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Question
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Solution
\[ \frac{2 + 3i}{4 + 5i}\]
\[ = \frac{2 + 3i}{4 + 5i} \times \frac{4 - 5i}{4 - 5i}\]
\[ = \frac{8 - 10i + 12i - 15 i^2}{16 - 25 i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{23 + 2i}{16 + 25}\]
\[ = \frac{23}{41} + \frac{2}{41}i\]
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