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hambasics:sections:test [2021/02/13 19:14] – created - external edit 127.0.0.1hambasics:sections:test [2024/11/24 12:57] (current) – va7fi
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 </WRAP> </WRAP>
  
-{{ggb>/howto/hambasics/sections/polar.ggb}}+{{ggb>/hambasics/sections/polar.ggb}}
  
 To convert between the Cartesian \$(a,b) \$ and the Polar \$ (r \angle \theta) \$ representations, only simple trigonometry and Pythagoras is needed. To convert between the Cartesian \$(a,b) \$ and the Polar \$ (r \angle \theta) \$ representations, only simple trigonometry and Pythagoras is needed.
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 The complex plane has many useful applications, but one of them allows us to visualize roots of the form \$ z^n = w \$.  For example, if we set \$ w = 9 \$ and \$ n = 2 \$ on the graph below, we'll see that the roots of \$z^2 = 9 \$ are \$ z = \pm 3 \$. The complex plane has many useful applications, but one of them allows us to visualize roots of the form \$ z^n = w \$.  For example, if we set \$ w = 9 \$ and \$ n = 2 \$ on the graph below, we'll see that the roots of \$z^2 = 9 \$ are \$ z = \pm 3 \$.
  
-{{ggb>/howto/hambasics/sections/complexroots.ggb 850,500}}+{{ggb>/hambasics/sections/complexroots.ggb 850,500}}
  
 Without using the graph above, what do you expect the solution(s) to \$ z^3 = 8 \$ will be?  That is, what number(s), when multiplied by itself three times gives 8? Without using the graph above, what do you expect the solution(s) to \$ z^3 = 8 \$ will be?  That is, what number(s), when multiplied by itself three times gives 8?
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-The lesson here is that since the polar representation uses exponents, and exponents turn multiplication into addition((\$ x^a \cdot x^b = x^{a+b} \$)), the polar representation is easiest for multiplication, division, exponentiation, and roots.  It's essentially why the [[howto/hambasics/sections/mathbasics#the_decibel |dB scale]] is so useful.  But addition and subtraction is intrinsically easier in Cartesian coordinates.+The lesson here is that since the polar representation uses exponents, and exponents turn multiplication into addition((\$ x^a \cdot x^b = x^{a+b} \$)), the polar representation is easiest for multiplication, division, exponentiation, and roots.  It's essentially why the [[hambasics/sections/mathbasics#the_decibel |dB scale]] is so useful.  But addition and subtraction is intrinsically easier in Cartesian coordinates.
 ===== Important Algebraic Results ===== ===== Important Algebraic Results =====
   * Use the Euler identity to get the following two useful results:   * Use the Euler identity to get the following two useful results:
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-This last result is the basis behind why [[howto/hambasics/sections/wavemodulationmath |modulating the amplitude]] of a carrier produces side bands.+This last result is the basis behind why [[hambasics/sections/wavemodulationmath |modulating the amplitude]] of a carrier produces side bands.
  
  
hambasics/sections/test.1613272457.txt.gz · Last modified: by 127.0.0.1