CDS Mathematics · Surds, Indices and Simplification

Continued Fractions and Nested Radicals

A finite continued fraction is evaluated from the bottom up and decomposed by repeatedly splitting off whole parts; an infinite nest equals itself, which gives a quadratic.

All Surds, Indices and Simplification notes CDS Mathematics strategy

Why this matters

Twelve PYQs. The finite continued fractions are pure bookkeeping — invert, take the whole part, repeat — and the infinite nested radicals all fall to one move: call the value x and notice that x sits inside itself.

Concept 1 of 2: Continued fractions

Evaluating

a + \cfrac{1}{b + \cfrac{1}{c}}

runs from the bottom up. Decomposing a fraction runs the other way: the whole part of the fraction is

a

; invert what is left and repeat.

Definition

Evaluate from the innermost level outward.
Decompose $\dfrac pq$ : write $\dfrac pq = a + \dfrac rq$ with $0 \le r < q$ , then continue with $\dfrac qr$ .
If a fraction $\dfrac{1}{a + \cdots}$ is given, first invert it.
A repeating continued fraction with $x$ at the bottom is a fixed-point equation: solve $x = f(x)$ and use the stated range.

One step of the decomposition

\dfrac pq = a + \cfrac{1}{\;q/r\;}, \quad a = \left\lfloor \tfrac pq \right\rfloor

Worked example

Write

\dfrac{43}{30}

a + \cfrac{1}{b + \cfrac{1}{c + \cfrac1d}}

with natural numbers, and find

a + b + c + d

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q82Moderate

Example 1 · Surds, Indices and Simplification · Continued Fractions and Nested Radicals

\frac{36}{11} = 3 + \cfrac{1}{x + \cfrac{1}{y + \cfrac{1}{z}}}

, where

x

y

and

z

are natural numbers, then what is

(x + y + z)

equal to ?

Invert before splitting

If the given value is

\dfrac{1}{a + \cdots} = \dfrac{16}{23}

, the first whole part comes from

\dfrac{23}{16}

, not from

\dfrac{16}{23}

(whose whole part is

0

Drill 7 more on continued fractions

Concept 2 of 2: Infinite nested radicals

An infinite nest looks the same after you peel off the outer layer. So if

x = \sqrt{a + \sqrt{a + \cdots}}

, then

x = \sqrt{a + x}

, a quadratic. The value is the positive root.

Definition

$x = \sqrt{a + \sqrt{a + \cdots}}$ gives $x^2 = a + x$ ; take the positive root.
$x = \sqrt{a\sqrt{a\sqrt{a\cdots}}}$ gives $x^2 = ax$ , so $x = a$ .
Read where the nest starts: $2 + \sqrt{2 + \sqrt{\cdots}}$ has a $2$ OUTSIDE the first root.

Self-similar nest

x = \sqrt{a + x} \;\Rightarrow\; x^2 - x - a = 0

Worked example

Evaluate

\sqrt{12 + \sqrt{12 + \sqrt{12 + \cdots}}}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q1Moderate

Example 2 · Surds, Indices and Simplification · Continued Fractions and Nested Radicals

x^2 - 20 = \sqrt{20 + \sqrt{20 + \sqrt{20 + \sqrt{20 + \ldots}}}}

infinite terms, then what is x equal to ?

Drop the negative root

x^2 - x - 12 = 0

has roots

4

and

-3

; a square root is never negative, so the value is

4

. Likewise

x^2 = 4x

gives

x = 4

, not

0

Drill 3 more on infinite nested radicals

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Continued fractions
One step of the decomposition
$\dfrac pq = a + \cfrac{1}{\;q/r\;}, \quad a = \left\lfloor \tfrac pq \right\rfloor$
Infinite nested radicals
Self-similar nest
$x = \sqrt{a + x} \;\Rightarrow\; x^2 - x - a = 0$

Watch out for (2)

Invert before splitting
→ Continued fractions
Drop the negative root
→ Infinite nested radicals

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