What is the quotient of \(\frac { 1+ i} { 3+ 4i} \)?

1. Multiply the numerator and the denominator by the conjugate of the denominator:
   \[\frac { 1+ i} { 3+ 4i} \cdot \frac { 3- 4i} { 3- 4i} = \frac { ( 1+ i) ( 3- 4i) } { ( 3+ 4i) ( 3- 4i) } \]
2. Simplify the denominator:
   \[( 3+ 4i) ( 3- 4i) = 3^ 2 - ( 4i) ^ 2 = 9 - 16( - 1) = 9 + 16 = 25\]
3. Expand the numerator:
   \[( 1+ i) ( 3- 4i) = 1 \cdot 3 + 1 \cdot ( - 4i) + i \cdot 3 + i \cdot ( - 4i) \]
   \[= 3 - 4i + 3i - 4i^ 2\]
   \[= 3 - 4i + 3i - 4( - 1) \]
   \[= 3 - 4i + 3i + 4\]
   \[= 7 - i\]
4. Combine the results:
   \[\frac { 7 - i} { 25} = \frac { 7} { 25} - \frac { i} { 25} \]

Supplemental Knowledge

Dividing complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\). This process eliminates the imaginary part in the denominator, making it easier to simplify.
Given two complex numbers \(\frac { a+ bi} { c+ di} \), you multiply both numerator and denominator by the conjugate of the denominator \(( c- di) \):
\[\frac { a+ bi} { c+ di} \times \frac { c- di} { c- di} = \frac { ( a+ bi) ( c- di) } { ( c+ di) ( c- di) } \]
The denominator simplifies to \(c^ 2 + d^ 2\), a real number, while the numerator needs to be expanded and simplified.

Real-World Applications 

Understanding complex numbers and their operations is crucial in various fields:

• Electrical Engineering: Complex numbers are used to analyze AC circuits.
• Quantum Physics: They help describe wave functions and probabilities.
• Signal Processing: Used in Fourier transforms for analyzing frequencies.

Imagine you're an electrical engineer working to design circuits. Knowing how to manipulate complex numbers allows you to calculate impedance - essential in making sure your circuit works optimally.

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