Find the modulus and amplitude of the following complex numbers and hence express them into polar form
`((3+4i) (4+ 5i))/((4+3i) (6+7i))`

Answer

To find the modulus and amplitude of the complex number given by the expression \(\frac{(3 + 4i)(4 + 5i)}{(4 + 3i)(6 + 7i)}\), we will follow these steps:

### Step 1: Multiply the Numerator
First, we will multiply the complex numbers in the numerator:
\[ (3 + 4i)(4 + 5i) \]
Using the distributive property:
\[ = 3 \cdot 4 + 3 \cdot 5i + 4i \cdot 4 + 4i \cdot 5i \]
Calculating each term:
\[ = 12 + 15i + 16i + 20i^2 \]
Since \(i^2 = -1\):
\[ = 12 + 15i + 16i - 20 \]
Combining like terms:
\[ = -8 + 31i \]

### Step 2: Multiply the Denominator
Next, we will multiply the complex numbers in the denominator:
\[ (4 + 3i)(6 + 7i) \]
Using the distributive property:
\[ = 4 \cdot 6 + 4 \cdot 7i + 3i \cdot 6 + 3i \cdot 7i \]
Calculating each term:
\[ = 24 + 28i + 18i + 21i^2 \]
Again, since \(i^2 = -1\):
\[ = 24 + 28i + 18i - 21 \]
Combining like terms:
\[ = 3 + 46i \]

### Step 3: Form the Complex Number
Now we can write our complex number:
\[ z = \frac{-8 + 31i}{3 + 46i} \]

### Step 4: Rationalize the Denominator
To simplify \(z\), we multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{(-8 + 31i)(3 - 46i)}{(3 + 46i)(3 - 46i)} \]

Calculating the denominator:
\[ = 3^2 - (46i)^2 = 9 + 2116 = 2125 \]

Calculating the numerator:
\[ = -24 + 368i + 93i - 1426i^2 \]
Since \(i^2 = -1\):
\[ = -24 + 368i + 93i + 1426 \]
Combining like terms:
\[ = 1402 + 461i \]

So we have:
\[ z = \frac{1402 + 461i}{2125} \]

### Step 5: Separate Real and Imaginary Parts
This gives us the real and imaginary parts:
\[ \text{Real part} = \frac{1402}{2125}, \quad \text{Imaginary part} = \frac{461}{2125} \]

### Step 6: Calculate the Modulus
The modulus of \(z\) is given by:
\[ |z| = \sqrt{\left(\frac{1402}{2125}\right)^2 + \left(\frac{461}{2125}\right)^2} \]
Calculating:
\[ = \frac{1}{2125}\sqrt{1402^2 + 461^2} \]
Calculating \(1402^2 + 461^2\):
\[ = 1965604 + 212521 = 2178125 \]
Thus,
\[ |z| = \frac{\sqrt{2178125}}{2125} \]

### Step 7: Calculate the Argument (Amplitude)
The argument (or amplitude) \(\theta\) is given by:
\[ \theta = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) = \tan^{-1}\left(\frac{\frac{461}{2125}}{\frac{1402}{2125}}\right) = \tan^{-1}\left(\frac{461}{1402}\right) \]

### Step 8: Write in Polar Form
The polar form of \(z\) is:
\[ z = |z| \left(\cos(\theta) + i\sin(\theta)\right) \]

### Summary of Results
- **Modulus**: \(|z| = \frac{\sqrt{2178125}}{2125}\)
- **Argument**: \(\theta = \tan^{-1}\left(\frac{461}{1402}\right)\)
- **Polar Form**: \(z = |z| \left(\cos(\theta) + i\sin(\theta)\right)\)