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Phi — Definition, Formula & Examples

Phi  Φ


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Key Formula

φ=1+52≈1.6180339887…\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\ldots
Where:
  • φ\varphi = The golden ratio, often written as φ (lowercase phi) or Φ (uppercase phi)

Worked Example

Problem: Verify that phi satisfies the property φ² = φ + 1.
Step 1: Write the exact value of phi.
φ=1+52\varphi = \frac{1 + \sqrt{5}}{2}
Step 2: Square it by squaring the numerator and denominator.
φ2=(1+5)24=1+25+54=6+254=3+52\varphi^2 = \frac{(1 + \sqrt{5})^2}{4} = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2}
Step 3: Now compute φ + 1 and check that it equals φ².
φ+1=1+52+1=1+5+22=3+52\varphi + 1 = \frac{1 + \sqrt{5}}{2} + 1 = \frac{1 + \sqrt{5} + 2}{2} = \frac{3 + \sqrt{5}}{2}
Answer: Both expressions equal 3+52\frac{3 + \sqrt{5}}{2}, confirming that φ2=φ+1\varphi^2 = \varphi + 1. This self-referential property is what makes phi unique.

Why It Matters

Phi appears throughout mathematics in the Fibonacci sequence, where the ratio of consecutive terms approaches φ as the sequence grows. It also shows up in geometry — a regular pentagon's diagonal-to-side ratio equals φ. Artists and architects have used the golden ratio for centuries as a guide for aesthetically pleasing proportions.

Common Mistakes

Mistake: Confusing phi (φ ≈ 1.618) with pi (π ≈ 3.14159).
Correction: Pi is the ratio of a circle's circumference to its diameter. Phi is the golden ratio. They are completely different constants despite their similar-sounding Greek-letter names.

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