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Sagot :
To solve the puzzle \#2, we need to find integers [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] from the set [tex]\(\{-9, -8, ..., -1, 1, 2, ..., 9\}\)[/tex] that satisfy the following equation when multiplying complex numbers:
[tex]\[ (a + bi)(c + di) = 34 + 8i \][/tex]
First, let's recall the formula for multiplying two complex numbers [tex]\((a + bi)(c + di)\)[/tex]:
[tex]\[ (a + bi)(c + di) = ac - bd + (ad + bc)i \][/tex]
Given this, our goal is to find values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] such that:
[tex]\[ ac - bd = 34 \quad \text{(Real Part)} \][/tex]
[tex]\[ ad + bc = 8 \quad \text{(Imaginary Part)} \][/tex]
Here are the steps to find the solutions:
### Step-by-Step Solution
1. Identify the Constraint Set:
We only consider non-zero integers from [tex]\(-9\)[/tex] to [tex]\(9\)[/tex], excluding [tex]\(0\)[/tex].
2. Search for Combinations:
We iterate through all possible values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] to find ones that satisfy both equations above.
After exploring all valid combinations, we find the solutions that satisfy both equations. The correct solutions are:
### Solutions
The combinations [tex]\((a, b, c, d)\)[/tex] that meet the requirements are:
[tex]\[ (-6, -5, -4, 2) \][/tex]
[tex]\[ (-5, 6, -2, -4) \][/tex]
[tex]\[ (-4, 2, -6, -5) \][/tex]
[tex]\[ (-2, -4, -5, 6) \][/tex]
[tex]\[ (2, 4, 5, -6) \][/tex]
[tex]\[ (4, -2, 6, 5) \][/tex]
[tex]\[ (5, -6, 2, 4) \][/tex]
[tex]\[ (6, 5, 4, -2) \][/tex]
### Verification
Let's verify one of the solutions to ensure it is correct. Let's verify [tex]\((a, b, c, d) = (6, 5, 4, -2)\)[/tex]:
Calculate the real part:
[tex]\[ ac - bd = 6 \cdot 4 - 5 \cdot (-2) = 24 + 10 = 34 \][/tex]
Calculate the imaginary part:
[tex]\[ ad + bc = 6 \cdot (-2) + 5 \cdot 4 = -12 + 20 = 8 \][/tex]
The calculations match our target result [tex]\(34 + 8i\)[/tex].
### Reflection:
Regarding the process of multiplying complex numbers, we see that the real part of the product results from subtracting the product of imaginary parts from the product of the real parts. The imaginary part comes from summing the cross-products of the real and imaginary parts. This process emphasizes the importance of correctly applying the distributive property of multiplication and combining like terms.
Reflecting on [tex]\(i^n\)[/tex], while solving these puzzles, we predominantly used the property that [tex]\(i^2 = -1\)[/tex]. However, for higher powers of [tex]\(i\)[/tex]:
[tex]\[ i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 \][/tex]
These cyclical properties were not explicitly needed for solving this puzzle, but they underpin our understanding of complex number multiplication.
### Objectives:
Through this puzzle, the objective was to:
1. Demonstrate the multiplication process for complex numbers.
2. Apply logical reasoning to find values that satisfy given conditions.
3. Reinforce how imaginary units behave under multiplication.
In conclusion, the solutions listed indeed meet the required conditions for multiplying complex numbers to yield [tex]\(34 + 8i\)[/tex].
[tex]\[ (a + bi)(c + di) = 34 + 8i \][/tex]
First, let's recall the formula for multiplying two complex numbers [tex]\((a + bi)(c + di)\)[/tex]:
[tex]\[ (a + bi)(c + di) = ac - bd + (ad + bc)i \][/tex]
Given this, our goal is to find values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] such that:
[tex]\[ ac - bd = 34 \quad \text{(Real Part)} \][/tex]
[tex]\[ ad + bc = 8 \quad \text{(Imaginary Part)} \][/tex]
Here are the steps to find the solutions:
### Step-by-Step Solution
1. Identify the Constraint Set:
We only consider non-zero integers from [tex]\(-9\)[/tex] to [tex]\(9\)[/tex], excluding [tex]\(0\)[/tex].
2. Search for Combinations:
We iterate through all possible values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] to find ones that satisfy both equations above.
After exploring all valid combinations, we find the solutions that satisfy both equations. The correct solutions are:
### Solutions
The combinations [tex]\((a, b, c, d)\)[/tex] that meet the requirements are:
[tex]\[ (-6, -5, -4, 2) \][/tex]
[tex]\[ (-5, 6, -2, -4) \][/tex]
[tex]\[ (-4, 2, -6, -5) \][/tex]
[tex]\[ (-2, -4, -5, 6) \][/tex]
[tex]\[ (2, 4, 5, -6) \][/tex]
[tex]\[ (4, -2, 6, 5) \][/tex]
[tex]\[ (5, -6, 2, 4) \][/tex]
[tex]\[ (6, 5, 4, -2) \][/tex]
### Verification
Let's verify one of the solutions to ensure it is correct. Let's verify [tex]\((a, b, c, d) = (6, 5, 4, -2)\)[/tex]:
Calculate the real part:
[tex]\[ ac - bd = 6 \cdot 4 - 5 \cdot (-2) = 24 + 10 = 34 \][/tex]
Calculate the imaginary part:
[tex]\[ ad + bc = 6 \cdot (-2) + 5 \cdot 4 = -12 + 20 = 8 \][/tex]
The calculations match our target result [tex]\(34 + 8i\)[/tex].
### Reflection:
Regarding the process of multiplying complex numbers, we see that the real part of the product results from subtracting the product of imaginary parts from the product of the real parts. The imaginary part comes from summing the cross-products of the real and imaginary parts. This process emphasizes the importance of correctly applying the distributive property of multiplication and combining like terms.
Reflecting on [tex]\(i^n\)[/tex], while solving these puzzles, we predominantly used the property that [tex]\(i^2 = -1\)[/tex]. However, for higher powers of [tex]\(i\)[/tex]:
[tex]\[ i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 \][/tex]
These cyclical properties were not explicitly needed for solving this puzzle, but they underpin our understanding of complex number multiplication.
### Objectives:
Through this puzzle, the objective was to:
1. Demonstrate the multiplication process for complex numbers.
2. Apply logical reasoning to find values that satisfy given conditions.
3. Reinforce how imaginary units behave under multiplication.
In conclusion, the solutions listed indeed meet the required conditions for multiplying complex numbers to yield [tex]\(34 + 8i\)[/tex].
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