Multiply \[$(-1+i)(-2+3i)\$\].Write Your Answer As A Complex Number In Standard Form.

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 mathematics, physics, and engineering. In this article, we will focus on multiplying complex numbers, specifically the expression ${(-1+i)(-2+3i)}$. We will break down the process into manageable steps and provide a clear explanation of each step.

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 ${i}$.

Multiplying Complex Numbers

To multiply two complex numbers, we can use the distributive property, which states that for any complex numbers ${a+bi}$ and ${c+di}$, the product is given by:

${(a+bi)(c+di) = ac + adi + bci + bdi^2}$

Since ${i^2 = -1}$, we can simplify the expression to:

${(a+bi)(c+di) = ac - bd + (ad + bc)i}$

Step 1: Multiply the Real Parts

To multiply the complex numbers ${(-1+i)(-2+3i)}$, we first multiply the real parts:

${(-1)(-2) = 2}$

Step 2: Multiply the Imaginary Parts

Next, we multiply the imaginary parts:

${(i)(3i) = 3i^2 = -3}$

Step 3: Multiply the Cross Terms

Now, we multiply the cross terms:

${(-1)(3i) = -3i}$

${(i)(-2) = -2i}$

Step 4: Combine the Terms

Finally, we combine the terms:

${2 - 3 + (-3i + -2i) = -1 - 5i}$

Conclusion

In this article, we have shown how to multiply complex numbers using the distributive property. We have broken down the process into manageable steps and provided a clear explanation of each step. By following these steps, you can multiply complex numbers with ease.

Example Problems

Here are a few example problems to help you practice multiplying complex numbers:

• ${(-2+3i)(-1+i)}$
• ${(3+4i)(2-5i)}$
• ${(-1-2i)(-3+4i)}$

Tips and Tricks

Here are a few tips and tricks to help you multiply complex numbers:

• Use the distributive property to multiply complex numbers.
• Multiply the real parts first, then the imaginary parts.
• Multiply the cross terms last.
• Combine the terms to get the final result.

Conclusion

Introduction

In our previous article, we discussed how to multiply complex numbers using the distributive property. We broke down the process into manageable steps and provided a clear explanation of each step. In this article, we will answer some frequently asked questions about multiplying complex numbers.

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

A: The main difference between multiplying complex numbers and multiplying real numbers is that complex numbers have an imaginary part, which is multiplied by ${i}$. When multiplying complex numbers, we need to consider the imaginary part and multiply it by ${i}$, whereas when multiplying real numbers, we only need to consider the real part.

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

A: When multiplying complex numbers with different signs, we need to consider the signs of both numbers. If one number has a positive sign and the other has a negative sign, the result will have a negative sign. For example, ${(-1+i)(-2+3i) = 2 - 3 + (-3i + -2i) = -1 - 5i}$.

Q: Can I multiply complex numbers with zero?

A: Yes, you can multiply complex numbers with zero. When multiplying a complex number with zero, the result will be zero. For example, ${(2+3i)(0) = 0}$.

Q: How do I multiply complex numbers with fractions?

A: When multiplying complex numbers with fractions, we need to multiply the numerators and denominators separately. For example, ${\frac{2+3i}{4} \times \frac{3-4i}{5} = \frac{(2+3i)(3-4i)}{20} = \frac{6-8i+9i-12i^2}{20} = \frac{18+i}{20}}$.

Q: Can I multiply complex numbers with imaginary numbers only?

A: Yes, you can multiply complex numbers with imaginary numbers only. When multiplying imaginary numbers only, we need to consider the signs of both numbers and multiply them. For example, ${(3i)(-2i) = -6i^2 = 6}$.

Q: How do I multiply complex numbers with complex conjugates?

A: When multiplying complex numbers with complex conjugates, we need to consider the conjugate of the second number. The conjugate of a complex number ${a+bi}$ is ${a-bi}$. For example, ${(2+3i)(2-3i) = 2^2 - (3i)^2 = 4 + 9 = 13}$.

Q: Can I multiply complex numbers with negative imaginary numbers?

A: Yes, you can multiply complex numbers with negative imaginary numbers. When multiplying complex numbers with negative imaginary numbers, we need to consider the signs of both numbers and multiply them. For example, ${(-2-3i)(-1+i) = 2 + 3 + (3i + 2i) = 5 + 5i}$.

Conclusion

Multiplying complex numbers is a fundamental concept in mathematics, and it has numerous applications in various fields. By understanding the rules and procedures for multiplying complex numbers, you can tackle more complex problems and become proficient in this area. Remember to use the distributive property, multiply the real parts first, then the imaginary parts, multiply the cross terms last, and combine the terms to get the final result. With practice, you will become proficient in multiplying complex numbers and be able to tackle more complex problems.

Example Problems

Here are a few example problems to help you practice multiplying complex numbers:

• ${(-2+3i)(-1+i)}$
• ${(3+4i)(2-5i)}$
• ${(-1-2i)(-3+4i)}$
• ${(2+3i)(2-3i)}$
• ${(-2-3i)(-1+i)}$

Tips and Tricks

Here are a few tips and tricks to help you multiply complex numbers:

• Use the distributive property to multiply complex numbers.
• Multiply the real parts first, then the imaginary parts.
• Multiply the cross terms last.
• Combine the terms to get the final result.
• Consider the signs of both numbers and multiply them.
• Use the conjugate of the second number when multiplying complex numbers with complex conjugates.