Exploring nth Roots of Unity via Inversion

2013-12-07 14:01

This applet visually demonstrates the geometric properties of nth roots of unity through inversion in the complex plane. It calculates and visualizes the sum of the expression 1/(1-ω) + 1/(1-ω^2) + ... + 1/(1-ω^(n-1)), where ω represents the roots of x^n-1=0. Using dynamic polygons, vectors, and rays, it helps users deeply understand the algebraic and geometric connections of complex roots of unit

#complex-number #polygon #vector #geometry-construction

In collections 複數

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Exploring nth Roots of Unity via Inversion

Loading the GeoGebra board and drawing, please wait…

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<expression label="showRoots" exp="(step ≤ 2) ∨ (step ≥ 5)" />
<element type="boolean" label="showRoots">
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<expression label="showOneMinusRoots" exp="(3 ≤ step) ∧ (step ≤ 5)" />
<element type="boolean" label="showOneMinusRoots">
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<command name="Sequence">
	<input a0="Text[&quot;ω^&quot; + (k), (1 + (0.6 * dr); (k * θ)) + Off, true, true]" a1="k" a2="1" a3="n - 1"/>
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</command>
<element type="list" label="txtRoots">
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	<condition showObject="(step ≤ 2) ∨ (step ≥ 5)"/>
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</element>
<command name="Sequence">
	<input a0="&quot;\frac{1}{1-ω^&quot; + (k) + &quot;}&quot;" a1="k" a2="1" a3="n - 1"/>
	<output a0="terms"/>
</command>
<element type="list" label="terms">
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<expression label="expr" exp="&quot;\frac{&quot; + (n - 1) + &quot;}{2}=&quot; + (Element[terms, 1]) + &quot;&quot; + (Sum[Sequence[&quot;+&quot; + (Element[terms, k]), k, 2, n - 1]]) + &quot;&quot;" />
<element type="text" label="expr">
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<element type="point" label="OffExpr">
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	<objColor r="0" g="0" b="255" alpha="0.0"/>
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	<pointStyle val="0"/>
</element>
<command name="Text">
	<input a0="expr" a1="Corner[1] + OffExpr" a2="true" a3="true"/>
	<output a0="txtExpr"/>
</command>
<element type="text" label="txtExpr">
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<expression label="text2" exp="&quot;方根次數&quot;"/>
<element type="text" label="text2">
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	<labelMode val="0"/>
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</element>
<expression label="text3" exp="&quot;顯示步驟&quot;"/>
<element type="text" label="text3">
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	<labelMode val="0"/>
	<absoluteScreenLocation  x="516" y="43"/>
</element>
</construction>
</geogebra>