2+\left(\frac{a}{2}\right)^2\\ \left(b+\frac{a}{2}\right)^2&=a^2+\frac{a^2}{4}\\ b+\frac{a}{2}&=±\sqrt{\frac{5a^2}{4}}\\ b&=\frac{\left(1±\sqrt5\right)a}{2} \end{aligned}

ϕ=ab=a(1±5)a2=2a1±5a=2(15)15=1±52\begin{aligned} \phi&=\frac{a}{b}\\ &=\frac{a}{\frac{\left(1±\sqrt5\right)a}{2}}\\ &=\frac{2a}{1±\sqrt5a}\\ &=\frac{2\left(1∓\sqrt5\right)}{1-5}\\ &=\boxed{\frac{1±\sqrt5}{2}} \end{aligned}

Note by Gandoff Tan
1 year, 10 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

Why is the plus-minus sign flipped upside down? @Gandoff Tan.

A Former Brilliant Member - 1 year, 1 month ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...