1. `A=5+6i,B=-2+3i,C=1-3i`
Find reciprocal(A)Solution:Here `A=5+6i,B=-2+3i,C=1-3i`
For a complex number `z=a+bi`, the reciprocal is `1/z=(a-bi)/(a^2+b^2)`
`z=a+bi`
`:. 1/z=1/(a+bi)`
`:. 1/z=1/((a+bi))*((a-bi))/((a-bi))`
`:. 1/z=(a-bi)/(a^2+b^2)`
`A=5+6i`
`:. 1/(A)=1/(5+6i)`
`=1/((5+6i))*((5-6i))/((5-6i))`
`=(5-6i)/(5^2+6^2)`
`=(5-6i)/(25+36)`
`=(5-6i)/61`
`=0.082-0.0984i`
`:.` The reciprocal of `5+6i` is `0.082-0.0984i`
2. `A=5+6i,B=-2+3i,C=1-3i`
Find reciprocal(B)Solution:Here `A=5+6i,B=-2+3i,C=1-3i`
For a complex number `z=a+bi`, the reciprocal is `1/z=(a-bi)/(a^2+b^2)`
`z=a+bi`
`:. 1/z=1/(a+bi)`
`:. 1/z=1/((a+bi))*((a-bi))/((a-bi))`
`:. 1/z=(a-bi)/(a^2+b^2)`
`B=-2+3i`
`:. 1/(B)=1/(-2+3i)`
`=1/((-2+3i))*((-2-3i))/((-2-3i))`
`=(-2-3i)/((-2)^2+3^2)`
`=(-2-3i)/(4+9)`
`=(-2-3i)/13`
`=-0.1538-0.2308i`
`:.` The reciprocal of `-2+3i` is `-0.1538-0.2308i`
3. `A=5+6i,B=-2+3i,C=1-3i`
Find reciprocal(C)Solution:Here `A=5+6i,B=-2+3i,C=1-3i`
For a complex number `z=a+bi`, the reciprocal is `1/z=(a-bi)/(a^2+b^2)`
`z=a+bi`
`:. 1/z=1/(a+bi)`
`:. 1/z=1/((a+bi))*((a-bi))/((a-bi))`
`:. 1/z=(a-bi)/(a^2+b^2)`
`C=1-3i`
`:. 1/(C)=1/(1-3i)`
`=1/((1-3i))*((1+3i))/((1+3i))`
`=(1+3i)/(1^2+3^2)`
`=(1+3i)/(1+9)`
`=(1+3i)/10`
`=0.1+0.3i`
`:.` The reciprocal of `1-3i` is `0.1+0.3i`
This material is intended as a summary. Use your textbook for detail explanation.
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