Multiply. Write The Answer In Standard Form.\[$(3 - 2i)(4 + I) = \$\] \[$\square\$\]
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 they have numerous applications in various fields, including algebra, geometry, and calculus.
Multiplication of Complex Numbers
The multiplication of complex numbers is a fundamental operation in complex analysis. Given two complex numbers z1 = a + bi and z2 = c + di, where a, b, c, and d are real numbers, the product of z1 and z2 is defined as:
z1 × z2 = (a + bi) × (c + di)
To multiply two complex numbers, we can use the distributive property of multiplication over addition, which states that:
(a + bi) × (c + di) = ac + adi + bci + bdi^2
Since i^2 = -1, we can simplify the expression as:
ac + adi + bci - bd
Combining like terms, we get:
(ac - bd) + (ad + bc)i
This is the product of the two complex numbers z1 and z2.
Example: Multiplying Complex Numbers
Let's consider the example of multiplying the complex numbers (3 - 2i) and (4 + i).
(3 - 2i) × (4 + i) = ?
Using the formula for multiplying complex numbers, we get:
(3 - 2i) × (4 + i) = (3 × 4 - (-2) × 1) + (3 × 1 + (-2) × 4)i
Simplifying the expression, we get:
(12 + 2) + (3 - 8)i
Combine like terms:
14 - 5i
Therefore, the product of (3 - 2i) and (4 + i) is 14 - 5i.
Standard Form
The standard form of a complex number is a + bi, where a and b are real numbers and i is the imaginary unit. In the example above, the product of (3 - 2i) and (4 + i) is 14 - 5i, which is already in standard form.
Conclusion
In conclusion, the multiplication of complex numbers is a fundamental operation in complex analysis. By using the distributive property of multiplication over addition, we can multiply two complex numbers and simplify the expression to obtain the product in standard form. The example of multiplying the complex numbers (3 - 2i) and (4 + i) demonstrates this process.
Key Takeaways
- 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.
- The multiplication of complex numbers is defined as (a + bi) × (c + di) = (ac - bd) + (ad + bc)i.
- The product of two complex numbers can be simplified to obtain the product in standard form.
- The example of multiplying the complex numbers (3 - 2i) and (4 + i) demonstrates the process of multiplying complex numbers and simplifying the expression to obtain the product in standard form.
Further Reading
For further reading on complex numbers and their applications, we recommend the following resources:
- "Complex Analysis" by Elias M. Stein and Rami Shakarchi
- "Complex Numbers and Geometry" by John H. Hubbard and Barbara H. Burstin
- "Complex Analysis: A First Course" by Serge Lang
Introduction
In our previous article, we discussed the multiplication of complex numbers and provided an example of multiplying the complex numbers (3 - 2i) and (4 + i). In this article, we will answer some frequently asked questions about the multiplication of complex numbers.
Q: What is the formula for multiplying complex numbers?
A: The formula for multiplying complex numbers is:
z1 × z2 = (a + bi) × (c + di) = (ac - bd) + (ad + bc)i
Q: How do I multiply two complex numbers?
A: To multiply two complex numbers, you can use the distributive property of multiplication over addition. Multiply each term in the first complex number by each term in the second complex number, and then combine like terms.
Q: What is the product of (2 + 3i) and (4 - 2i)?
A: To find the product of (2 + 3i) and (4 - 2i), we can use the formula for multiplying complex numbers:
(2 + 3i) × (4 - 2i) = (2 × 4 - 3 × (-2)) + (2 × (-2) + 3 × 4)i
Simplifying the expression, we get:
(8 + 6) + (-4 + 12)i
Combine like terms:
14 + 8i
Therefore, the product of (2 + 3i) and (4 - 2i) is 14 + 8i.
Q: Can I multiply a complex number by a real number?
A: Yes, you can multiply a complex number by a real number. To do this, simply multiply the real number by each term in the complex number.
For example, to multiply (3 - 2i) by 4, we can multiply each term in the complex number by 4:
4 × (3 - 2i) = 4 × 3 - 4 × 2i
Simplifying the expression, we get:
12 - 8i
Therefore, the product of (3 - 2i) and 4 is 12 - 8i.
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 represented by the letter i. When multiplying complex numbers, you need to take into account the imaginary part and combine like terms.
Q: Can I use the commutative property of multiplication to multiply complex numbers?
A: Yes, you can use the commutative property of multiplication to multiply complex numbers. The commutative property states that the order of the factors does not change the result of the multiplication.
For example, to multiply (3 - 2i) and (4 + i), we can use the commutative property to write:
(3 - 2i) × (4 + i) = (4 + i) × (3 - 2i)
Using the formula for multiplying complex numbers, we get:
(3 - 2i) × (4 + i) = (3 × 4 - (-2) × 1) + (3 × 1 + (-2) × 4)i
Simplifying the expression, we get:
14 - 5i
Therefore, the product of (3 - 2i) and (4 + i) is 14 - 5i.
Conclusion
In conclusion, the multiplication of complex numbers is a fundamental operation in complex analysis. By using the distributive property of multiplication over addition, we can multiply two complex numbers and simplify the expression to obtain the product in standard form. We hope this Q&A article has provided you with a better understanding of the multiplication of complex numbers.
Key Takeaways
- The formula for multiplying complex numbers is z1 × z2 = (a + bi) × (c + di) = (ac - bd) + (ad + bc)i.
- To multiply two complex numbers, use the distributive property of multiplication over addition.
- The product of two complex numbers can be simplified to obtain the product in standard form.
- The commutative property of multiplication can be used to multiply complex numbers.
Further Reading
For further reading on complex numbers and their applications, we recommend the following resources:
- "Complex Analysis" by Elias M. Stein and Rami Shakarchi
- "Complex Numbers and Geometry" by John H. Hubbard and Barbara H. Burstin
- "Complex Analysis: A First Course" by Serge Lang
These resources provide a comprehensive introduction to complex numbers and their applications in various fields, including algebra, geometry, and calculus.