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Questions and Answers
What is the result of adding the complex numbers $2 + \sqrt{-2}$ and $3 - 2i$?
How do you add two complex numbers represented as $a + bi$?
What is the result of multiplying the complex numbers $(2 + 1i)$ and $(3 - 2i)$?
What is the conjugate of the complex number $2 + \sqrt{-2}$?
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What is the real part of the product of (2 + 1i) and (3 - 2i)?
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What does the product of a complex number and its conjugate always result in?
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What is the conjugate of (2 + sqrt(-2))?
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When multiplying two complex numbers, what does the imaginary part of the product depend on?
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Study Notes
Exploring the Conjugate of 2+√-2: Addition, Multiplication, and Conjugates of Complex Numbers
Complex numbers, with their real and imaginary parts, open up a world of mathematical wonder. The number we'll dive into today is the conjugate of (2 + \sqrt{-2}), a number that's common in higher-order algebra and calculus. Let's explore the fundamentals of complex number arithmetic, focusing on addition, multiplication, and conjugates of complex numbers.
Addition of Complex Numbers
Complex numbers are represented as (a + bi), where (a) is the real part and (b) is the imaginary part. To add two complex numbers, simply add their real parts and their imaginary parts separately:
[(a_1 + b_1i) + (a_2 + b_2i) = (a_1 + a_2) + (b_1 + b_2)i]
For example, let's add (2 + \sqrt{-2} = 2 + 1i) and (3 - 2i):
[ (2 + 1i) + (3 - 2i) = 5 + (-1 + 2)i = 5 + 1i]
Now that we can add complex numbers, let's see how to multiply them.
Multiplication of Complex Numbers
To multiply two complex numbers, multiply their real parts and their imaginary parts separately, and then combine them. The imaginary part of the product will contain the product of the real parts and the sum of the imaginary parts, while the real part will contain the sum of the real parts and the product of the imaginary parts:
[ (a_1 + b_1i)(a_2 + b_2i) = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i]
Applying this to our example:
[ (2 + 1i)(3 - 2i) = 6 + 2i - 3i - 2 = 3 + 1i]
Now that we can multiply complex numbers, we'll discover the concept of conjugates.
Conjugates of Complex Numbers
The conjugate of a complex number (a + bi) is the number (a - bi). When multiplied by the original number, the conjugate will always produce a real result:
[ (a + bi)(a - bi) = a^2 + b^2]
Using our example, we find the conjugate of (2 + \sqrt{-2}):
[ \text{Conjugate}(2 + \sqrt{-2}) = 2 - \sqrt{-2} = 2 + 1i \times (-1) = 2 - 1i]
Now you know how to add, multiply, and find conjugates of complex numbers. You're ready to tackle the conjugate of (2 + \sqrt{-2}) and the many other complex numbers that await your exploration!
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Description
Delve into the world of complex numbers by exploring the addition, multiplication, and conjugates of a specific complex number, 2 + √-2. Learn how to add and multiply complex numbers, and discover how to find the conjugate of a complex number to always yield a real result.
