Finding the Real Part of a Complex Number Expression

We are asked to find the real part of the complex number \(z = \frac{{5 + 2i}}{{2 - 5i}} - \frac{{3 - 4i}}{{4 + 3i}} - \frac{1}{i}\). To do this, we need to simplify each term in the expression for \(z\) and then combine them into a single complex number in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Simplifying Each Term of the Complex Number

Let's simplify each fraction and the last term separately using complex number operations.

Term 1: Simplifying \(\frac{{5 + 2i}}{{2 - 5i}}\)

To divide complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 - 5i\) is \(2 + 5i\).

\(\frac{{5 + 2i}}{{2 - 5i}} = \frac{{(5 + 2i)(2 + 5i)}}{{(2 - 5i)(2 + 5i)}}\)

Now, let's expand the numerator and the denominator:

• Numerator: \((5 + 2i)(2 + 5i) = 5(2) + 5(5i) + 2i(2) + 2i(5i) = 10 + 25i + 4i + 10i^2\). Since \(i^2 = -1\), this becomes \(10 + 29i - 10 = 29i\).
• Denominator: \((2 - 5i)(2 + 5i)\). This is in the form \((a-bi)(a+bi) = a^2 + b^2\). So, \(2^2 + (-5)^2 = 4 + 25 = 29\).

So, the first term simplifies to \(\frac{29i}{29} = i\).

Term 2: Simplifying \(\frac{{3 - 4i}}{{4 + 3i}}\)

Again, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(4 - 3i\).

\(\frac{{3 - 4i}}{{4 + 3i}} = \frac{{(3 - 4i)(4 - 3i)}}{{(4 + 3i)(4 - 3i)}}\)

Let's expand the numerator and the denominator:

• Numerator: \((3 - 4i)(4 - 3i) = 3(4) + 3(-3i) + (-4i)(4) + (-4i)(-3i) = 12 - 9i - 16i + 12i^2\). Since \(i^2 = -1\), this becomes \(12 - 25i - 12 = -25i\).
• Denominator: \((4 + 3i)(4 - 3i) = 4^2 + 3^2 = 16 + 9 = 25\).

So, the second term simplifies to \(\frac{-25i}{25} = -i\).

Term 3: Simplifying \(\frac{1}{i}\)

To simplify this term, we can multiply the numerator and denominator by \(-i\).

\(\frac{1}{i} = \frac{1 \times (-i)}{i \times (-i)} = \frac{-i}{-i^2}\). Since \(i^2 = -1\), \(-i^2 = -(-1) = 1\).

So, the third term simplifies to \(\frac{-i}{1} = -i\).

Combining the Simplified Terms

Now we substitute the simplified terms back into the expression for \(z\):

\(z = (\text{Term 1}) - (\text{Term 2}) - (\text{Term 3})\)

\(z = (i) - (-i) - (-i)\)

Simplifying the signs:

\(z = i + i + i\)

\(z = 3i\)

Identifying the Real Part of the Complex Number z

The complex number \(z\) is \(3i\). A complex number in the form \(a + bi\) has a real part \(a\) and an imaginary part \(b\). In our case, \(z = 0 + 3i\). Therefore, the real part of \(z\) is \(0\).

Understanding the real part of a complex number is fundamental in complex number operations. This problem involved simplifying a complex expression combining division and subtraction, ultimately leading to a purely imaginary number, where the real part is 0.

The real part of the given complex number \(z\) is 0.