Subtract. ( 3 − 4 I ) − ( 2 − 6 I (3 - 4i) - (2 - 6i ( 3 − 4 I ) − ( 2 − 6 I ]Write Your Answer As A Complex Number In Standard Form.

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Introduction

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Complex numbers are mathematical expressions that consist of a real and an imaginary part. They are used to represent points in a two-dimensional plane and have numerous applications in mathematics, physics, and engineering. In this article, we will focus on subtracting complex numbers, which is an essential operation in complex number arithmetic.

What are Complex Numbers?

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A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. The real part of a complex number is denoted by $a$, and the imaginary part is denoted by $b$. For example, the complex number $3 + 4i$ has a real part of $3$ and an imaginary part of $4$.

Subtracting Complex Numbers

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Subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. The formula for subtracting complex numbers is:

$(a + bi) - (c + di) = (a - c) + (b - d)i$

where $a$, $b$, $c$, and $d$ are real numbers.

Example: Subtracting Complex Numbers

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Let's consider the example of subtracting the complex numbers $3 - 4i$ and $2 - 6i$. To subtract these complex numbers, we need to subtract the real parts and the imaginary parts separately.

Step 1: Subtract the Real Parts

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The real part of the first complex number is $3$, and the real part of the second complex number is $2$. To subtract the real parts, we simply subtract $3$ from $2$.

$3 - 2 = 1$

Step 2: Subtract the Imaginary Parts

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The imaginary part of the first complex number is $-4i$, and the imaginary part of the second complex number is $-6i$. To subtract the imaginary parts, we simply subtract $-6i$ from $-4i$.

$-4i - (-6i) = -4i + 6i = 2i$

Step 3: Combine the Results

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Now that we have subtracted the real parts and the imaginary parts, we can combine the results to form the final answer.

$1 + 2i$

Therefore, the result of subtracting the complex numbers $3 - 4i$ and $2 - 6i$ is $1 + 2i$.

Conclusion

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Subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. By following the formula for subtracting complex numbers, we can easily perform this operation and obtain the correct result. In this article, we have demonstrated how to subtract complex numbers using a step-by-step approach.

Frequently Asked Questions

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Q: What is the formula for subtracting complex numbers?

A: The formula for subtracting complex numbers is $(a + bi) - (c + di) = (a - c) + (b - d)i$.

Q: How do I subtract the real parts of complex numbers?

A: To subtract the real parts, simply subtract the real part of the first complex number from the real part of the second complex number.

Q: How do I subtract the imaginary parts of complex numbers?

A: To subtract the imaginary parts, simply subtract the imaginary part of the first complex number from the imaginary part of the second complex number.

Q: What is the result of subtracting the complex numbers $3 - 4i$ and $2 - 6i$?

A: The result of subtracting the complex numbers $3 - 4i$ and $2 - 6i$ is $1 + 2i$.

References

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• [1] "Complex Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/complexnumbers.html
• [2] "Subtracting Complex Numbers" by Purplemath. Retrieved from https://www.purplemath.com/modules/complex.htm

Additional Resources

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• [1] "Complex Numbers" by Khan Academy. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-review/complex-numbers/v/complex-numbers
• [2] "Subtracting Complex Numbers" by IXL. Retrieved from https://www.ixl.com/math/complex-numbers/subtracting-complex-numbers
  

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Introduction

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Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. However, complex numbers can be challenging to understand, especially for beginners. In this article, we will address some of the most frequently asked questions about complex numbers.

Q&A

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Q: What is a complex number?

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A: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$.

Q: What is the imaginary unit?

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A: The imaginary unit, denoted by $i$, is a mathematical concept that satisfies the equation $i^2 = -1$. It is used to extend the real number system to the complex number system.

Q: How do I add complex numbers?

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A: To add complex numbers, simply add the real parts and the imaginary parts separately. The formula for adding complex numbers is:

$(a + bi) + (c + di) = (a + c) + (b + d)i$

Q: How do I subtract complex numbers?

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A: To subtract complex numbers, simply subtract the real parts and the imaginary parts separately. The formula for subtracting complex numbers is:

$(a + bi) - (c + di) = (a - c) + (b - d)i$

Q: How do I multiply complex numbers?

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A: To multiply complex numbers, use the distributive property and the fact that $i^2 = -1$. The formula for multiplying complex numbers is:

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Q: How do I divide complex numbers?

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A: To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The formula for dividing complex numbers is:

$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Q: What is the conjugate of a complex number?

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A: The conjugate of a complex number $a + bi$ is $a - bi$.

Q: How do I find the magnitude of a complex number?

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A: To find the magnitude of a complex number $a + bi$, use the formula:

$|a + bi| = \sqrt{a^2 + b^2}$

Q: How do I find the argument of a complex number?

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A: To find the argument of a complex number $a + bi$, use the formula:

$\arg(a + bi) = \tan^{-1}\left(\frac{b}{a}\right)$

Conclusion

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Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we have addressed some of the most frequently asked questions about complex numbers. We hope that this article has provided you with a better understanding of complex numbers and their properties.

Frequently Asked Questions

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Q: What is the difference between a complex number and a real number?

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A: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. A real number is a number that can be expressed in the form $a$, where $a$ is a real number.

Q: Can complex numbers be used in real-world applications?

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A: Yes, complex numbers have numerous applications in various fields, including physics, engineering, and computer science.

Q: How do I use complex numbers in mathematics?

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A: Complex numbers can be used to solve equations, find roots, and perform calculations in mathematics.

References

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• [1] "Complex Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/complexnumbers.html
• [2] "Subtracting Complex Numbers" by Purplemath. Retrieved from https://www.purplemath.com/modules/complex.htm

Additional Resources

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• [1] "Complex Numbers" by Khan Academy. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-review/complex-numbers/v/complex-numbers
• [2] "Subtracting Complex Numbers" by IXL. Retrieved from https://www.ixl.com/math/complex-numbers/subtracting-complex-numbers