Simplifying Complex Numbers How To Add And Express In A + Bi Form

This article provides a detailed explanation of how to add and simplify complex numbers, focusing on expressions involving square roots of negative numbers. We will break down the process step-by-step, ensuring a clear understanding of the underlying concepts. The core problem we address is simplifying the expression $(5+\sqrt{-49})+(7+\sqrt{-81})$. This involves understanding imaginary units, performing arithmetic operations with complex numbers, and expressing the final result in the standard form a + bi. By the end of this guide, you will be equipped with the knowledge and skills to confidently tackle similar complex number problems.

Understanding Complex Numbers

To effectively add and simplify complex numbers, it's crucial to first grasp what they are. 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. The imaginary unit i is defined as the square root of -1, i.e., i = ${\sqrt{-1}}$. The a part is called the real part, and the bi part is called the imaginary part. Complex numbers extend the real number system by including a dimension for imaginary numbers, allowing us to represent solutions to equations that have no real solutions, such as the square root of a negative number. This concept is pivotal in various fields including electrical engineering, quantum mechanics, and applied mathematics.

Complex numbers are not just abstract mathematical constructs; they have practical applications in diverse fields. In electrical engineering, complex numbers are used to analyze alternating current (AC) circuits. The impedance, which is the opposition to the flow of current in an AC circuit, is often expressed as a complex number. The real part represents resistance, and the imaginary part represents reactance. By using complex numbers, engineers can simplify calculations involving AC circuits and better understand circuit behavior. In quantum mechanics, complex numbers are fundamental to the mathematical formulation of the theory. The wave function, which describes the probability amplitude of a quantum particle, is a complex-valued function. The use of complex numbers allows physicists to accurately predict the behavior of quantum systems. Furthermore, complex numbers find applications in signal processing, control systems, and fluid dynamics. Their ability to represent two-dimensional quantities makes them invaluable tools in these areas. Understanding the nature and applications of complex numbers is thus essential for anyone studying mathematics, physics, or engineering.

The Imaginary Unit i

Central to simplifying expressions with complex numbers is the imaginary unit, denoted as i. As mentioned earlier, i is defined as the square root of -1. This means i² = -1. This seemingly simple definition unlocks a new realm of mathematical possibilities, allowing us to work with the square roots of negative numbers. For instance, ${\sqrt{-9}}$ can be rewritten as ${\sqrt{9 \cdot -1}}$, which simplifies to ${\sqrt{9} \cdot \sqrt{-1}}$ or 3i. Understanding how to manipulate i is crucial for performing operations with complex numbers.

To further elaborate on the properties of the imaginary unit i, let's explore its powers. We know that i¹ = i, i² = -1. What about i³? We can express i³ as i² * i, which is equal to -1 * i = -i. Similarly, i⁴ can be written as i² * i², which equals (-1) * (-1) = 1. This pattern continues, and we find that i⁵ = i, i⁶ = -1, i⁷ = -i, i⁸ = 1, and so on. This cyclic pattern of i, -1, -i, 1 is crucial for simplifying higher powers of i. For example, if we want to simplify i²⁵, we can divide 25 by 4 and find the remainder. The remainder is 1, so i²⁵ is equivalent to i¹, which is simply i. This technique is very useful when dealing with complex number expressions involving higher powers of i. Mastering the properties of the imaginary unit i is not only fundamental for complex number arithmetic but also lays the groundwork for more advanced topics in complex analysis.

Breaking Down the Problem

Now, let's simplify the given expression: $(5+\sqrt{-49})+(7+\sqrt{-81})$. The first step is to address the square roots of the negative numbers. We can rewrite ${\sqrt{-49}}$ as ${\sqrt{49 \cdot -1}}$ and ${\sqrt{-81}}$ as ${\sqrt{81 \cdot -1}}$. Using the property that ${\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}}$, we get ${\sqrt{49} \cdot \sqrt{-1}}$ and ${\sqrt{81} \cdot \sqrt{-1}}$, respectively. Since ${\sqrt{49} = 7}$ and ${\sqrt{81} = 9}$, and ${\sqrt{-1} = i}$, the expression becomes $(5+7i)+(7+9i)$. This transformation is the key to moving forward and simplifying the entire expression.

To further illustrate this breakdown, consider the process of converting square roots of negative numbers into their imaginary counterparts. This is a fundamental step in complex number simplification. Let’s examine ${\sqrt{-49}}$ in more detail. We can think of -49 as the product of 49 and -1. Mathematically, this is expressed as -49 = 49 * (-1). Now, applying the square root to both sides, we have ${\sqrt{-49} = \sqrt{49 \* -1}}$. Using the property that the square root of a product is the product of the square roots, we get ${\sqrt{49} \cdot \sqrt{-1}}$. Since ${\sqrt{49} = 7}$ and ${\sqrt{-1} = i}$, we substitute these values to obtain 7 * i, which is written as 7i. The same logic applies to ${\sqrt{-81}}$. We rewrite -81 as 81 * (-1), and then ${\sqrt{-81} = \sqrt{81 \* -1}}$. Applying the product of square roots property, we have ${\sqrt{81} \cdot \sqrt{-1}}$. Substituting ${\sqrt{81} = 9}$ and ${\sqrt{-1} = i}$, we get 9 * i, or 9i. This step-by-step process demystifies the conversion and makes it easier to handle similar problems in the future. With these conversions, the original expression $(5+\sqrt{-49})+(7+\sqrt{-81})$ transforms into a more manageable form for addition.

Adding Complex Numbers

Now that we have the expression $(5+7i)+(7+9i)$, we can proceed with the addition. To add complex numbers, we add the real parts together and the imaginary parts together. This is similar to combining like terms in algebraic expressions. So, we add 5 and 7 (the real parts) and 7i and 9i (the imaginary parts). This gives us (5+7) + (7i+9i). Performing the additions, we get 12 + 16i. This result is in the standard form a + bi, where a = 12 and b = 16.

The process of adding complex numbers is straightforward once the numbers are in the standard form a + bi. To further clarify this, let's consider a general case. Suppose we have two complex numbers, z₁ = a₁ + b₁i and z₂ = a₂ + b₂i. To add these two complex numbers, we simply add their real parts and their imaginary parts separately. The sum, z₁ + z₂, is given by (a₁ + a₂) + (b₁ + b₂)i. This can be visualized as adding the vectors corresponding to the complex numbers in the complex plane. The real parts are added along the real axis, and the imaginary parts are added along the imaginary axis. For example, if z₁ = 3 + 4i and z₂ = 2 - 1i, then z₁ + z₂ = (3 + 2) + (4 + (-1))i = 5 + 3i. Understanding this principle allows for efficient addition of multiple complex numbers or more complex expressions. The key is always to combine the real terms and the imaginary terms independently to arrive at the final answer in the form a + bi. This method ensures accuracy and clarity in dealing with complex number additions.

Final Result

Therefore, the simplified form of $(5+\sqrt{-49})+(7+\sqrt{-81})$ is 12 + 16i. This is our final answer, expressed in the form a + bi. This result demonstrates the process of adding complex numbers and simplifying expressions involving square roots of negative numbers. Remember, the key steps are to first convert square roots of negative numbers to their imaginary form, then add the real and imaginary parts separately.

In summary, simplifying complex number expressions involves several key steps. First, identify and convert the square roots of negative numbers into their imaginary equivalents using the definition i = ${\sqrt{-1}}$. This transformation is crucial for moving from expressions involving radicals to the standard complex number form a + bi. Second, rewrite the entire expression using these imaginary units. This often involves replacing ${\sqrt{-n}}$ with i${\sqrt{n}}$ where n is a positive real number. Third, perform the addition by combining the real parts and the imaginary parts separately. This involves adding the coefficients of the real terms and the coefficients of the imaginary terms independently. Finally, express the result in the standard form a + bi, where a represents the real part and b represents the imaginary part. By consistently following these steps, you can confidently simplify a wide range of complex number expressions. This methodical approach not only ensures accuracy but also enhances your understanding of complex number arithmetic. With practice, you'll be able to efficiently and effectively tackle more complex problems involving complex numbers.