Distribute the first complex number

Oct 13, 2025 by windailys.com 62 views

Introduction

In mathematics, complex numbers are numbers 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. Complex numbers 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 multiplying complex numbers, specifically the product of (2-7i) and (-1+4i).

What is Multiplication of Complex Numbers?

Multiplication of complex numbers is a fundamental operation in mathematics that involves multiplying two or more complex numbers. The multiplication of complex numbers is similar to the multiplication of binomials in algebra. When multiplying complex numbers, we use the distributive property and the fact that i^2 = -1 to simplify the expression.

Step-by-Step Guide to Multiplying Complex Numbers

To multiply complex numbers, we follow these steps:

Distribute the first complex number: Multiply the first complex number by each term in the second complex number.
Combine like terms: Combine the real and imaginary parts of the expression.
Simplify the expression: Simplify the expression by using the fact that i^2 = -1.

Multiplying (2-7i) and (-1+4i)

Now, let's apply the steps to multiply (2-7i) and (-1+4i).

Step 1: Distribute the first complex number

We multiply (2-7i) by (-1+4i):

(2-7i) × (-1+4i) = 2(-1+4i) - 7i(-1+4i)

Step 2: Distribute the terms

We distribute the terms:

2(-1+4i) = -2 + 8i -7i(-1+4i) = 7i - 28i^2

Step 3: Combine like terms

We combine the real and imaginary parts:

-2 + 8i + 7i - 28i^2

Step 4: Simplify the expression

We simplify the expression by using the fact that i^2 = -1:

-2 + 8i + 7i - 28(-1) = -2 + 8i + 7i + 28 = 26 + 15i

Conclusion

In conclusion, multiplying complex numbers involves distributing the first complex number, combining like terms, and simplifying the expression. By following these steps, we can multiply complex numbers and simplify the expression. In this article, we multiplied (2-7i) and (-1+4i) and obtained the result 26 + 15i.

Answer

The correct answer is:

A. 26 + 15i

Discussion

This problem is a great example of how to multiply complex numbers. The key is to distribute the first complex number, combine like terms, and simplify the expression. By following these steps, we can multiply complex numbers and simplify the expression.

Practice Problems

Multiply (3+4i) and (2-5i).
Multiply (1-2i) and (3+4i).
Multiply (4+3i) and (2-1i).

Solutions

(3+4i) × (2-5i) = 6 - 15i + 8i - 20i^2 = 6 - 7i + 20 = 26 - 7i
(1-2i) × (3+4i) = 3 + 4i - 6i - 8i^2 = 3 - 2i + 8 = 11 - 2i
(4+3i) × (2-1i) = 8 - 4i + 6i - 3i^2 = 8 + 2i + 3 = 11 + 2i
Multiplying Complex Numbers: A Q&A Guide =============================================

Introduction

In our previous article, we discussed how to multiply complex numbers. In this article, we will provide a Q&A guide to help you understand the concept of multiplying complex numbers better.

Q: What is the formula for multiplying complex numbers?

A: The formula for multiplying complex numbers is:

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

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

A: When multiplying complex numbers with different signs, you need to follow the same steps as multiplying complex numbers with the same signs. The only difference is that you need to be careful with the signs when combining like terms.

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 is always zero.

Q: How do I multiply complex numbers with fractions?

A: When multiplying complex numbers with fractions, you need to multiply the numerators and denominators separately. Then, simplify the expression by canceling out any common factors.

Q: Can I multiply complex numbers with decimals?

A: Yes, you can multiply complex numbers with decimals. When multiplying complex numbers with decimals, you need to follow the same steps as multiplying complex numbers with integers.

Q: How do I multiply complex numbers with negative exponents?

A: When multiplying complex numbers with negative exponents, you need to follow the same steps as multiplying complex numbers with positive exponents. The only difference is that you need to be careful with the signs when combining like terms.

Q: Can I multiply complex numbers with imaginary numbers?

A: Yes, you can multiply complex numbers with imaginary numbers. When multiplying complex numbers with imaginary numbers, you need to follow the same steps as multiplying complex numbers with real numbers.

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

A: When multiplying complex numbers with complex conjugates, you need to follow the same steps as multiplying complex numbers with real numbers. The only difference is that you need to be careful with the signs when combining like terms.

Q: Can I multiply complex numbers with matrices?

A: Yes, you can multiply complex numbers with matrices. When multiplying complex numbers with matrices, you need to follow the same steps as multiplying complex numbers with real numbers.

Q: How do I multiply complex numbers with vectors?

A: When multiplying complex numbers with vectors, you need to follow the same steps as multiplying complex numbers with real numbers. The only difference is that you need to be careful with the signs when combining like terms.

Conclusion

In conclusion, multiplying complex numbers is a fundamental operation in mathematics that involves multiplying two or more complex numbers. By following the steps outlined in this article, you can multiply complex numbers and simplify the expression. Remember to be careful with the signs when combining like terms.

Practice Problems

Multiply (3+4i) and (2-5i).
Multiply (1-2i) and (3+4i).
Multiply (4+3i) and (2-1i).

Solutions

(3+4i) × (2-5i) = 6 - 15i + 8i - 20i^2 = 6 - 7i + 20 = 26 - 7i
(1-2i) × (3+4i) = 3 + 4i - 6i - 8i^2 = 3 - 2i + 8 = 11 - 2i
(4+3i) × (2-1i) = 8 - 4i + 6i - 3i^2 = 8 + 2i + 3 = 11 + 2i

Glossary

Complex Number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
Imaginary Unit: A number that satisfies the equation i^2 = -1.
Real Number: A number that can be expressed in the form a, where a is a real number.
Matrix: A rectangular array of numbers.
Vector: A quantity with both magnitude and direction.