To divide the complex numbers [tex]\(\frac{1+2i}{5+4i}\)[/tex], we need to express the result in the standard form [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers.
### Step-by-Step Solution:
1. Conjugate of the Denominator:
The first step in dividing complex numbers is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(5 + 4i\)[/tex] is [tex]\(5 - 4i\)[/tex].
[tex]\[
\frac{1+2i}{5+4i} \times \frac{5-4i}{5-4i} = \frac{(1+2i)(5-4i)}{(5+4i)(5-4i)}
\][/tex]
2. Multiply the Numerator:
We distribute the terms in the numerator:
[tex]\[
(1+2i)(5-4i) = 1 \cdot 5 + 1 \cdot (-4i) + 2i \cdot 5 + 2i \cdot (-4i)
\][/tex]
Simplify each term:
[tex]\[
= 5 - 4i + 10i - 8i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], replace [tex]\(i^2\)[/tex] with [tex]\(-1\)[/tex]:
[tex]\[
= 5 - 4i + 10i + 8
= 13 + 6i
\][/tex]
3. Multiply the Denominator:
To simplify the denominator, recognize that we are multiplying a complex number by its conjugate, which results in the sum of the squares of the real and imaginary parts:
[tex]\[
(5+4i)(5-4i) = 5^2 - (4i)^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], this becomes:
[tex]\[
= 25 - 16(-1)
= 25 + 16
= 41
\][/tex]
4. Combine the Results:
Now, we have:
[tex]\[
\frac{13 + 6i}{41}
\][/tex]
We can separate this into real and imaginary parts:
[tex]\[
= \frac{13}{41} + \frac{6}{41}i
\][/tex]
### Final Answer:
Thus, the result in standard form is:
[tex]\[
\frac{1+2i}{5+4i} = \frac{13}{41} + \frac{6}{41}i
\][/tex]
In decimal form, the result simplifies to approximately:
[tex]\[
0.3170731707317074 + 0.14634146341463417i
\][/tex]
This is the final simplified form of the division of complex numbers [tex]\( \frac{1+2i}{5+4i} \)[/tex].
