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Problem 38

The Last Safe Positive Real Number

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The Last Safe Positive Real Number

Complete to collectJade CabochonLevel 3 · 5 Glints

A complex set, one real slice

The full Mandelbrot set is built by choosing a complex number , starting at , and repeating

If some iterate has , the orbit escapes and is outside the set. This problem studies only positive real values of . Then every is real, so all the work below uses ordinary real-number algebra.

A mathematically computed Mandelbrot set with its horizontal real axis drawn in gold and a question mark at the positive real-axis endpoint. Lobes above and below the axis extend farther right.
The full set reaches farther right off this line. Here you will prove only the positive real endpoint.

What is the greatest positive real in the set?

Write a complete proof on separate paper.

A greatest-value proof has two jobs: include the candidate, then exclude every larger value. The two certificates below divide those jobs.

  1. Build a safe zone. Let . Assume a current value satisfies . Prove that the next value also satisfies . Then use to explain why this orbit never escapes.

  2. Page 2 · Exclude every larger value

    Move just to the right. Let , where . Rewrite the step increase as a square plus :

    Use your rewrite to prove that at every step.

  3. Force an escape. Explain why the guaranteed increase gives . Then choose a whole-number step that makes .

  4. Close both sides. State the greatest positive real value of in the Mandelbrot set. Explain how Part 1 proves that value is included and Parts 2–3 rule out every larger positive real value.

At a glance

About this problem

These tags show what the problem is about and the math you’ll use. Choose one to find similar problems.

Skills

Helpful first

Grade

Math area

Topics

Is this a good fit?

Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 10 · Grade 11

Uses HS A1 math.

Thinking

Stretch challenge

A nontrivial plan using familiar mathematical ideas.

Time & response

About 14–34 minutes

Moderate amount of written work · Proof response.

How to support: The full set uses complex numbers, but this problem deliberately stays on the real axis. Ask what traps the boundary orbit and what a fixed minimum increase eventually forces; avoid supplying the completed square.

Worksheet arc · Synthesis · 4 of 5 on Complete arc

Edge of Escape

Synthesize recurrence and safe-zone reasoning to prove the exact positive real-axis endpoint.

How did it feel?

Pick the option that best matches this challenge. Recommendations use topic, math area, grade, story, and thinking challenge.

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Technical reference

Rating, skills & curriculum

Designed for Grade 10 · Grade 11 · Uses HS A1 math · About 14–34 minutes

Grade 10 · Grade 11 · Algebraic structure and equations · Expressions and symbolic structure

MJ HS:A1.6 · C3 · W2

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ HS:A1.6

High school

HS A1 · point 6

Likely range
MJ HS:A1.5–HS:A1.7
Confidence
80%

Challenge

C3 · Stretch

A nontrivial plan using familiar mathematical ideas.

  • Method selection2
  • Reasoning depth3
  • Novelty2
  • Constraint management2
  • Proof or justification3

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
14–34 minutes · typically 22
Response
Proof
Known-method steps
18
  • Mechanical execution2
  • Written output3
  • Representation production1
  • Bookkeeping1

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Rewrite expressions to reveal zeros, extrema, growth, or other features

    The learner can rewrite expressions to reveal zeros, extrema, growth, or other features. This capability is distinct from mere exposure to or isolated use of factoring; completing square.

    • Developing mastery
    • 100% focus
    • MJ HS:A1.6 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Define and analyze simple recursive processes and recurrence relations

    The learner can define and analyze simple recursive processes and recurrence relations. This capability is distinct from mere exposure to or isolated use of sequences; functions.

    • Developing readiness
    • Required
    • MJ HS:DISC.4 milestone

  • Add, subtract, and multiply polynomials

    The learner can add, subtract, and multiply polynomials. This capability is distinct from mere exposure to or isolated use of distributive property; exponent laws.

    • Independent readiness
    • Required
    • MJ HS:A1.6 milestone

  • Write and graph one-variable inequalities such as x>c

    The learner can write and graph one-variable inequalities such as x>c. This capability is distinct from mere exposure to or isolated use of number line; comparison.

    • Fluent readiness
    • Required
    • MJ 6.9 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Common Core State Standards for Mathematics2010 · 1 location
Clusters1 location · Representative: Write expressions in equivalent forms to solve problems