What Is The Standard Form Of This Complex Number? 3 I + ( 3 4 + 2 I ) − ( 3 + 3 I 3i + \left(\frac{3}{4} + 2i\right) - (3 + 3i 3 I + ( 4 3 ​ + 2 I ) − ( 3 + 3 I ]A. $ $ B. 47 4 I \frac{47}{4} I 4 47 ​ I C. 9 4 − 2 I \frac{9}{4} - 2i 4 9 ​ − 2 I D. 15 4 + 8 I \frac{15}{4} + 8i 4 15 ​ + 8 I

Understanding Complex Numbers

Complex numbers are mathematical expressions that consist of a real number part and an imaginary number part. The imaginary number part is denoted by the letter 'i', where i is the square root of -1. Complex numbers are used to represent points in a two-dimensional plane, and they have numerous applications in mathematics, physics, and engineering.

The Given Complex Number

The given complex number is $3i + \left(\frac{3}{4} + 2i\right) - (3 + 3i)$. To simplify this expression, we need to combine like terms and perform the necessary operations.

Simplifying the Complex Number

To simplify the given complex number, we need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Combine like terms.
3. Perform any necessary operations.

Let's start by evaluating the expressions inside the parentheses:

$\left(\frac{3}{4} + 2i\right) - (3 + 3i)$

$= \frac{3}{4} + 2i - 3 - 3i$

Now, let's combine like terms:

$= \frac{3}{4} - 3 + 2i - 3i$

$= \frac{3}{4} - 3 - i$

Next, let's simplify the expression by combining the real number parts:

$= \frac{3}{4} - \frac{12}{4} - i$

$= \frac{3 - 12}{4} - i$

$= \frac{-9}{4} - i$

Now, let's add the $3i$ term to the expression:

$= \frac{-9}{4} - i + 3i$

$= \frac{-9}{4} + 2i$

The Standard Form of the Complex Number

The standard form of a complex number is $a + bi$, where $a$ is the real number part and $b$ is the imaginary number part. In this case, the real number part is $\frac{-9}{4}$ and the imaginary number part is $2$.

Conclusion

The standard form of the complex number $3i + \left(\frac{3}{4} + 2i\right) - (3 + 3i)$ is $\frac{-9}{4} + 2i$. This is the correct answer.

Comparison with the Options

Let's compare the standard form of the complex number with the given options:

A. $\frac{47}{4} i$ B. $\frac{9}{4} - 2i$ C. $\frac{15}{4} + 8i$ D. $\frac{-9}{4} + 2i$

The correct answer is option D, $\frac{-9}{4} + 2i$.

Final Answer

The final answer is $\boxed{\frac{-9}{4} + 2i}$.

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and computer science. In this article, we will provide a comprehensive Q&A guide to help you understand the basics of complex numbers.

Q1: What is a Complex Number?

A complex number is a mathematical expression that consists of a real number part and an imaginary number part. The imaginary number part is denoted by the letter 'i', where i is the square root of -1.

Q2: How Do I Represent a Complex Number?

A complex number can be represented in the form $a + bi$, where $a$ is the real number part and $b$ is the imaginary number part.

Q3: What is the Imaginary Unit 'i'?

The imaginary unit 'i' is a mathematical concept that is defined as the square root of -1. It is used to represent the imaginary part of a complex number.

Q4: How Do I Add Complex Numbers?

To add complex numbers, you need to add the real number parts and the imaginary number parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, the sum is $(a + c) + (b + d)i$.

Q5: How Do I Subtract Complex Numbers?

To subtract complex numbers, you need to subtract the real number parts and the imaginary number parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, the difference is $(a - c) + (b - d)i$.

Q6: How Do I Multiply Complex Numbers?

To multiply complex numbers, you need to follow the distributive property and multiply the real number parts and the imaginary number parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, the product is $(ac - bd) + (ad + bc)i$.

Q7: How Do I Divide Complex Numbers?

To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$. For example, if you have two complex numbers $a + bi$ and $c + di$, the quotient is $\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$.

Q8: What is the Modulus of a Complex Number?

The modulus of a complex number $a + bi$ is the distance from the origin to the point $(a, b)$ in the complex plane. It is denoted by $|a + bi|$ and is calculated as $\sqrt{a^2 + b^2}$.

Q9: What is the Argument of a Complex Number?

The argument of a complex number $a + bi$ is the angle between the positive real axis and the line segment joining the origin to the point $(a, b)$ in the complex plane. It is denoted by $\arg(a + bi)$ and is calculated as $\tan^{-1}\left(\frac{b}{a}\right)$.

Q10: What is the Conjugate of a Complex Number?

The conjugate of a complex number $a + bi$ is $a - bi$. It is denoted by $\overline{a + bi}$.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand the basics of complex numbers. We have covered topics such as representing complex numbers, adding and subtracting complex numbers, multiplying and dividing complex numbers, and calculating the modulus and argument of a complex number.

Final Answer

The final answer is that complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the basics of complex numbers, you can solve a wide range of mathematical problems and apply complex numbers to real-world situations.

Additional Resources

If you want to learn more about complex numbers, we recommend checking out the following resources:

• Khan Academy: Complex Numbers
• MIT OpenCourseWare: Complex Analysis
• Wolfram MathWorld: Complex Numbers

We hope this article has been helpful in understanding the basics of complex numbers. If you have any further questions or need additional clarification, please don't hesitate to ask.