Simplifying Expressions Involving Complex Numbers Addition And Multiplication

In the realm of mathematics, complex numbers extend the familiar real number system by incorporating an imaginary unit, typically denoted as i, which is defined as the square root of -1. Complex numbers are expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The real part of the complex number is a, and the imaginary part is b. Operations with complex numbers involve combining both real and imaginary parts, adhering to specific rules that govern their behavior.

This article delves into the simplification of expressions involving complex numbers, focusing on addition and multiplication. We will explore the fundamental principles that underpin these operations and demonstrate how to apply them effectively. By understanding these concepts, you can confidently manipulate complex numbers and solve a wide range of mathematical problems.

Adding Complex Numbers

When adding complex numbers, the process involves combining the real parts and the imaginary parts separately. This is analogous to combining like terms in algebraic expressions. Given two complex numbers, a + bi and c + di, their sum is obtained by adding their real parts (a and c) and their imaginary parts (b and d). The result is a new complex number expressed as (a + c) + (b + d)i.

To illustrate this, let's consider the example provided: $(5 + i√6) + (5 + 7i√6)$. Here, we have two complex numbers: 5 + i√6 and 5 + 7i√6. To add these numbers, we follow the rule of combining real and imaginary parts:

• Real parts: 5 + 5 = 10
• Imaginary parts: i√6 + 7i√6 = (1 + 7)i√6 = 8i√6

Therefore, the sum of the two complex numbers is 10 + 8i√6. This result demonstrates the straightforward process of adding complex numbers by treating the real and imaginary components as distinct terms.

The addition of complex numbers adheres to the commutative and associative properties, similar to real number addition. The commutative property states that the order of addition does not affect the result, i.e., (a + bi) + (c + di) = (c + di) + (a + bi). The associative property allows us to group complex numbers in any order when adding more than two numbers, i.e., [(a + bi) + (c + di)] + (e + fi) = (a + bi) + [(c + di) + (e + fi)].

Understanding these properties further solidifies the understanding of complex number addition and allows for flexible manipulation of expressions.

Example Breakdown

Let's break down the provided example in more detail to ensure clarity. The expression is $(5 + i√6) + (5 + 7i√6)$. We identify the real parts as 5 and 5, and the imaginary parts as i√6 and 7i√6.

1. Combine the real parts: 5 + 5 = 10. This gives us the real component of the resulting complex number.
2. Combine the imaginary parts: i√6 + 7i√6. We can factor out the i√6 term, resulting in (1 + 7)i√6 = 8i√6. This is the imaginary component of the resulting complex number.

Combining the real and imaginary parts, we get the final answer: 10 + 8i√6. This step-by-step approach highlights the simplicity of complex number addition.

In summary, adding complex numbers involves a straightforward process of combining like terms. The real parts are added together, and the imaginary parts are added together, resulting in a new complex number. This operation follows the same fundamental principles as algebraic addition, making it an accessible concept for those familiar with basic algebra.

Multiplying Complex Numbers

Multiplying complex numbers involves a process similar to multiplying binomials, using the distributive property (also known as the FOIL method). Given two complex numbers, a + bi and c + di, their product is obtained by multiplying each term in the first complex number by each term in the second complex number. This results in four terms: ac, adi, bci, and bdi². The key to simplifying this expression lies in the fact that i² = -1.

Using the i² = -1 identity, the term bdi² becomes -bd. The product of the two complex numbers can then be written as ac + adi + bci - bd. By grouping the real terms (ac and -bd) and the imaginary terms (adi and bci), we arrive at the simplified form: (ac - bd) + (ad + bc)i. This is the general formula for multiplying two complex numbers.

Let's apply this to the given example: $(5 + i√6) * (5 + 7i√6)$. We follow the distributive property:

• 5 * 5 = 25
• 5 * 7i√6 = 35i√6
• i√6 * 5 = 5i√6
• i√6 * 7i√6 = 7i² * 6 = 7 * (-1) * 6 = -42

Combining these terms, we get: 25 + 35i√6 + 5i√6 - 42. Now, we group the real and imaginary parts:

• Real parts: 25 - 42 = -17
• Imaginary parts: 35i√6 + 5i√6 = 40i√6

Therefore, the product of the two complex numbers is -17 + 40i√6. This example illustrates how the distributive property and the i² = -1 identity are used to multiply and simplify complex numbers.

The multiplication of complex numbers also adheres to the commutative, associative, and distributive properties. The commutative property states that the order of multiplication does not affect the result, i.e., (a + bi)(c + di) = (c + di)(a + bi). The associative property allows us to group complex numbers in any order when multiplying more than two numbers. The distributive property, as discussed, is crucial in expanding the product of two complex numbers.

Step-by-Step Multiplication Example

To further clarify the multiplication process, let's revisit the example $(5 + i√6) * (5 + 7i√6)$ and break it down into individual steps:

1. Apply the distributive property (FOIL):
   • First terms: 5 * 5 = 25
   • Outer terms: 5 * 7i√6 = 35i√6
   • Inner terms: i√6 * 5 = 5i√6
   • Last terms: i√6 * 7i√6 = 7i² * 6
2. Simplify the last term using i² = -1: 7i² * 6 = 7 * (-1) * 6 = -42
3. Combine all terms: 25 + 35i√6 + 5i√6 - 42
4. Group real and imaginary parts: (25 - 42) + (35i√6 + 5i√6)
5. Simplify real parts: 25 - 42 = -17
6. Simplify imaginary parts: 35i√6 + 5i√6 = 40i√6
7. Write the final result: -17 + 40i√6

This detailed breakdown illustrates the methodical approach to complex number multiplication, highlighting the importance of each step in arriving at the correct answer.

In conclusion, multiplying complex numbers involves applying the distributive property, simplifying using the i² = -1 identity, and combining like terms. This process, while initially appearing complex, becomes straightforward with practice and a clear understanding of the underlying principles.

Summary

In this article, we have explored the fundamental operations of addition and multiplication with complex numbers. We have seen that adding complex numbers involves combining the real and imaginary parts separately, while multiplying complex numbers utilizes the distributive property and the i² = -1 identity. These operations are essential for manipulating complex numbers and solving mathematical problems in various fields.

The key takeaways from this discussion are:

• Complex numbers are expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
• Adding complex numbers involves combining the real parts and the imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.
• Multiplying complex numbers involves using the distributive property and the i² = -1 identity: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

By mastering these operations, you can confidently work with complex numbers and tackle more advanced mathematical concepts that rely on them. Complex numbers are not just abstract mathematical entities; they have practical applications in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. A solid understanding of complex number operations is therefore invaluable for anyone pursuing studies or careers in these areas.

This concludes our exploration of adding and multiplying complex numbers. By understanding the underlying principles and practicing these operations, you can develop a strong foundation for further mathematical studies and applications.