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MATH
[1251]

Examples from Complex arithmetic

All examples from Complex arithmetic.

Back to Complex arithmetic

Calculate with Cartesian forms

Let z1=1+iz_1=1+i and z2=2−πiz_2=\sqrt2-\pi i. Then

z1+z2=(1+2)+(1−π)i,z_1+z_2=(1+\sqrt2)+(1-\pi)i,

and

z1z2=(1+i)(2−πi)=(2+π)+(2−π)i.\begin{aligned} z_1z_2 &=(1+i)(\sqrt2-\pi i)\\ &=(\sqrt2+\pi)+(\sqrt2-\pi)i. \end{aligned}

For subtraction,

z1−z2=(1−2)+(1+π)i.z_1-z_2=(1-\sqrt2)+(1+\pi)i.

For division, multiply numerator and denominator by

z2‾=2+πi:\overline{z_2}=\sqrt2+\pi i:z1z2=(1+i)(2+πi)(2−πi)(2+πi)=(2−π)+(2+π)i2+π2.\frac{z_1}{z_2} =\frac{(1+i)(\sqrt2+\pi i)}{(\sqrt2-\pi i)(\sqrt2+\pi i)} =\frac{(\sqrt2-\pi)+(\sqrt2+\pi)i}{2+\pi^2}.

[routine]

  1. [1]:Add, multiply and divide complex numbers

Compare solutions across number systems

The equation 2x+4=82x+4=8 has the solution x=2x=2 in every number system listed above. The equation 2x+4=−82x+4=-8 first becomes solvable in Z\mathbb Z, where x=−6x=-6, while 2x+4=72x+4=7 first becomes solvable in Q\mathbb Q, where x=3/2x=3/2. The equation 2x2+4=72x^2+4=7 has real solutions x=±3/2x=\pm\sqrt{3/2}, which are irrational, whereas

2x2+4=3⟹x=±i22x^2+4=3 \quad\Longrightarrow\quad x=\pm\frac{i}{\sqrt2}

requires C\mathbb C.

A solution in one system also belongs to every larger system in the chain, so each comparison identifies the smallest system that contains the solution.

[routine]

  1. [1]:Complex number

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Calculate with Cartesian formsCompare solutions across number systems