Melissa And Tomas Are Playing A Game With Complex Numbers. If Melissa Has A Score Of $5-4i$ And Tomas Has A Score Of $3+2i$, What Is Their Total Score?A. $8-6i$ B. $8+6i$ C. $8-2i$ D. $8+2i$

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Introduction

In the world of mathematics, complex numbers play a vital role in various fields, including algebra, geometry, and calculus. 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. In this article, we will explore how to add complex numbers, using the scores of Melissa and Tomas in a game as an example.

What are Complex Numbers?

Complex numbers are numbers that have both real and imaginary parts. The real part is denoted by a, and the imaginary part is denoted by bi. For example, the complex number 3 + 4i has a real part of 3 and an imaginary part of 4i.

Adding Complex Numbers

To add complex numbers, we simply add the real parts and the imaginary parts separately. For example, to add the complex numbers 2 + 3i and 4 + 5i, we get:

(2 + 3i) + (4 + 5i) = (2 + 4) + (3i + 5i) = 6 + 8i

Melissa and Tomas's Scores

Melissa has a score of 5 - 4i, and Tomas has a score of 3 + 2i. To find their total score, we need to add these two complex numbers.

Step 1: Add the Real Parts

The real part of Melissa's score is 5, and the real part of Tomas's score is 3. To add these two real numbers, we get:

5 + 3 = 8

Step 2: Add the Imaginary Parts

The imaginary part of Melissa's score is -4i, and the imaginary part of Tomas's score is 2i. To add these two imaginary numbers, we get:

-4i + 2i = -2i

Step 3: Combine the Real and Imaginary Parts

Now that we have added the real and imaginary parts separately, we can combine them to get the total score.

Total Score = 8 - 2i

Conclusion

In conclusion, to add complex numbers, we simply add the real parts and the imaginary parts separately. Using the scores of Melissa and Tomas in a game as an example, we found their total score to be 8 - 2i.

Answer

The correct answer is:

C. 8−2i8-2i

Discussion

This problem is a great example of how complex numbers can be used in real-world applications. In this case, we used complex numbers to represent the scores of Melissa and Tomas in a game. By adding these complex numbers, we were able to find their total score.

Related Topics

  • Complex Numbers: Adding and Subtracting
  • Complex Numbers: Multiplying and Dividing
  • Complex Numbers: Conjugates and Modulus

References

  • "Complex Numbers" by Khan Academy
  • "Complex Numbers" by Math Is Fun
  • "Complex Numbers" by Wolfram MathWorld

Further Reading

  • "Complex Analysis" by Walter Rudin
  • "Complex Numbers and Geometry" by David A. Brannan
  • "Complex Numbers and Algebra" by Michael Artin
    Complex Numbers: Q&A =========================

Introduction

In our previous article, we explored how to add complex numbers using the scores of Melissa and Tomas in a game as an example. In this article, we will answer some frequently asked questions about complex numbers.

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

A: A real number is a number that can be expressed without any imaginary part, such as 3 or -4. A complex number, on the other hand, is a number that has both real and imaginary parts, such as 3 + 4i or 2 - 5i.

Q: What is the imaginary unit i?

A: The imaginary unit i is a number that satisfies the equation i^2 = -1. It is used to represent the imaginary part of a complex number.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, to add the complex numbers 2 + 3i and 4 + 5i, you get:

(2 + 3i) + (4 + 5i) = (2 + 4) + (3i + 5i) = 6 + 8i

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, to subtract the complex numbers 4 + 5i and 2 + 3i, you get:

(4 + 5i) - (2 + 3i) = (4 - 2) + (5i - 3i) = 2 + 2i

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and the fact that i^2 = -1. For example, to multiply the complex numbers 2 + 3i and 4 + 5i, you get:

(2 + 3i)(4 + 5i) = 2(4 + 5i) + 3i(4 + 5i) = 8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i

Q: How do I divide complex numbers?

A: To divide complex numbers, you can use the conjugate of the denominator and the fact that i^2 = -1. For example, to divide the complex numbers 4 + 5i and 2 + 3i, you get:

(4 + 5i)/(2 + 3i) = (4 + 5i)(2 - 3i)/(2 + 3i)(2 - 3i) = (8 - 12i + 10i - 15i^2)/(4 - 9i^2) = (8 - 2i + 15)/(4 + 9) = 23/13 - 2i/13

Q: What is the modulus of a complex number?

A: The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It is calculated using the formula:

|a + bi| = √(a^2 + b^2)

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is the complex number with the opposite sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

Conclusion

In conclusion, complex numbers are a powerful tool for representing and manipulating mathematical expressions. By understanding how to add, subtract, multiply, and divide complex numbers, you can solve a wide range of problems in mathematics and other fields.

Related Topics

  • Complex Numbers: Adding and Subtracting
  • Complex Numbers: Multiplying and Dividing
  • Complex Numbers: Conjugates and Modulus

References

  • "Complex Numbers" by Khan Academy
  • "Complex Numbers" by Math Is Fun
  • "Complex Numbers" by Wolfram MathWorld

Further Reading

  • "Complex Analysis" by Walter Rudin
  • "Complex Numbers and Geometry" by David A. Brannan
  • "Complex Numbers and Algebra" by Michael Artin