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Sagot :
Sure! Let's solve the given complex number division and express the result in the form [tex]\(a + jb\)[/tex].
The problem is to simplify the expression:
[tex]\[ \frac{3 - j}{2 - j} \][/tex]
### Step 1: Rationalize the Denominator
To simplify, we need to rationalize the denominator. This involves multiplying both numerator and denominator by the complex conjugate of the denominator. The complex conjugate of [tex]\(2 - j\)[/tex] is [tex]\(2 + j\)[/tex].
### Step 2: Multiply by the Conjugate
We multiply both the numerator and the denominator by [tex]\(2 + j\)[/tex]:
[tex]\[ \frac{(3 - j)(2 + j)}{(2 - j)(2 + j)} \][/tex]
### Step 3: Expand the Numerator
First, expand the numerator:
[tex]\[ (3 - j)(2 + j) = (3 \cdot 2) + (3 \cdot j) - (j \cdot 2) - (j \cdot j) \][/tex]
[tex]\[ = 6 + 3j - 2j - j^2 \][/tex]
We know that [tex]\(j^2 = -1\)[/tex], so substituting this in:
[tex]\[ = 6 + j - (-1) \][/tex]
[tex]\[ = 6 + j + 1 \][/tex]
[tex]\[ = 7 + j \][/tex]
### Step 4: Expand the Denominator
Next, expand the denominator:
[tex]\[ (2 - j)(2 + j) = (2 \cdot 2) + (2 \cdot j) - (j \cdot 2) - (j \cdot j) \][/tex]
[tex]\[ = 4 + 2j - 2j - j^2 \][/tex]
Again, substituting [tex]\(j^2 = -1\)[/tex]:
[tex]\[ = 4 + 0 - (-1) \][/tex]
[tex]\[ = 4 + 1 \][/tex]
[tex]\[ = 5 \][/tex]
### Step 5: Form the Fraction
Now we have:
[tex]\[ \frac{7 + j}{5} \][/tex]
We can separate this into real and imaginary parts:
[tex]\[ = \frac{7}{5} + \frac{j}{5} \][/tex]
### Step 6: Express in Standard Form
The fraction [tex]\(\frac{7}{5}\)[/tex] simplifies to [tex]\(1.4\)[/tex]:
[tex]\[ \frac{7}{5} = 1.4 \][/tex]
And the fraction [tex]\(\frac{1}{5}\)[/tex] simplifies to [tex]\(0.2\)[/tex]:
[tex]\[ \frac{j}{5} = 0.2j \][/tex]
### Final Result
Putting it all together, we get:
[tex]\[ \frac{3 - j}{2 - j} = 1.4 + 0.2j \][/tex]
Therefore, [tex]\(a = 1.4\)[/tex] and [tex]\(b = 0.2\)[/tex], so:
[tex]\[ \frac{3 - j}{2 - j} = 1.4 + j(0.2) \][/tex]
The problem is to simplify the expression:
[tex]\[ \frac{3 - j}{2 - j} \][/tex]
### Step 1: Rationalize the Denominator
To simplify, we need to rationalize the denominator. This involves multiplying both numerator and denominator by the complex conjugate of the denominator. The complex conjugate of [tex]\(2 - j\)[/tex] is [tex]\(2 + j\)[/tex].
### Step 2: Multiply by the Conjugate
We multiply both the numerator and the denominator by [tex]\(2 + j\)[/tex]:
[tex]\[ \frac{(3 - j)(2 + j)}{(2 - j)(2 + j)} \][/tex]
### Step 3: Expand the Numerator
First, expand the numerator:
[tex]\[ (3 - j)(2 + j) = (3 \cdot 2) + (3 \cdot j) - (j \cdot 2) - (j \cdot j) \][/tex]
[tex]\[ = 6 + 3j - 2j - j^2 \][/tex]
We know that [tex]\(j^2 = -1\)[/tex], so substituting this in:
[tex]\[ = 6 + j - (-1) \][/tex]
[tex]\[ = 6 + j + 1 \][/tex]
[tex]\[ = 7 + j \][/tex]
### Step 4: Expand the Denominator
Next, expand the denominator:
[tex]\[ (2 - j)(2 + j) = (2 \cdot 2) + (2 \cdot j) - (j \cdot 2) - (j \cdot j) \][/tex]
[tex]\[ = 4 + 2j - 2j - j^2 \][/tex]
Again, substituting [tex]\(j^2 = -1\)[/tex]:
[tex]\[ = 4 + 0 - (-1) \][/tex]
[tex]\[ = 4 + 1 \][/tex]
[tex]\[ = 5 \][/tex]
### Step 5: Form the Fraction
Now we have:
[tex]\[ \frac{7 + j}{5} \][/tex]
We can separate this into real and imaginary parts:
[tex]\[ = \frac{7}{5} + \frac{j}{5} \][/tex]
### Step 6: Express in Standard Form
The fraction [tex]\(\frac{7}{5}\)[/tex] simplifies to [tex]\(1.4\)[/tex]:
[tex]\[ \frac{7}{5} = 1.4 \][/tex]
And the fraction [tex]\(\frac{1}{5}\)[/tex] simplifies to [tex]\(0.2\)[/tex]:
[tex]\[ \frac{j}{5} = 0.2j \][/tex]
### Final Result
Putting it all together, we get:
[tex]\[ \frac{3 - j}{2 - j} = 1.4 + 0.2j \][/tex]
Therefore, [tex]\(a = 1.4\)[/tex] and [tex]\(b = 0.2\)[/tex], so:
[tex]\[ \frac{3 - j}{2 - j} = 1.4 + j(0.2) \][/tex]
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