Dividing (Leaving Cert Mathematics): Revision Notes

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Dividing

To divide complex numbers, we eliminate the imaginary part from the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Example

Note

Simplify the following :

1+i2+i\frac{1+i}{2+i}

First, derive the conjugate of the denominator, 2+i‾=2−i\overline{2+i}=2-i.

1+i2+i=1+i2+i⋅2−i2−i=2−i+2i−i24−2i+2i−i2=2−i+2i−(−1)4−2i+2i−(−1)=3+i5\begin{align*} \frac{1+i}{2+i}&=\frac{1+i}{2+i}\cdot \frac{2-i}{2-i} \\\\ &= \frac{2-i+2i-i^2}{4-2i+2i-i^2} \\\\ &=\frac{2-i+2i-(-1)}{4-2i+2i-(-1)} \\\\ &=\frac{3+i}{5} \end{align*}

Write in the form a+bia+bi :

3+i5=35+15i\begin{align*} \frac{3+i}{5}&=\frac{3}{5}+\frac{1}{5}i \end{align*}
Note

In all cases, multiplying a complex number zz with it's conjugate z‾\overline{z} will eliminate the imaginary component.

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