Solve $\sqrt{-144}$.

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Introduction

Mathematics is a vast and fascinating subject that deals with numbers, quantities, and shapes. It is a fundamental tool for problem-solving and critical thinking. In mathematics, we often encounter problems that involve solving equations and expressions, including those with square roots. In this article, we will focus on solving the equation $\sqrt{-144}$.

Understanding Square Roots

Before we dive into solving the equation $\sqrt{-144}$, let's briefly review what square roots are. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$. Similarly, the square root of 25 is 5, because $5 \times 5 = 25$.

The Challenge of Negative Numbers

However, when we encounter negative numbers, things get a bit more complicated. By definition, the square of any real number is always non-negative. This means that there is no real number that can be squared to give a negative number. Therefore, the square root of a negative number is not a real number.

Complex Numbers

To solve the equation $\sqrt{-144}$, we need to introduce complex numbers. Complex numbers are numbers that have both real and imaginary parts. They are denoted by the letter $i$, where $i = \sqrt{-1}$. Using complex numbers, we can extend the real number system to include numbers that can be squared to give negative values.

Solving $\sqrt{-144}$

Now that we have introduced complex numbers, we can solve the equation $\sqrt{-144}$. To do this, we can rewrite the equation as $\sqrt{-144} = \sqrt{(-1) \times 144}$. Using the property of square roots, we can rewrite this as $\sqrt{-144} = \sqrt{-1} \times \sqrt{144}$.

Simplifying the Expression

We know that $\sqrt{-1} = i$, so we can substitute this into the expression to get $\sqrt{-144} = i \times \sqrt{144}$. Now, we can simplify the expression by evaluating the square root of 144. Since $144 = 12^2$, we have $\sqrt{144} = 12$.

The Final Answer

Putting it all together, we have $\sqrt{-144} = i \times 12$. Therefore, the final answer is $\boxed{12i}$.

Conclusion

In this article, we have solved the equation $\sqrt{-144}$ using complex numbers. We have shown that the square root of a negative number is not a real number, but rather a complex number. By introducing complex numbers, we can extend the real number system to include numbers that can be squared to give negative values. This is a fundamental concept in mathematics, and it has many applications in fields such as physics, engineering, and computer science.

Additional Resources

For those who want to learn more about complex numbers and their applications, here are some additional resources:

• Wikipedia: Complex Numbers
• Khan Academy: Complex Numbers
• MIT OpenCourseWare: Complex Analysis

Frequently Asked Questions

• Q: What is the square root of a negative number? A: The square root of a negative number is not a real number, but rather a complex number.
• Q: How do I solve the equation $\sqrt{-144}$? A: To solve the equation $\sqrt{-144}$, you can rewrite it as $\sqrt{(-1) \times 144}$ and then simplify the expression using the property of square roots.
• Q: What is the final answer to the equation $\sqrt{-144}$? A: The final answer to the equation $\sqrt{-144}$ is $\boxed{12i}$.

Final Thoughts

Solving the equation $\sqrt{-144}$ is a great example of how complex numbers can be used to extend the real number system. By introducing complex numbers, we can solve equations that would otherwise be impossible to solve. This is a fundamental concept in mathematics, and it has many applications in fields such as physics, engineering, and computer science.

Introduction

Complex numbers are a fundamental concept in mathematics that have many applications in fields such as physics, engineering, and computer science. In our previous article, we solved the equation $\sqrt{-144}$ using complex numbers. In this article, we will provide a Q&A section to help you better understand complex numbers and their applications.

Q&A

Q: What is a complex number?

A: A complex number is a number that has both real and imaginary parts. It is denoted by the letter $z$, where $z = a + bi$, and $a$ and $b$ are real numbers.

Q: What is the imaginary unit?

A: The imaginary unit is a complex number that is defined as $i = \sqrt{-1}$. It is used to extend the real number system to include numbers that can be squared to give negative values.

Q: How do I add complex numbers?

A: To add complex numbers, you can simply add the real parts and the imaginary parts separately. For example, if $z_1 = a + bi$ and $z_2 = c + di$, then $z_1 + z_2 = (a + c) + (b + d)i$.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you can simply subtract the real parts and the imaginary parts separately. For example, if $z_1 = a + bi$ and $z_2 = c + di$, then $z_1 - z_2 = (a - c) + (b - d)i$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and the fact that $i^2 = -1$. For example, if $z_1 = a + bi$ and $z_2 = c + di$, then $z_1 \times z_2 = (ac - bd) + (ad + bc)i$.

Q: How do I divide complex numbers?

A: To divide complex numbers, you can use the fact that $z_1 \div z_2 = \frac{z_1 \times \overline{z_2}}{|z_2|^2}$, where $\overline{z_2}$ is the conjugate of $z_2$ and $|z_2|$ is the magnitude of $z_2$.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $z = a + bi$ is $\overline{z} = a - bi$.

Q: What is the magnitude of a complex number?

A: The magnitude of a complex number $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$.

Q: How do I find the square root of a complex number?

A: To find the square root of a complex number, you can use the fact that $\sqrt{z} = \pm \left(\frac{\sqrt{a + \sqrt{a^2 + b^2}}}{2} + \frac{b}{2\sqrt{a + \sqrt{a^2 + b^2}}}\right) + \frac{b}{2\sqrt{a + \sqrt{a^2 + b^2}}}i$.

Q: What is the polar form of a complex number?

A: The polar form of a complex number $z = a + bi$ is $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$ and $\theta = \tan^{-1} \left(\frac{b}{a}\right)$.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you can use the fact that $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$ and $\theta = \tan^{-1} \left(\frac{b}{a}\right)$.

Q: How do I convert a complex number from polar form to rectangular form?

A: To convert a complex number from polar form to rectangular form, you can use the fact that $z = a + bi = r \cos \theta + ir \sin \theta$.

Conclusion

In this Q&A article, we have provided answers to some of the most common questions about complex numbers. We hope that this article has helped you better understand complex numbers and their applications. If you have any further questions, please don't hesitate to ask.

Additional Resources

For those who want to learn more about complex numbers and their applications, here are some additional resources:

• Wikipedia: Complex Numbers
• Khan Academy: Complex Numbers
• MIT OpenCourseWare: Complex Analysis

Final Thoughts

Complex numbers are a fundamental concept in mathematics that have many applications in fields such as physics, engineering, and computer science. By understanding complex numbers, you can solve equations that would otherwise be impossible to solve. We hope that this Q&A article has helped you better understand complex numbers and their applications.