Solve 1/frac{1{6.343+15.632i}+0.6-1.2i}

$$\frac { 1 } { \frac { 1 } { 6.343 + 15.632 i } + 0.6 - 1.2 i }$$

Evaluate

$$\frac{885494219}{2791979657}+\frac{1785718438}{2791979657}i\approx 0.317156401+0.639588628i$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{6.343+15.632i} by the complex conjugate of the denominator, 6.343-15.632i.

$$\frac{1}{\frac{1\left(6.343-15.632i\right)}{\left(6.343+15.632i\right)\left(6.343-15.632i\right)}+0.6-1.2i}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{1}{\frac{1\left(6.343-15.632i\right)}{6.343^{2}-15.632^{2}i^{2}}+0.6-1.2i}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{1}{\frac{1\left(6.343-15.632i\right)}{284.593073}+0.6-1.2i}$$

Multiply 1 and 6.343-15.632i to get 6.343-15.632i.

$$\frac{1}{\frac{6.343-15.632i}{284.593073}+0.6-1.2i}$$

Divide 6.343-15.632i by 284.593073 to get \frac{6343000}{284593073}-\frac{15632000}{284593073}i.

$$\frac{1}{\frac{6343000}{284593073}-\frac{15632000}{284593073}i+0.6-1.2i}$$

Combine the real and imaginary parts in \frac{6343000}{284593073}-\frac{15632000}{284593073}i+0.6-1.2i.

$$\frac{1}{\frac{6343000}{284593073}+0.6+\left(-\frac{15632000}{284593073}-1.2\right)i}$$

Do the additions in \frac{6343000}{284593073}+0.6+\left(-\frac{15632000}{284593073}-1.2\right)i.

$$\frac{1}{\frac{885494219}{1422965365}-\frac{1785718438}{1422965365}i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i.

$$\frac{1\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)}{\left(\frac{885494219}{1422965365}-\frac{1785718438}{1422965365}i\right)\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{1\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)}{\frac{885494219}{1422965365}^{2}-\frac{1785718438}{1422965365}^{2}i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{1\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)}{\frac{2791979657}{1422965365}}$$

Multiply 1 and \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i to get \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i.

$$\frac{\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i}{\frac{2791979657}{1422965365}}$$

Divide \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i by \frac{2791979657}{1422965365} to get \frac{885494219}{2791979657}+\frac{1785718438}{2791979657}i.

$$\frac{885494219}{2791979657}+\frac{1785718438}{2791979657}i$$

Real Part

$$\frac{885494219}{2791979657} = 0.3171564007566836$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{6.343+15.632i} by the complex conjugate of the denominator, 6.343-15.632i.

$$Re(\frac{1}{\frac{1\left(6.343-15.632i\right)}{\left(6.343+15.632i\right)\left(6.343-15.632i\right)}+0.6-1.2i})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{1}{\frac{1\left(6.343-15.632i\right)}{6.343^{2}-15.632^{2}i^{2}}+0.6-1.2i})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{1}{\frac{1\left(6.343-15.632i\right)}{284.593073}+0.6-1.2i})$$

Multiply 1 and 6.343-15.632i to get 6.343-15.632i.

$$Re(\frac{1}{\frac{6.343-15.632i}{284.593073}+0.6-1.2i})$$

Divide 6.343-15.632i by 284.593073 to get \frac{6343000}{284593073}-\frac{15632000}{284593073}i.

$$Re(\frac{1}{\frac{6343000}{284593073}-\frac{15632000}{284593073}i+0.6-1.2i})$$

Combine the real and imaginary parts in \frac{6343000}{284593073}-\frac{15632000}{284593073}i+0.6-1.2i.

$$Re(\frac{1}{\frac{6343000}{284593073}+0.6+\left(-\frac{15632000}{284593073}-1.2\right)i})$$

Do the additions in \frac{6343000}{284593073}+0.6+\left(-\frac{15632000}{284593073}-1.2\right)i.

$$Re(\frac{1}{\frac{885494219}{1422965365}-\frac{1785718438}{1422965365}i})$$

Multiply both numerator and denominator of \frac{1}{\frac{885494219}{1422965365}-\frac{1785718438}{1422965365}i} by the complex conjugate of the denominator, \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i.

$$Re(\frac{1\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)}{\left(\frac{885494219}{1422965365}-\frac{1785718438}{1422965365}i\right)\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{1\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)}{\frac{885494219}{1422965365}^{2}-\frac{1785718438}{1422965365}^{2}i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{1\left(\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i\right)}{\frac{2791979657}{1422965365}})$$

Multiply 1 and \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i to get \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i.

$$Re(\frac{\frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i}{\frac{2791979657}{1422965365}})$$

Divide \frac{885494219}{1422965365}+\frac{1785718438}{1422965365}i by \frac{2791979657}{1422965365} to get \frac{885494219}{2791979657}+\frac{1785718438}{2791979657}i.

$$Re(\frac{885494219}{2791979657}+\frac{1785718438}{2791979657}i)$$

The real part of \frac{885494219}{2791979657}+\frac{1785718438}{2791979657}i is \frac{885494219}{2791979657}.

$$\frac{885494219}{2791979657}$$