Expressing Square Root Of Negative 144 In Terms Of I

Jul 15, 2025 by ADMIN 53 views

Understanding Imaginary Numbers Expressed with 'i'

In the realm of mathematics, particularly when dealing with complex numbers, imaginary units play a crucial role. The imaginary unit, denoted as i, is defined as the square root of -1. This concept allows us to express the square roots of negative numbers, which are not possible within the real number system. Let’s delve into how we can express $-\sqrt{-144}$ in terms of i. This exploration will provide a solid understanding of imaginary numbers and their application in simplifying mathematical expressions.

What are Imaginary Numbers?

Imaginary numbers are a fascinating extension of the real number system. At their core lies the imaginary unit, i, which, as mentioned earlier, is defined as $\sqrt{-1}$ . The need for imaginary numbers arises when we encounter the square root of a negative number. In the real number system, the square of any number is always non-negative. For instance, the square of 3 is 9, and the square of -3 is also 9. Therefore, there is no real number that, when squared, gives a negative result. This is where imaginary numbers step in to fill the gap.

When we encounter an expression like $\sqrt{-a}$ , where a is a positive real number, we can rewrite it using the imaginary unit i. Specifically, $\sqrt{-a}$ can be expressed as $\sqrt{a} * \sqrt{-1}$ , which simplifies to $\sqrt{a}i$ . This transformation allows us to handle the square roots of negative numbers systematically. For example, $\sqrt{-9}$ can be rewritten as $\sqrt{9} * \sqrt{-1}$ , which equals 3i. The imaginary unit i thus serves as a bridge, enabling us to work with these otherwise undefined expressions within the real number system.

Complex numbers, a broader category, encompass both real and imaginary numbers. A complex number is typically expressed in the form a + bi, where a is the real part and bi is the imaginary part. When a is 0, we have a purely imaginary number. When b is 0, the complex number simplifies to a real number. This structure allows complex numbers to represent a wide array of mathematical and physical phenomena, including alternating currents in electrical engineering and quantum mechanics in physics.

Understanding imaginary numbers is pivotal in various fields of mathematics, including algebra, calculus, and complex analysis. They are not just abstract concepts but powerful tools that enable us to solve equations and model systems that would be impossible to address using real numbers alone. The introduction of i opens up a new dimension in the number system, enhancing our ability to describe and analyze the world around us.

Step-by-Step Solution for $-\sqrt{-144}$

To simplify $-\sqrt{-144}$ and express it in terms of i, we need to follow a step-by-step approach that leverages the definition of imaginary numbers and the properties of square roots. This process ensures clarity and accuracy in our calculation. Here’s how we can break it down:

Rewrite the Expression Using the Imaginary Unit i: The first step in simplifying $-\sqrt{-144}$ is to recognize that we are dealing with the square root of a negative number. As we know, the square root of -1 is defined as the imaginary unit i. Therefore, we can rewrite the expression as follows:

$-\sqrt{-144} = -\sqrt{144 * -1}$

This step is crucial as it separates the negative sign from the number, allowing us to apply the imaginary unit i correctly.
Apply the Product Rule of Square Roots: We can use the property that the square root of a product is the product of the square roots, i.e., $\sqrt{ab} = \sqrt{a} * \sqrt{b}$ . Applying this rule, we get:
```
$\sqrt{144 * -1} = \sqrt{144} * \sqrt{-1}$
```
This step helps us to isolate the square root of -1, which we can then replace with i.
Simplify the Square Root of 144: The square root of 144 is a well-known perfect square. We know that 12 multiplied by itself gives 144. Therefore, $\sqrt{144} = 12$ . Substituting this value into our expression, we have:
```
$\sqrt{144} * \sqrt{-1} = 12 * \sqrt{-1}$
```
This simplification makes the expression easier to manage and brings us closer to the final answer.
Replace $\sqrt{-1}$ with i: Now, we replace $\sqrt{-1}$ with its equivalent, the imaginary unit i. This substitution is the key to expressing the original expression in terms of i. So, we have:
```
$12 * \sqrt{-1} = 12i$
```
This step directly applies the definition of i and transforms the expression into a purely imaginary number.
Account for the Negative Sign: Remember that our original expression was $-\sqrt{-144}$ . We have simplified $\sqrt{-144}$ to 12i, but we still need to account for the negative sign outside the square root. Multiplying our simplified expression by -1 gives us:
```
$-1 * 12i = -12i$
```
This final step completes the simplification, giving us the answer in the required form.

By following these steps, we have successfully expressed $-\sqrt{-144}$ in terms of i, arriving at the final answer of -12i. This methodical approach not only provides the solution but also reinforces the understanding of how imaginary numbers and the imaginary unit i are used in simplifying mathematical expressions.

Expressing the Result in the Form a + bi

In complex number theory, a complex number is typically expressed in the form a + bi, where a represents the real part and b represents the imaginary part. This standard form allows for a clear and consistent representation of complex numbers, making them easier to work with in mathematical operations and applications. In our case, we need to express the simplified form of $-\sqrt{-144}$ , which we found to be -12i, in this a + bi format. This process will help solidify our understanding of complex number representation.

Understanding the a + bi Form

The a + bi form is a fundamental concept in complex numbers. The real part, a, is a real number that represents the component of the complex number along the real number line. The imaginary part, bi, consists of the imaginary unit i multiplied by a real number b, representing the component of the complex number along the imaginary axis. Together, a and bi define a complex number that can be visualized on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

When a complex number is expressed in the a + bi form, it becomes straightforward to perform operations such as addition, subtraction, multiplication, and division. For instance, adding two complex numbers (a + bi) and (c + di) involves adding their real parts and their imaginary parts separately: (a + c) + (b + d)i. Similarly, multiplication involves applying the distributive property and the fact that i² = -1.

The a + bi form also makes it easy to identify special cases of complex numbers. If b is 0, then the complex number a + bi simplifies to a, which is a real number. If a is 0, then the complex number becomes bi, which is a purely imaginary number. Thus, the a + bi form serves as a unifying representation that includes both real and imaginary numbers as subsets.

Applying the Form to Our Result

Our result from simplifying $-\sqrt{-144}$ is -12i. To express this in the a + bi form, we need to identify the real part (a) and the imaginary part (b). In this case, we can rewrite -12i as:

$0 + (-12)i$

Here, the real part a is 0, and the imaginary part b is -12. This is because -12i has no real component; it is purely imaginary. Thus, expressing -12i in the a + bi form clarifies its nature as a complex number with a zero real part and a negative imaginary part.

This representation is not just a formality; it provides a complete picture of the complex number. By stating that the complex number is 0 + (-12)i, we explicitly show that it lies entirely on the imaginary axis of the complex plane, at the point -12i. This form is particularly useful in contexts where complex numbers are used in calculations or graphical representations, as it ensures clarity and consistency.

Final Answer

Therefore, the final answer, expressed in the form a + bi, is:

$0 - 12i$

This completes our process of simplifying $-\sqrt{-144}$ and representing it as a complex number in the standard a + bi format. This exercise underscores the importance of understanding imaginary units and the structure of complex numbers in mathematics.

Conclusion

In conclusion, expressing $-\sqrt{-144}$ in terms of i involves understanding the fundamental concept of imaginary numbers and the standard form of complex numbers. By breaking down the problem into manageable steps—rewriting the expression using the imaginary unit, applying the product rule of square roots, simplifying the square root of 144, replacing $\sqrt{-1}$ with i, and accounting for the negative sign—we methodically arrived at the simplified form -12i. Furthermore, expressing this result in the a + bi form, as 0 - 12i, reinforces the concept of complex numbers and their representation.

This exercise highlights the significance of imaginary numbers in expanding our mathematical toolkit. Imaginary numbers, symbolized by the imaginary unit i, allow us to deal with the square roots of negative numbers, which are not defined within the realm of real numbers. This capability is crucial in various branches of mathematics and its applications, including electrical engineering, physics, and computer science.

The step-by-step approach we employed underscores the importance of clarity and precision in mathematical problem-solving. Each step, from rewriting the expression to accounting for the negative sign, builds upon the previous one, ensuring that we arrive at the correct solution. This methodical approach not only aids in solving the specific problem at hand but also enhances our overall problem-solving skills in mathematics.

The representation of complex numbers in the a + bi form is a key concept in complex number theory. It provides a standardized way to express complex numbers, making them easier to manipulate and understand. The a + bi form explicitly shows the real and imaginary components of a complex number, which is essential for performing operations and visualizing complex numbers on the complex plane.

Ultimately, the ability to express mathematical expressions in different forms is a hallmark of mathematical proficiency. In this case, transforming $-\sqrt{-144}$ into -12i and further representing it as 0 - 12i demonstrates a comprehensive understanding of imaginary and complex numbers. This understanding is invaluable for anyone pursuing advanced studies in mathematics or related fields.

In summary, the process of expressing $-\sqrt{-144}$ in terms of i serves as a practical illustration of complex number concepts and problem-solving strategies. It reinforces the importance of understanding the definitions, properties, and representations of mathematical entities. By mastering these fundamentals, we can tackle more complex problems and deepen our appreciation for the elegance and power of mathematics.