The Formula for Subtracting Complex Numbers

Oct 13, 2025 by miodeya.com 88 views

Introduction

Complex numbers are an extension of the real number system, which includes numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i^2 = -1. In this article, we will focus on subtracting complex numbers, which is a fundamental operation in algebra and calculus.

What is Subtraction in Complex Numbers?

Subtraction in complex numbers is similar to subtraction in real numbers, but with an additional consideration for the imaginary part. When subtracting two complex numbers, we subtract the real parts and the imaginary parts separately.

The Formula for Subtracting Complex Numbers

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 given in the problem statement:

Subtract ${$ (3+2i)$}$ from ${$ (-9-8i)$}$.

Using the formula above, we can subtract the complex numbers as follows:

(-9 - 8i) - (3 + 2i) = (-9 - 3) + (-8 - 2)i

Simplifying the expression, we get:

-12 - 10i

Answer

Therefore, the correct answer is:

${$ -12-10i$}$

Why is this the Correct Answer?

To understand why this is the correct answer, let's analyze the subtraction process step by step.

We subtract the real parts: -9 - 3 = -12
We subtract the imaginary parts: -8 - 2 = -10
We combine the results: -12 - 10i

This is the correct answer because it follows the formula for subtracting complex numbers.

Conclusion

In conclusion, subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. By following the formula for subtracting complex numbers, we can perform this operation with ease. In this article, we have seen how to subtract complex numbers using the formula and have applied this to a specific example.

Common Mistakes to Avoid

When subtracting complex numbers, it's essential to remember the following:

Subtract the real parts and the imaginary parts separately.
Use the formula for subtracting complex numbers: (a + bi) - (c + di) = (a - c) + (b - d)i
Simplify the expression by combining the results.

By following these guidelines, you can avoid common mistakes and perform complex number subtraction with confidence.

Practice Problems

To reinforce your understanding of subtracting complex numbers, try the following practice problems:

Subtract ${$ (5+3i)$}$ from ${$ (-2-4i)$}$.
Subtract ${$ (7-2i)$}$ from ${$ (3+5i)$}$.
Subtract ${$ (9+6i)$}$ from ${$ (-1-3i)$}$.

Answer Key

${$ -7-7i$}$
${$ -4+2i$}$
${ $10+9i\$}$

Final Thoughts

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

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

Q: How do I subtract complex numbers?

A: To subtract complex numbers, follow these steps:

Subtract the real parts: a - c
Subtract the imaginary parts: b - d
Combine the results: (a - c) + (b - d)i

Q: What if the complex numbers have different signs?

A: If the complex numbers have different signs, you can still subtract them by following the formula. For example:

(3 + 2i) - (-9 - 8i) = (3 - (-9)) + (2 - (-8))i

Simplifying the expression, we get:

12 + 10i

Q: Can I subtract complex numbers with zero imaginary part?

A: Yes, you can subtract complex numbers with zero imaginary part. For example:

(3 + 0i) - (2 + 0i) = (3 - 2) + (0 - 0)i

Simplifying the expression, we get:

1 + 0i

Q: What if I have a complex number with a negative sign?

A: If you have a complex number with a negative sign, you can still subtract it by following the formula. For example:

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

Simplifying the expression, we get:

-5 - 5i

Q: Can I subtract complex numbers with different magnitudes?

A: Yes, you can subtract complex numbers with different magnitudes. For example:

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

Simplifying the expression, we get:

1 - i

Q: What if I have a complex number with a fractional part?

A: If you have a complex number with a fractional part, you can still subtract it by following the formula. For example:

(3/2 + 1/4i) - (1/3 + 2/5i) = (3/2 - 1/3) + (1/4 - 2/5)i

Simplifying the expression, we get:

(13/6) + (-7/20)i

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 in the same form and cannot be subtracted directly.

Q: What if I have a complex number with a negative exponent?

A: If you have a complex number with a negative exponent, you can still subtract it by following the formula. For example:

(2^(-1) + 3^(-1)i) - (4^(-1) + 5^(-1)i) = (2^(-1) - 4^(-1)) + (3^(-1) - 5^(-1))i

Simplifying the expression, we get:

(-1/2) + (-2/3)i

Conclusion

In conclusion, subtracting complex numbers involves following a specific formula and simplifying the expression. By understanding the formula and practicing with different examples, you can become proficient in subtracting complex numbers. Remember to subtract the real parts and the imaginary parts separately and use the formula for subtracting complex numbers. With practice, you will become confident in your ability to subtract complex numbers and tackle more complex problems with ease.