Check Each True Statement About The Number 7 − 3 I 7 - \sqrt{3}i 7 − 3 ​ I .A. 7 Is The Real Part Of The Number.B. 3 \sqrt{3} 3 ​ Is The Imaginary Part Of The Number.C. 7 − 3 7-\sqrt{3} 7 − 3 ​ Is The Coefficient Of I I I .D. This Number Is The Sum Of

$$

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. 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, which satisfies the equation $i^2 = -1$. In this article, we will explore the properties of complex numbers and examine the true statements about the number $7 - \sqrt{3}i$.

What is a Complex Number?

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 real part of a complex number is the part that is not multiplied by $i$, and it is denoted by $a$. The imaginary part of a complex number is the part that is multiplied by $i$, and it is denoted by $b$. For example, the complex number $3 + 4i$ has a real part of $3$ and an imaginary part of $4$.

The Number $7 - \sqrt{3}i$

The number $7 - \sqrt{3}i$ is a complex number that can be expressed in the form $a + bi$. In this case, the real part of the number is $7$, and the imaginary part is $-\sqrt{3}$. The coefficient of $i$ is the real part of the number, which is $7$. Therefore, statement C is true.

Analyzing the Statements

Now that we have a better understanding of complex numbers and the number $7 - \sqrt{3}i$, let's analyze the statements.

A. 7 is the real part of the number.

As we discussed earlier, the real part of a complex number is the part that is not multiplied by $i$. In the case of the number $7 - \sqrt{3}i$, the real part is indeed $7$. Therefore, statement A is true.

B. $\sqrt{3}$ is the imaginary part of the number.

The imaginary part of a complex number is the part that is multiplied by $i$. In the case of the number $7 - \sqrt{3}i$, the imaginary part is indeed $-\sqrt{3}$. Therefore, statement B is false.

C. $7-\sqrt{3}$ is the coefficient of $i$.

The coefficient of $i$ is the real part of the number, which is $7$. Therefore, statement C is true.

D. This number is the sum of

This statement is incomplete, and it does not provide enough information to determine whether it is true or false.

Conclusion

In conclusion, the true statements about the number $7 - \sqrt{3}i$ are:

• A. 7 is the real part of the number.
• C. $7-\sqrt{3}$ is the coefficient of $i$.

The false statement is:

• B. $\sqrt{3}$ is the imaginary part of the number.

The incomplete statement is:

• D. This number is the sum of

We hope this article has provided a better understanding of complex numbers and the number $7 - \sqrt{3}i$. If you have any questions or need further clarification, please don't hesitate to ask.

Additional Resources

• Complex Numbers
• Imaginary Unit
• Complex Number Operations

Related Topics

• Complex Conjugates
• Complex Number Inequalities
• Complex Number Equations
  Frequently Asked Questions about Complex Numbers =====================================================

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will answer some frequently asked questions about complex numbers.

Q: What is a complex number?

A: 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, which satisfies the equation $i^2 = -1$.

Q: What is the real part of a complex number?

A: The real part of a complex number is the part that is not multiplied by $i$. It is denoted by $a$ in the expression $a + bi$.

Q: What is the imaginary part of a complex number?

A: The imaginary part of a complex number is the part that is multiplied by $i$. It is denoted by $b$ in the expression $a + bi$.

Q: What is the imaginary unit?

A: The imaginary unit is a number that satisfies the equation $i^2 = -1$. It is denoted by $i$.

Q: How do I add complex numbers?

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

Q: How do I subtract complex numbers?

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

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that $i^2 = -1$. For example, if you have two complex numbers $a + bi$ and $c + di$, their product is $(ac - bd) + (ad + bc)i$.

Q: How do I divide complex numbers?

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

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of $a + bi$ is $a - bi$.

Q: What is the modulus of a complex number?

A: The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It is denoted by $|a + bi|$ and is equal to $\sqrt{a^2 + b^2}$.

Q: What is the argument of a complex number?

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

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

A: The polar form of a complex number is a way of expressing the complex number in terms of its modulus and argument. It is denoted by $r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the argument.

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

A: The exponential form of a complex number is a way of expressing the complex number in terms of its modulus and argument. It is denoted by $re^{i\theta}$, where $r$ is the modulus and $\theta$ is the argument.

Conclusion

In conclusion, complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. We hope this article has provided a better understanding of complex numbers and their properties. If you have any questions or need further clarification, please don't hesitate to ask.

Additional Resources

• Complex Numbers
• Imaginary Unit
• Complex Number Operations

Related Topics

• Complex Conjugates
• Complex Number Inequalities
• Complex Number Equations