Subtract. ( − 4 + 7 I ) − ( 3 − 3 I (-4+7i) - (3-3i ( − 4 + 7 I ) − ( 3 − 3 I ]

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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, including algebra, geometry, and calculus. 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

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

Let's consider the example of subtracting (-4 + 7i) and (3 - 3i).

(-4 + 7i) - (3 - 3i)

To subtract these complex numbers, we need to subtract the real parts and the imaginary parts separately.

Real part: -4 - 3 = -7 Imaginary part: 7i - (-3i) = 7i + 3i = 10i

Therefore, the result of subtracting (-4 + 7i) and (3 - 3i) is:

-7 + 10i

Properties of Complex Number Subtraction

Complex number subtraction has several properties that are similar to those of real number subtraction.

  • Commutativity: The order of the complex numbers being subtracted does not affect the result. In other words, (a + bi) - (c + di) = (c + di) - (a + bi).
  • Associativity: The order in which complex numbers are subtracted does not affect the result. In other words, ((a + bi) - (c + di)) - (e + fi) = (a + bi) - ((c + di) - (e + fi)).
  • Distributivity: Complex number subtraction distributes over addition. In other words, (a + bi) - (c + di) + (e + fi) = (a + bi) - (c + di) + (e + fi).

Conclusion

Subtracting complex numbers is a fundamental operation in complex number arithmetic. By understanding the properties of complex number subtraction, we can perform complex number arithmetic with ease. In this article, we have discussed the formula for subtracting complex numbers, provided an example of subtracting complex numbers, and explored the properties of complex number subtraction.

Frequently Asked Questions

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: Can you provide an example of subtracting complex numbers?

A: Yes, let's consider the example of subtracting (-4 + 7i) and (3 - 3i). The result of subtracting these complex numbers is -7 + 10i.

Q: What are the properties of complex number subtraction?

A: Complex number subtraction has several properties, including commutativity, associativity, and distributivity.

References

Further Reading

  • "Complex Numbers: A Beginner's Guide" by Coursera
  • "Complex Analysis" by MIT OpenCourseWare

Note: The references and further reading section are for additional resources and are not part of the main content.

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Introduction

Subtracting complex numbers is a fundamental operation in complex number arithmetic. In this article, we will provide a comprehensive Q&A section to help you understand complex number subtraction better.

Q&A

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: Can you provide an example of subtracting complex numbers?

A: Yes, let's consider the example of subtracting (-4 + 7i) and (3 - 3i). The result of subtracting these complex numbers is -7 + 10i.

Q: What are the properties of complex number subtraction?

A: Complex number subtraction has several properties, including commutativity, associativity, and distributivity.

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 an imaginary part, which is multiplied by i. When subtracting complex numbers, we need to subtract the real parts and the imaginary parts separately.

Q: Can you provide more examples of subtracting complex numbers?

A: Yes, here are a few more examples:

  • (-2 + 5i) - (1 - 2i) = (-2 - 1) + (5i + 2i) = -3 + 7i
  • (4 - 3i) - (2 + 4i) = (4 - 2) + (-3i - 4i) = 2 - 7i
  • (-1 + 2i) - (-3 + 4i) = (-1 + 3) + (2i - 4i) = 2 - 2i

Q: How do I simplify complex number expressions?

A: To simplify complex number expressions, you can use the following steps:

  1. Combine like terms.
  2. Simplify the real and imaginary parts separately.
  3. Use the properties of complex numbers, such as commutativity and associativity.

Q: Can you provide more information on complex number properties?

A: Yes, here are some additional properties of complex numbers:

  • Commutativity: The order of the complex numbers being added or subtracted does not affect the result.
  • Associativity: The order in which complex numbers are added or subtracted does not affect the result.
  • Distributivity: Complex number addition and subtraction distribute over multiplication.
  • Multiplicative Identity: The multiplicative identity of complex numbers is 1.
  • Multiplicative Inverse: The multiplicative inverse of a complex number a + bi is (a - bi) / (a^2 + b^2).

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the following formula:

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

Q: Can you provide an example of multiplying complex numbers?

A: Yes, let's consider the example of multiplying (2 + 3i) and (4 - 2i).

(2 + 3i) * (4 - 2i) = (2 * 4 - 3 * -2) + (2 * -2 + 3 * 4)i = (8 + 6) + (-4 + 12)i = 14 + 8i

Conclusion

Subtracting complex numbers is a fundamental operation in complex number arithmetic. By understanding the properties of complex number subtraction and how to simplify complex number expressions, you can perform complex number arithmetic with ease. In this article, we have provided a comprehensive Q&A section to help you understand complex number subtraction better.

Frequently Asked Questions

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: Can you provide an example of subtracting complex numbers?

A: Yes, let's consider the example of subtracting (-4 + 7i) and (3 - 3i). The result of subtracting these complex numbers is -7 + 10i.

Q: What are the properties of complex number subtraction?

A: Complex number subtraction has several properties, including commutativity, associativity, and distributivity.

References

Further Reading

  • "Complex Numbers: A Beginner's Guide" by Coursera
  • "Complex Analysis" by MIT OpenCourseWare

Note: The references and further reading section are for additional resources and are not part of the main content.