Whole Earth Curriculum WEC

Lesson 04 — Complex Numbers: Why New Numbers Arise

Learners discover why a number whose square is negative was invented — not as a trick but as a need that arose from real problems — and learn to write, plot, and operate with complex numbers a + bi. They connect the story to the history of algebra across cultures and explain, in their own words, why new kinds of numbers arise.

D05 P3: Intellectual & Cognitive Awareness D05.S1 55 minutes Draft

Objectives

Essential question

Why do we invent a new kind of number whose square is negative, and what can it do that the old numbers cannot?

imaginary unitcomplex numberreal partimaginary partcomplex conjugatecomplex plane

Materials

Standard materials

Low-tech / no-cost

Enriched / lab & device

Works in different contexts

A complex plane with a horizontal real axis and vertical imaginary axis, the point a + bi plotted, and a small table showing that i squared equals negative one and that multiplying (a + bi) by its conjugate (a - bi) gives a squared plus b squared
A complex plane with a horizontal real axis and vertical imaginary axis, the point a + bi plotted, and a small table showing that i squared equals negative one and that multiplying (a + bi) by its conjugate (a - bi) gives a squared plus b squared

Lesson 4 — Complex Numbers: Why New Numbers Arise

Summary

Learners see that a number whose square is negative was needed, not invented for fun — it arose from real problems about solving equations — and learn to write, plot, and operate with complex numbers a + bi. They connect this to the history of algebra across cultures and explain why new kinds of numbers arise.

Objectives

  • Extend number reasoning to complex numbers and explain why new kinds of numbers arise from real problems. (D05.S1.10.02)

Connection

Every new kind of number was once a scandal. Zero, negative numbers, and fractions each had to be argued for, because they answered questions the old numbers could not. Now ask: what number times itself gives −1? No ordinary number works — positive squares are positive, negative squares are positive too. For a long time mathematicians hit this wall and turned back. But the wall kept appearing inside real problems — like finding where a curve crosses a line — and the people who finally walked through it found that a “number whose square is −1” opens a whole second dimension of arithmetic. Today we meet that number, call it i, and see that a complex number is just a place on a plane — two ordinary numbers bundled together.

Materials

  • Complex-number map
  • Math journal

Preparation

  • Copy or draw the complex-number map.
  • Retrieval: from Lesson 2, roots and the idea that not every length is a fraction; from Grade 9, solving quadratics (D05.S2.09.02). Today: what if a square is negative?
  • Prepare worked examples of complex addition, multiplication, and conjugates.

Facilitator note

This lesson is written to the learner (“you”). The idea to land: the imaginary unit i (with i² = −1) extends number to a second dimension, and it was needed — it arose from real problems, not from a rule someone liked. Teach the arithmetic explicitly with worked examples (S-011), and let the “why” story carry the motivation.

The global lens is central: the idea that new numbers are needed to solve equations has deep roots in algebra’s own history. The word “algebra” comes from al-jabr in the 9th-century work of the scholar al-Khwarizmi (S-293, S-431), who solved quadratics by completing the square and justified them geometrically; the Persian poet-mathematician Omar Khayyam (11th–12th century) solved cubic equations by intersecting curves geometrically (S-296). In 16th-century Italy, solving the cubic forced mathematicians to let square roots of negatives appear mid-calculation, even when the final answer was an ordinary real number — the standard account of how complex numbers became respectable (S-240, as a general authority; hold the “first” claim lightly). The intellectual lens: a new number is defined by a consistent rule (i² = −1), and everything else follows. The critical-thinking lens: ask whether a new kind of number is consistent and useful — that, not familiarity, is the test. The technology lens: complex numbers are not an oddity — they are the daily language of alternating current, signal processing, and the wave mathematics behind screens and radio. Preview: Lesson 5 turns from numbers to functions.

Procedure

  1. Recall (5 min). From Grade 9: solve x² = 4 and x² = −4. The first has answers 2 and −2; the second has none among the numbers you know. Is “no answer” the end, or a door?
  2. Meet i (10 min). Define the imaginary unit i so that i² = −1. Then x² = −4 has answers 2i and −2i, because (2i)² = 4·(−1) = −4. A complex number is a + bi, where a and b are real: a is the real part, b the imaginary part. Real numbers are the special case b = 0.
  3. Plot it (10 min). On the complex-number map, put the real part on the horizontal axis and the imaginary part on the vertical axis. Plot 3 + 4i as the point (3, 4). Notice: a complex number is a place, like a coordinate — two ordinary numbers bundled into one.
  4. Operate (15 min). Worked examples, then guided practice:
    • Add: (2 + 3i) + (1 + 5i) = 3 + 8i (add real parts, add imaginary parts).
    • Multiply: (2 + i)(1 + 3i) = 2 + 6i + i + 3i² = 2 + 7i − 3 = −1 + 7i, because i² = −1.
    • Conjugate: the complex conjugate of a + bi is a − bi. Their product is always a real number: (a + bi)(a − bi) = a² + b².
  5. Explain why (10 min). In your journal, answer in your own words: why do new kinds of numbers arise? Use zero, negative numbers, or complex numbers as your evidence, and name one real problem each new number solved.
  6. Close (5 min). Say one thing a complex number can do that a real number alone cannot.

Differentiation

  • Support: Plot only first, then add; give the “why” explanation as a sentence starter (“A new number arises when…”).
  • Extension: Solve a quadratic with a negative discriminant (e.g. x² + 1 = 0) and express the answers in a + bi form.

Assessment

  • Formative (self + peer): Can the learner add, multiply, and conjugate complex numbers, and explain why complex numbers arose?
  • Portfolio artifact (unit): The complex-number map with the written “why new numbers arise” explanation, as the unit’s “number extension” page.

Home connection

Tell someone the story of i: “a number whose square is −1 was needed to finish real problems, so we made one — and it turned out to be a place on a plane.” Ask them if they can think of another time people invented a new tool to finish a job.

Resources