What is the 11th Fibonacci number?

F(11) = 89. It comes from adding the two terms before it: F(10) + F(9) = 55 + 34 = 89. It has 2 digits, and the terms either side are 55 and 144.

Is 89 a prime number?

Yes — 89 is a Fibonacci prime. Its index 11 is also prime, which is necessary for any Fibonacci prime above F(4). Whether the sequence contains infinitely many primes is still unproved.

What is F(11) divided by F(10)?

89 ÷ 55 = 1.618181818. The golden ratio is φ = 1.618033989, so this is off by 1.48e-4 — above it, since the ratios alternate on either side as they close in.

What are the Lucas number and running total at index 11?

L(11) = 199, which is F(10) + F(12) = 55 + 144. The running total F(1) + … + F(11) is 232, which equals F(13) − 1 = 233 − 1 — an identity that holds at every index, not just this one. The squares up to here sum to F(11) × F(12) = 12,816.

Do Fibonacci numbers really turn up in nature?

Some of the claims hold and some do not. Spiral counts in sunflower heads, pinecones and pineapples genuinely are Fibonacci numbers, with a real mechanism behind it: placing each new growth about 137.5° round from the last packs seeds most efficiently, and that packing produces Fibonacci counts. The nautilus shell, however, is a logarithmic spiral but not a golden one, and golden-ratio claims about the Parthenon or the Mona Lisa are retrofitted rather than documented.

What is the sum of the first n Fibonacci numbers?

F(n+2) − 1, always. Add the first ten and you get 143, which is 144 − 1. The identity is easy to see once stated: each term is a difference of two later ones, so the sum collapses. The sum of the squares is just as tidy — F(1)² + … + F(n)² = F(n) × F(n+1) — which is the algebraic statement of the square tiling that builds the spiral.

What is Cassini’s identity?

F(n)² − F(n−1)F(n+1) = (−1)ⁿ⁻¹ — the square of any term differs from the product of its neighbours by exactly one, alternating in sign. It is the basis of a well-known dissection puzzle in which a square appears to be rearranged into a rectangle of a different area; the missing unit hides along a thin sliver, and the sliver is exactly this identity made visible.

Is the Fibonacci spiral the golden spiral?

No. The familiar picture is built from quarter circles, one per square, each of constant radius, so its curvature jumps at every join. A true golden spiral is a smooth logarithmic curve whose radius grows by a factor of φ every quarter turn. They look nearly identical because the Fibonacci ratios approach φ so quickly, and the approximation improves with every square, but they are different curves.

Where is the sequence actually used?

In more places than the folklore suggests, and mostly unglamorous ones. Fibonacci heaps give the fastest known asymptotic bounds for several graph algorithms. Fibonacci search divides a sorted array using these numbers rather than halves. Fibonacci coding produces a self-synchronising variable-length encoding, and the Zeckendorf representation is what makes it work. Agile teams estimate in Fibonacci numbers because the widening gaps discourage false precision.

The 11th Fibonacci Number

The 11th Fibonacci number is 89. Index 11 is a term whose value 89 is prime, and whose reciprocal hides the sequence: 1 ÷ 89 = 0.011235955…, the Fibonacci numbers running into each other as they overflow their decimal places. It follows from the rule the whole sequence runs on: add the two terms bef...

BY POSITION

INDEX n

WORTH TRYING

F(11)

RATIO TO F(10)

1.618182

2 digits · previous 55 · next 144 · φ is 1.618034, so this is off by 1.5e-4

Digits

Lucas L(n)

199

Sum of F(1)…F(n)

232

Ratio to previous

1.618181818

Exact in a double?

yes

Index n

AROUND F(11)

F(n)

DIGITS

144

233

377

STEPS

Every term is the sum of the two before it, starting from F(0) = 0 and F(1) = 1

F(11) = F(10) + F(9) = 55 + 34

2 digits, small enough to hold exactly in ordinary arithmetic

F(11) = 89

GOLDEN SPIRAL

The squares build the sequence. Each new square takes the whole long side of what is already there, so its side is the sum of the two before it — which is the Fibonacci rule stated in geometry rather than arithmetic. The rectangle is always F(k) by F(k+1), and the quarter arcs join end to end into the spiral. It approaches the golden spiral without ever being one: this is built from circular arcs, and the true golden spiral is a smooth logarithmic curve.

THE RATIO CLOSING ON φ

n = 3

2.00000000

n = 5

1.66666667

n = 8

1.61538462

n = 12

1.61797753

n = 20

1.61803396

n = 40

1.61803399

Bar length is the distance from φ = 1.618033989, so shorter is closer. The ratio alternates above and below it and the error falls by roughly a factor of φ² each step, which is why n = 40 already matches φ to the last digit a double can hold.

Exact, not floating point. F(78) is the last term that fits exactly in ordinary double arithmetic; F(79) is 14,472,334,024,676,221, which is past the limit. Everything here is computed with exact big integers, so F(500) is right to the final digit rather than rounded. Indices run to 1,000.

Live simulation · drawn to scale · updates as you type

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HOW TO USE

1
The answer is already here: F(11) = 89. This page opens with index 11 loaded, so nothing needs typing.
2
The STEPS panel shows where it comes from: F(10) + F(9) = 55 + 34 = 89.
3
The six tiles carry the derived quantities — digit count, the Lucas number, the running total, the ratio to the previous term, and whether the value still fits exactly in ordinary computer arithmetic.
4
Change the index to carry on, or switch to IS IT ONE? to test whether some other number appears in the sequence at all.

THE FORMULAS

The ruleF(n) = F(n−1) + F(n−2)

The seedsF(0) = 0, F(1) = 1

Golden ratioφ = (1 + √5) ÷ 2

Binet's formulaF(n) = (φⁿ − ψⁿ) ÷ √5

Ratio limitF(n) ÷ F(n−1) → φ

Sum of the first nF(1) + … + F(n) = F(n+2) − 1

Sum of squaresF(1)² + … + F(n)² = F(n)·F(n+1)

Cassini identityF(n)² − F(n−1)F(n+1) = (−1)ⁿ⁻¹

Greatest common divisorgcd(F(m), F(n)) = F(gcd(m,n))

Lucas numbersL(n) = F(n−1) + F(n+1)

Fibonacci test5N² ± 4 is a perfect square

Zeckendorfa unique sum of non-consecutive terms

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Last updated: July 29, 2026 · Formula verified · Exact big-integer arithmetic, so terms past F(78) are right to the final digit · Eagle-eyed accuracy for every calculation.