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Problem 36

The Orbit of i

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The Orbit of i

Complete to collectSmoky Quartz PointLevel 2 · 4 Glints
A brass telescope in a moonlit observatory points toward a luminous looping trail across a violet-and-blue star field.

Page 1 · Build the tools

Complex numbers, one idea at a time

The new number is defined by . A complex number has the form , with a real part and an -part .

Three moves you will use

1

Turn a number into a point

Plot at . For example, goes at .

2

Square by expanding

The last step uses . Example: .

3

Measure distance from 0

The point is units from 0, by the Pythagorean theorem.

Page 2 · Follow the orbit of

Write complex numbers in the form when useful.

Start with and repeat . This list of points is an orbit. If an orbit repeats while every point has distance at most 2 from 0, it stays bounded and belongs to the Mandelbrot set.

A circular observatory lens contains a blank complex coordinate plane from negative two to two on both axes. The horizontal axis is labeled real and the vertical axis is labeled i-part.
Plot each distinct orbit point in the lens.

  1. For , calculate . Show enough squaring to make each step checkable, complete the table, and plot each distinct point in the lens.

    Point
    00(0, 0)
    1
    2
    3
    4

  2. Which values form a repeating cycle? Use the recurrence to predict and .

  3. Find the distance from 0 of each value in the cycle. Explain why this proves that belongs to the Mandelbrot set.

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 A2 math.

Thinking

Standard challenge

The solver chooses and coordinates familiar methods.

Time & response

About 11–28 minutes

Moderate amount of written work · Extended Explanation response.

How to support: Everything needed about complex numbers is introduced on the page. Encourage careful use of i squared equals negative 1 and ask what an exact repeated value guarantees about every later step.

Worksheet arc · Guided Practice · 2 of 5 on Complete arc

Edge of Escape

Open the complex-orbit branch by introducing complex arithmetic, magnitude, and a bounded repeating orbit.

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 A2 math · About 11–28 minutes

Grade 10 · Grade 11 · Number and quantity · Complex numbers and vectors

MJ HS:A2.2 · C2 · 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:A2.2

High school

HS A2 · point 2

Likely range
MJ HS:A2.1–HS:A2.3
Confidence
82%

Challenge

C2 · Standard

The solver chooses and coordinates familiar methods.

  • Method selection1
  • Reasoning depth2
  • Novelty1
  • Constraint management2
  • Proof or justification1

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
11–28 minutes · typically 18
Response
Extended Explanation
Known-method steps
17
  • Mechanical execution2
  • Written output2
  • Representation production2
  • Bookkeeping1

Learning focus

Skills this problem practices

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

  • Perform arithmetic with complex numbers

    The learner can perform arithmetic with complex numbers. This capability is distinct from mere exposure to or isolated use of real numbers; square roots.

    • Developing mastery
    • 100% focus
    • MJ HS:A2.2 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • 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

  • 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.

    • Introduced readiness
    • Required
    • MJ HS:DISC.3 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: Perform arithmetic operations with complex numbers.