Subtract The Following Complex Numbers:\[$(4+4i)-(13+17i)\$\]A. \[$9-13i\$\] B. \[$9+21i\$\] C. \[$-9+21i\$\] D. \[$-9-13i\$\]

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Introduction

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

What are Complex Numbers?

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 the part that is not multiplied by i, while the imaginary part is the part that is multiplied by i.

Subtracting Complex Numbers

To subtract complex numbers, we need to follow the same rules as subtracting real numbers. We simply subtract the real parts and the imaginary parts separately.

Example: Subtracting Complex Numbers

Let's consider the complex numbers (4 + 4i) and (13 + 17i). We want to subtract the second complex number from the first one.

Step 1: Subtract the Real Parts

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

4 - 13 = -9

Step 2: Subtract the Imaginary Parts

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

4i - 17i = -13i

Step 3: Combine the Results

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

-9 - 13i

Conclusion

Subtracting complex numbers is a straightforward process that involves subtracting the real parts and the imaginary parts separately. By following the steps outlined in this article, you can easily subtract complex numbers and solve problems involving complex number arithmetic.

Answer

The correct answer is:

-9 - 13i

Comparison with Options

Let's compare our answer with the options provided:

A. 9 - 13i B. 9 + 21i C. -9 + 21i D. -9 - 13i

Our answer, -9 - 13i, matches option D.

Tips and Tricks

Here are some tips and tricks to help you subtract complex numbers:

  • Make sure to subtract the real parts and the imaginary parts separately.
  • Use the correct order of operations (PEMDAS) to evaluate the expressions.
  • Check your answer by plugging it back into the original equation.

Practice Problems

Here are some practice problems to help you practice subtracting complex numbers:

  1. Subtract (3 + 5i) from (7 + 2i).
  2. Subtract (2 - 3i) from (5 + 4i).
  3. Subtract (1 + 2i) from (4 - 3i).

Conclusion

Q: What is the difference between subtracting complex numbers and subtracting real numbers?

A: The main difference between subtracting complex numbers and subtracting real numbers is that complex numbers have both a real part and an imaginary part. When subtracting complex numbers, we need to subtract the real parts and the imaginary parts separately.

Q: How do I subtract complex numbers with different signs?

A: When subtracting complex numbers with different signs, we need to follow the rules of subtraction. For example, if we have (3 + 4i) - (2 - 5i), we need to subtract the real parts and the imaginary parts separately. In this case, we would get (3 - 2) + (4i + 5i) = 1 + 9i.

Q: Can I subtract complex numbers with the same sign?

A: Yes, you can subtract complex numbers with the same sign. For example, if we have (3 + 4i) - (2 + 5i), we need to subtract the real parts and the imaginary parts separately. In this case, we would get (3 - 2) + (4i - 5i) = 1 - i.

Q: How do I handle complex numbers with zero imaginary part?

A: If a complex number has a zero imaginary part, it is essentially a real number. For example, if we have (3 + 0i) - (2 + 0i), we can simply subtract the real parts, which would give us 3 - 2 = 1.

Q: Can I subtract complex numbers with different magnitudes?

A: Yes, you can subtract complex numbers with different magnitudes. For example, if we have (3 + 4i) - (2 + 5i), we need to subtract the real parts and the imaginary parts separately. In this case, we would get (3 - 2) + (4i - 5i) = 1 - i.

Q: How do I handle complex numbers with negative imaginary part?

A: If a complex number has a negative imaginary part, it is simply a complex number with a negative sign. For example, if we have (3 - 4i) - (2 + 5i), we need to subtract the real parts and the imaginary parts separately. In this case, we would get (3 - 2) + (-4i - 5i) = 1 - 9i.

Q: Can I subtract complex numbers with different exponents?

A: No, you cannot subtract complex numbers with different exponents. Complex numbers with different exponents are not comparable, and you cannot subtract them.

Q: How do I handle complex numbers with complex coefficients?

A: If a complex number has complex coefficients, it is a complex number with complex coefficients. For example, if we have (3 + 4i) - (2 + 5i), we need to subtract the real parts and the imaginary parts separately. In this case, we would get (3 - 2) + (4i - 5i) = 1 - i.

Q: Can I subtract complex numbers with different bases?

A: No, you cannot subtract complex numbers with different bases. Complex numbers with different bases are not comparable, and you cannot subtract them.

Conclusion

Subtracting complex numbers can be a bit tricky, but with practice and patience, you can master this operation. Remember to subtract the real parts and the imaginary parts separately, and use the correct order of operations to evaluate the expressions. If you have any more questions or need further clarification, feel free to ask!