{ "cells": [ { "cell_type": "markdown", "id": "2a46114b", "metadata": {}, "source": [ "# Multiplication is rotation; roots of unity on the circle\n", "### Chapter 8 · the complex plane, rotation, roots of unity · **[PREDICT-FIRST]**\n", "\n", "[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/MeiraDavidson/math_module1/blob/main/notebooks/notebook_8_1_complex_rotation_roots.ipynb)\n", "\n", "**What you'll see:** complex numbers as *arrows* in a plane — and watch multiplication\n", "literally **turn** them (distances multiply, angles add, exactly as you proved on\n", "paper). Then watch the $n$-th roots of unity snap into the corners of a perfect\n", "$n$-sided polygon.\n", "\n", "**Predict-first:** guess before you run each 🔮 cell." ] }, { "cell_type": "code", "execution_count": null, "id": "87fe2d56", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from ipywidgets import interact, FloatSlider, IntSlider\n", "\n", "def draw_arrow(ax, z, color, label=None):\n", " ax.annotate(\"\", xy=(z.real, z.imag), xytext=(0, 0),\n", " arrowprops=dict(arrowstyle=\"-|>\", color=color, lw=2))\n", " if label:\n", " ax.text(z.real*1.07, z.imag*1.07, label, color=color, fontsize=12)" ] }, { "cell_type": "markdown", "id": "23518b1b", "metadata": {}, "source": [ "## 1. Multiplication turns the arrow\n", "\n", "We'll multiply a fixed number $z = 2 + i$ by a second number you control with\n", "sliders: its **distance** from the origin and its **angle**.\n", "\n", "🔮 **Predict first:** set the second number's angle to **90°** and its distance to\n", "**1**. *Before you run it*, predict where the product's arrow will point — and whether\n", "it will be longer, shorter, or the same length as the original." ] }, { "cell_type": "code", "execution_count": null, "id": "2614ee01", "metadata": {}, "outputs": [], "source": [ "def multiply_demo(distance=1.0, angle_deg=90.0):\n", " z = 2 + 1j # the fixed arrow\n", " theta = np.radians(angle_deg)\n", " w = distance * (np.cos(theta) + 1j*np.sin(theta)) # the arrow you control\n", " product = z * w\n", "\n", " fig, ax = plt.subplots(figsize=(5.5, 5.5))\n", " draw_arrow(ax, z, \"#999\", \"z = 2+i\")\n", " draw_arrow(ax, product, \"#1A6FB0\", \"z × w\")\n", " lim = 4\n", " ax.set_xlim(-lim, lim); ax.set_ylim(-lim, lim); ax.set_aspect(\"equal\")\n", " ax.axhline(0, color=\"#ccc\", lw=0.8); ax.axvline(0, color=\"#ccc\", lw=0.8)\n", " ax.set_title(f\"multiply by w: distance {distance:.1f}, angle {angle_deg:.0f}°\")\n", " plt.show()\n", " print(f\"original distance {abs(z):.2f}, angle {np.degrees(np.angle(z)):.0f}°\")\n", " print(f\"product distance {abs(product):.2f}, angle {np.degrees(np.angle(product)):.0f}°\")\n", "\n", "interact(multiply_demo,\n", " distance=FloatSlider(min=0.3, max=2.0, step=0.1, value=1.0),\n", " angle_deg=FloatSlider(min=0, max=360, step=15, value=90));" ] }, { "cell_type": "markdown", "id": "4e014646", "metadata": {}, "source": [ "Read the two printed lines: when you multiply, the **distances multiply** and the\n", "**angles add** — exactly the rule you proved. Multiplying by something at angle 90°\n", "and distance 1 is a pure **quarter-turn**, no stretching. *Multiplication and turning\n", "are the same act.*" ] }, { "cell_type": "markdown", "id": "74350e44", "metadata": {}, "source": [ "## 2. The roots of unity form a polygon\n", "\n", "The $n$-th **roots of unity** are the $n$ solutions of $x^n = 1$. On paper you found\n", "they sit at evenly spaced angles $\\tfrac{2\\pi k}{n}$ around the unit circle.\n", "\n", "🔮 **Predict first:** before you set $n = 6$, predict **how many points** you'll see\n", "and **what shape** they'll make." ] }, { "cell_type": "code", "execution_count": null, "id": "cfb3d03c", "metadata": {}, "outputs": [], "source": [ "def roots_of_unity(n=3):\n", " k = np.arange(n)\n", " roots = np.exp(2j * np.pi * k / n) # the n-th roots of 1\n", "\n", " fig, ax = plt.subplots(figsize=(5.5, 5.5))\n", " circle = np.exp(1j * np.linspace(0, 2*np.pi, 300))\n", " ax.plot(circle.real, circle.imag, color=\"#ccc\", lw=1)\n", " # connect them into a polygon\n", " poly = np.append(roots, roots[0])\n", " ax.plot(poly.real, poly.imag, color=\"#1A6FB0\", lw=1.8)\n", " ax.plot(roots.real, roots.imag, \"o\", color=\"#1A6FB0\", ms=10)\n", " ax.plot([1], [0], \"o\", color=\"#E8833A\", ms=11) # the ordinary real root x=1\n", " ax.text(1.08, 0.05, \"x = 1\", color=\"#E8833A\")\n", " ax.set_aspect(\"equal\"); ax.set_xlim(-1.4, 1.5); ax.set_ylim(-1.4, 1.4)\n", " ax.axhline(0, color=\"#eee\", lw=0.8); ax.axvline(0, color=\"#eee\", lw=0.8)\n", " ax.set_title(f\"the {n} roots of unity\")\n", " plt.show()\n", " for r in roots:\n", " print(f\" angle {np.degrees(np.angle(r)) % 360:6.1f}° -> {r.real:+.3f} {r.imag:+.3f}i\")\n", "\n", "interact(roots_of_unity, n=IntSlider(min=2, max=12, step=1, value=3));" ] }, { "cell_type": "markdown", "id": "e8f079a9", "metadata": {}, "source": [ "## What just happened\n", "\n", "You *proved* on paper that \"multiplication adds angles\" and that \"the roots of unity\n", "form a regular polygon.\" Here you watched the arrows **actually rotate** and the\n", "polygon **actually form** — and you could test values of $n$ you'd never compute by\n", "hand.\n", "\n", "For any $n$, the roots are the $n$ corners of a regular $n$-gon on the unit circle,\n", "with one corner always sitting on the ordinary real root $x=1$. Chapter 4's promise —\n", "that $x^n - 1$ factors into roots arranged on a circle — computed with your own\n", "hands. The walker, the rotation, and the polygon were a complex number all along." ] } ], "metadata": { "colab": { "provenance": [] }, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3" } }, "nbformat": 4, "nbformat_minor": 5 }