Subtract $(3+2i$\] From $(-9-8i$\].A. $-17-5i$B. $-6-6i$C. $-12-10i$D. $12+10i$

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Understanding Complex Numbers

Complex numbers are mathematical expressions that consist of a real number part and an imaginary number part. They are often represented in the form a+bia + bi, where aa is the real part and bb is the imaginary part. In this case, we have two complex numbers: (3+2i)(3+2i) and (−9−8i)(-9-8i). We are asked to subtract the first complex number from the second.

Subtracting Complex Numbers

To subtract complex numbers, we need to subtract the real parts and the imaginary parts separately. This means we will subtract the real part of the first complex number from the real part of the second complex number, and the imaginary part of the first complex number from the imaginary part of the second complex number.

Step 1: Subtract the Real Parts

The real part of the first complex number is 33, and the real part of the second complex number is −9-9. To subtract these, we simply subtract 33 from −9-9.

Step 2: Subtract the Imaginary Parts

The imaginary part of the first complex number is 2i2i, and the imaginary part of the second complex number is −8i-8i. To subtract these, we simply subtract 2i2i from −8i-8i.

Calculating the Result

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

Real Part Calculation

−9−3=−12-9 - 3 = -12

Imaginary Part Calculation

−8i−2i=−10i-8i - 2i = -10i

Combining the Results

The final answer is obtained by combining the real part and the imaginary part. Therefore, the result of subtracting (3+2i)(3+2i) from (−9−8i)(-9-8i) is −12−10i-12-10i.

Conclusion

In this problem, we have subtracted one complex number from another. We have followed the rules of subtracting complex numbers, which involve subtracting the real parts and the imaginary parts separately. The result of this subtraction is −12−10i-12-10i.

Answer Choice

The correct answer is C. −12−10i-12-10i.

Additional Information

Complex numbers are used in many areas of mathematics and science, including algebra, geometry, trigonometry, calculus, and physics. They are used to represent quantities that have both real and imaginary parts, such as the position and velocity of an object in a two-dimensional or three-dimensional space.

Example Use Case

Complex numbers are used in electrical engineering to represent the impedance of a circuit. The impedance of a circuit is a complex number that represents the opposition to the flow of current in the circuit. It has both a real part and an imaginary part, which represent the resistance and reactance of the circuit, respectively.

Real-World Applications

Complex numbers have many real-world applications, including:

  • Electrical engineering: Complex numbers are used to represent the impedance of a circuit and to analyze the behavior of electrical circuits.
  • Signal processing: Complex numbers are used to represent the frequency and amplitude of a signal.
  • Control systems: Complex numbers are used to represent the transfer function of a control system.
  • Quantum mechanics: Complex numbers are used to represent the wave function of a quantum system.

Conclusion

In conclusion, subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. The result of this subtraction is a new complex number. Complex numbers have many real-world applications, including electrical engineering, signal processing, control systems, and quantum mechanics.

Understanding Complex Numbers

Complex numbers are mathematical expressions that consist of a real number part and an imaginary number part. They are often represented in the form a+bia + bi, where aa is the real part and bb is the imaginary part.

Q&A: Complex Numbers

Q: What is a complex number?

A: A complex number is a mathematical expression that consists of a real number part and an imaginary number part. It is often represented in the form a+bia + bi, where aa is the real part and bb is the imaginary part.

Q: How do I add complex numbers?

A: To add complex numbers, you need to add the real parts and the imaginary parts separately. This means you will add the real part of the first complex number to the real part of the second complex number, and the imaginary part of the first complex number to the imaginary part of the second complex number.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you need to subtract the real parts and the imaginary parts separately. This means you will subtract the real part of the first complex number from the real part of the second complex number, and the imaginary part of the first complex number from the imaginary part of the second complex number.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you need to follow the distributive property and multiply the real part of the first complex number by the real part of the second complex number, the real part of the first complex number by the imaginary part of the second complex number, the imaginary part of the first complex number by the real part of the second complex number, and the imaginary part of the first complex number by the imaginary part of the second complex number.

Q: How do I divide complex numbers?

A: To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 3+4i3 + 4i is 3−4i3 - 4i.

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∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}.

Q: What is the argument of a complex number?

A: The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point representing the complex number in the complex plane. It is calculated using the formula arg⁡(a+bi)=tan⁡−1(ba)\arg(a + bi) = \tan^{-1}\left(\frac{b}{a}\right).

Q: What are the applications of complex numbers?

A: Complex numbers have many real-world applications, including electrical engineering, signal processing, control systems, and quantum mechanics.

Q: How do I represent complex numbers in the complex plane?

A: Complex numbers can be represented in the complex plane using the x-axis and the y-axis. The real part of the complex number is represented on the x-axis, and the imaginary part is represented on the y-axis.

Q: What is the relationship between complex numbers and trigonometry?

A: Complex numbers and trigonometry are closely related. The argument of a complex number is related to the angle between the positive real axis and the line segment connecting the origin to the point representing the complex number in the complex plane.

Q: What is the relationship between complex numbers and geometry?

A: Complex numbers and geometry are closely related. The modulus of a complex number is related to the distance from the origin to the point representing the complex number in the complex plane.

Conclusion

In conclusion, complex numbers are mathematical expressions that consist of a real number part and an imaginary number part. They have many real-world applications, including electrical engineering, signal processing, control systems, and quantum mechanics. Understanding complex numbers is essential for many areas of mathematics and science.