Multiplying Complex Numbers: Simplify And Relate

Let's dive into the fascinating world of complex numbers! In this article, we'll explore how to multiply complex numbers, simplify the results, and understand the algebraic relationships between different forms. We'll specifically focus on the example you provided: multiplying a complex number by a fraction involving its conjugate, and then relating the simplified result to the original complex number and the complex number (1+i). Get ready to boost your math skills!

Understanding Complex Numbers

Before we jump into the multiplication, let's refresh our understanding of complex numbers. 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, defined as the square root of -1 (i.e., i² = -1). The 'a' part is called the real part, and the 'b' part is called the imaginary part. Complex numbers are essential in various fields, including mathematics, physics, engineering, and computer science. They allow us to solve equations that have no solutions within the real number system, and they provide a powerful tool for analyzing phenomena involving oscillations, waves, and alternating currents.

The set of complex numbers extends the set of real numbers by including the imaginary unit 'i'. This inclusion allows us to represent numbers that are not on the traditional number line. Geometrically, complex numbers can be visualized on a complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. Each complex number corresponds to a unique point on this plane, and we can perform algebraic operations on complex numbers just like we do with real numbers. Addition, subtraction, multiplication, and division are all well-defined operations for complex numbers, and they follow specific rules that take into account the properties of the imaginary unit 'i'. Understanding complex numbers is the cornerstone for advanced topics in mathematics and its applications. In quantum mechanics, for example, the wave functions that describe the state of a particle are complex-valued functions, and their properties are essential for understanding the behavior of quantum systems. Similarly, in electrical engineering, complex numbers are used to analyze alternating current circuits, where the impedance of a circuit element can be represented as a complex number. The ability to manipulate and interpret complex numbers is therefore crucial for anyone working in these fields.

Multiplying Complex Numbers: A Step-by-Step Guide

Now, let's get to the core of our problem: multiplying complex numbers. The key here is to treat 'i' as a variable initially, but remember that i² = -1. We'll use the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) to multiply the terms. Consider two complex numbers, (a + bi) and (c + di). Their product is calculated as follows:

(a + bi)(c + di) = a(c + di) + bi(c + di) = ac + adi + bci + bdi² Since i² = -1, we can substitute -1 for i²: = ac + adi + bci + bd(-1) = ac + adi + bci - bd Now, we group the real and imaginary terms: = (ac - bd) + (ad + bc)i So, the product of (a + bi) and (c + di) is (ac - bd) + (ad + bc)i. This formula provides a general method for multiplying any two complex numbers. It's important to remember the steps involved: distribute, substitute i² with -1, and then combine like terms. Let's illustrate this process with a numerical example. Suppose we want to multiply (2 + 3i) and (1 - i). Following the formula, we have:

(2 + 3i)(1 - i) = 2(1) + 2(-i) + 3i(1) + 3i(-i) = 2 - 2i + 3i - 3i² = 2 - 2i + 3i - 3(-1) = 2 - 2i + 3i + 3 = (2 + 3) + (-2i + 3i) = 5 + i Thus, the product of (2 + 3i) and (1 - i) is 5 + i. This example demonstrates the practical application of the formula and reinforces the importance of correctly applying the distributive property and substituting i² with -1. With practice, multiplying complex numbers becomes second nature, and you'll be able to handle more complex calculations with ease. The ability to perform these calculations accurately is crucial for many applications, such as solving complex equations, analyzing circuits, and understanding quantum mechanics.

Applying it to Our Problem: Simplifying the Expression

Let's apply this knowledge to the specific expression you provided:

(1/(1+i)) * ((1-i)/(1-i))

Notice that we're multiplying by ((1-i)/(1-i)), which is essentially multiplying by 1. This is a clever trick because it allows us to get rid of the complex number in the denominator. The term (1 - i) is called the conjugate of (1 + i). Multiplying a complex number by its conjugate always results in a real number. To see why, let's perform the multiplication:

(1 + i)(1 - i) = 1(1) + 1(-i) + i(1) + i(-i) = 1 - i + i - i² = 1 - (-1) (Since i² = -1) = 1 + 1 = 2 So, the product of a complex number and its conjugate is a real number, which makes it easier to work with. Now, let's return to our original expression and simplify it:

(1/(1+i)) * ((1-i)/(1-i)) = (1 * (1-i)) / ((1+i) * (1-i)) = (1 - i) / (1 - i²) = (1 - i) / (1 - (-1)) = (1 - i) / 2 = 1/2 - (1/2)i Thus, the simplified form of the expression is 1/2 - (1/2)i. This result is a complex number with a real part of 1/2 and an imaginary part of -1/2. The process of multiplying by the conjugate is a powerful technique in complex number arithmetic, and it's often used to simplify expressions and solve equations. This technique not only eliminates the imaginary part from the denominator but also helps in expressing complex fractions in a more manageable form. Understanding and mastering this technique is crucial for any further exploration of complex number theory and its applications. In many fields, such as electrical engineering and physics, complex numbers appear frequently in calculations, and the ability to manipulate them efficiently is essential. The process of rationalizing the denominator by multiplying by the conjugate is analogous to rationalizing the denominator of a fraction involving square roots, and it's a fundamental skill in complex number algebra.

Relating the Simplified Result to (1 + i)

Now, let's explore how our simplified result, 1/2 - (1/2)i, is related to the original complex number (1 + i). We can express this relationship algebraically:

1/2 - (1/2)i = (1/2)(1 - i) Notice that (1/2 - (1/2)i) is simply half of (1 - i). Furthermore, we know that (1 - i) is the conjugate of (1 + i). Therefore, our simplified result is half of the conjugate of (1 + i). This relationship highlights the symmetry and interconnectedness of complex numbers. The conjugate of a complex number is its mirror image across the real axis in the complex plane, and scaling a complex number by a real factor changes its magnitude but not its direction. In this case, multiplying (1 + i) by its conjugate gives us a real number, and then scaling the conjugate by 1/2 gives us the simplified complex number. This algebraic connection is crucial for understanding the properties of complex numbers and how they behave under different operations. For example, this relationship can be used to quickly find the reciprocal of a complex number. The reciprocal of a complex number (a + bi) is given by 1/(a + bi), which can be simplified by multiplying the numerator and denominator by the conjugate (a - bi). The resulting expression will have a real denominator, making it easier to work with. Understanding these relationships is not just about manipulating numbers; it's about gaining a deeper insight into the structure and behavior of complex numbers.

General Relationship and Conclusion

In general, if you have a complex number z = a + bi, then 1/z can be simplified by multiplying the numerator and denominator by the conjugate of z, which is a - bi:

1/z = 1/(a + bi) = (a - bi) / ((a + bi)(a - bi)) = (a - bi) / (a² + b²) So, 1/z is a scaled version of the conjugate of z. This general relationship is a powerful tool in complex number arithmetic and has numerous applications in various fields.

In conclusion, we've successfully multiplied complex numbers, simplified the result by using the conjugate, and related the simplified form back to the original complex number (1 + i). This process not only reinforces our understanding of complex number arithmetic but also highlights the beautiful algebraic relationships that exist within the complex number system. Keep practicing, and you'll become a complex number master!

For further exploration into the world of complex numbers and their applications, you might find valuable resources on Khan Academy's Complex Numbers Section.