Perform The Indicated Operations And Simplify.1. Add: \[$(7 - 11i) + (25 + 21i)\$\] Sum = \[$\square\$\]2. Subtract: \[$(7 - 11i) - (25 + 21i)\$\] Difference = \[$\square\$\]

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Complex numbers are mathematical expressions that consist of a real number and an imaginary number. They are used to represent points in a two-dimensional plane and are essential in various fields, including mathematics, physics, and engineering. In this article, we will perform two operations with complex numbers: addition and subtraction.

Addition of Complex Numbers

Complex numbers are added by combining their real and imaginary parts separately. The formula for adding two complex numbers is:

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

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

Example 1: Adding Complex Numbers

Add the complex numbers (7 - 11i) and (25 + 21i).

To add these complex numbers, we combine their real and imaginary parts separately:

Real part: 7 + 25 = 32 Imaginary part: -11i + 21i = 10i

Therefore, the sum of the complex numbers (7 - 11i) and (25 + 21i) is:

32 + 10i

Example 2: Subtracting Complex Numbers

Subtract the complex numbers (7 - 11i) and (25 + 21i).

To subtract these complex numbers, we combine their real and imaginary parts separately:

Real part: 7 - 25 = -18 Imaginary part: -11i - 21i = -32i

Therefore, the difference of the complex numbers (7 - 11i) and (25 + 21i) is:

-18 - 32i

Simplifying Complex Numbers

Complex numbers can be simplified by combining like terms. For example, we can combine the real and imaginary parts of a complex number separately.

Example 3: Simplifying a Complex Number

Simplify the complex number 3 + 4i.

To simplify this complex number, we can combine the real and imaginary parts separately:

Real part: 3 Imaginary part: 4i

Therefore, the simplified complex number is:

3 + 4i

Properties of Complex Numbers

Complex numbers have several properties that are useful in mathematics and other fields. Some of these properties include:

  • Commutative property: The order of the complex numbers does not affect the result of the operation.
  • Associative property: The order in which we perform the operations does not affect the result.
  • Distributive property: We can distribute a complex number to the real and imaginary parts of another complex number.

Example 4: Demonstrating the Commutative Property

Demonstrate the commutative property of complex numbers by adding (7 - 11i) and (25 + 21i) in a different order.

To demonstrate the commutative property, we can add the complex numbers in a different order:

(25 + 21i) + (7 - 11i) = (25 + 7) + (21i - 11i) = 32 + 10i

As we can see, the result is the same as in Example 1.

Applications of Complex Numbers

Complex numbers have many applications in mathematics and other fields. Some of these applications include:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent signals and analyze their behavior.
  • Navigation: Complex numbers are used to represent positions and velocities in navigation systems.

Example 5: Using Complex Numbers in Electrical Engineering

Use complex numbers to represent an AC circuit and analyze its behavior.

To represent an AC circuit using complex numbers, we can use the following formula:

V = IZ

where V is the voltage, I is the current, and Z is the impedance.

Suppose we have an AC circuit with a voltage of 10 + 20i and an impedance of 3 + 4i. We can use complex numbers to represent the current and analyze the behavior of the circuit.

I = V/Z = (10 + 20i) / (3 + 4i)

To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator:

I = ((10 + 20i) * (3 - 4i)) / ((3 + 4i) * (3 - 4i))

I = (30 - 40i + 60i - 80i^2) / (9 - 16i^2)

I = (30 + 20i + 80) / (9 + 16)

I = 110 + 20i / 25

I = 4.4 + 0.8i

Therefore, the current in the AC circuit is 4.4 + 0.8i.

Conclusion

In this article, we performed two operations with complex numbers: addition and subtraction. We also discussed the properties of complex numbers and their applications in mathematics and other fields. Complex numbers are a powerful tool for representing and analyzing complex phenomena, and they have many applications in various fields.

References

  • "Complex Numbers" by Math Is Fun
  • "Complex Numbers" by Khan Academy
  • "Complex Numbers in Electrical Engineering" by IEEE

Further Reading

  • "Complex Analysis" by Walter Rudin
  • "Complex Numbers and Geometry" by John Stillwell
  • "Complex Numbers in Signal Processing" by IEEE

In this article, we will answer some frequently asked questions about complex numbers.

Q: What is a complex number?

A complex number is a mathematical expression that consists of a real number and an imaginary number. It is written in the form a + bi, where a is the real part and bi is the imaginary part.

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

A real number is a number that can be expressed without any imaginary part, such as 3 or -4. An imaginary number is a number that can be expressed with an imaginary part, such as 3i or -4i.

Q: How do you add complex numbers?

To add complex numbers, you combine their real and imaginary parts separately. For example, to add (3 + 4i) and (2 - 3i), you would add the real parts (3 + 2 = 5) and the imaginary parts (4i - 3i = i).

Q: How do you subtract complex numbers?

To subtract complex numbers, you combine their real and imaginary parts separately. For example, to subtract (3 + 4i) from (2 - 3i), you would subtract the real parts (2 - 3 = -1) and the imaginary parts (3i - 4i = -i).

Q: What is the conjugate of a complex number?

The conjugate of a complex number is a complex number with the same real part but the opposite imaginary part. For example, the conjugate of (3 + 4i) is (3 - 4i).

Q: Why do we need complex numbers?

Complex numbers are used to represent points in a two-dimensional plane and are essential in various fields, including mathematics, physics, and engineering. They are used to solve equations that cannot be solved using real numbers alone.

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

Yes, complex numbers have many real-world applications, including:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent signals and analyze their behavior.
  • Navigation: Complex numbers are used to represent positions and velocities in navigation systems.

Q: How do you simplify complex numbers?

To simplify complex numbers, you can combine like terms. For example, to simplify (3 + 4i) + (2 - 3i), you would combine the real parts (3 + 2 = 5) and the imaginary parts (4i - 3i = i).

Q: What is the modulus of a complex number?

The modulus of a complex number is the distance from the origin to the point represented by the complex number in the complex plane. It is calculated using the formula |a + bi| = √(a^2 + b^2).

Q: What is the argument of a complex number?

The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point represented by the complex number in the complex plane. It is calculated using the formula arg(a + bi) = arctan(b/a).

Q: Can complex numbers be used to solve equations?

Yes, complex numbers can be used to solve equations that cannot be solved using real numbers alone. For example, the equation x^2 + 1 = 0 has no real solutions, but it has complex solutions.

Q: How do you represent complex numbers in polar form?

To represent complex numbers in polar form, you use the formula a + bi = r(cosθ + isinθ), where r is the modulus and θ is the argument.

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

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 represented by the complex number in the complex plane.

Q: Can complex numbers be used in computer science?

Yes, complex numbers have many applications in computer science, including:

  • Graphics: Complex numbers are used to represent points in a two-dimensional plane and are essential in graphics rendering.
  • Signal processing: Complex numbers are used to represent signals and analyze their behavior.
  • Navigation: Complex numbers are used to represent positions and velocities in navigation systems.

Conclusion

In this article, we answered some frequently asked questions about complex numbers. Complex numbers are a powerful tool for representing and analyzing complex phenomena, and they have many applications in various fields.

References

  • "Complex Numbers" by Math Is Fun
  • "Complex Numbers" by Khan Academy
  • "Complex Numbers in Electrical Engineering" by IEEE

Further Reading

  • "Complex Analysis" by Walter Rudin
  • "Complex Numbers and Geometry" by John Stillwell
  • "Complex Numbers in Signal Processing" by IEEE