Evaluate The Expression ( − 4 + 9 I ) ( 11 − 11 I (-4+9i)(11-11i ( − 4 + 9 I ) ( 11 − 11 I ] And Write The Result In The Form A + B I A+bi A + Bi .

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Introduction

In this article, we will evaluate the expression $(-4+9i)(11-11i)$ and write the result in the form $a+bi$. This involves multiplying two complex numbers and simplifying the result. We will use the distributive property and the fact that $i^2 = -1$ to simplify the expression.

What are Complex Numbers?

Before we begin, let's review what complex numbers 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, which satisfies $i^2 = -1$. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but with some additional rules.

Multiplying Complex Numbers

To multiply two complex numbers, we use the distributive property, which states that for any complex numbers $a+bi$ and $c+di$, we have:

$$(a+bi)(c+di) = ac + adi + bci + bdi^2$$

Using the fact that $i^2 = -1$, we can simplify this expression to:

$$(a+bi)(c+di) = (ac - bd) + (ad + bc)i$$

Evaluating the Expression

Now, let's evaluate the expression $(-4+9i)(11-11i)$. We can use the formula we derived above to simplify the expression:

$$(-4+9i)(11-11i) = (-4 \cdot 11 - 9 \cdot -11) + (-4 \cdot -11 + 9 \cdot 11)i$$

Simplifying this expression, we get:

$$(-4+9i)(11-11i) = (-44 + 99) + (44 + 99)i$$

$$(-4+9i)(11-11i) = 55 + 143i$$

Conclusion

In this article, we evaluated the expression $(-4+9i)(11-11i)$ and wrote the result in the form $a+bi$. We used the distributive property and the fact that $i^2 = -1$ to simplify the expression. The result is $55 + 143i$.

Applications of Complex Numbers

Complex numbers have many applications in mathematics and science. They are used to represent points in the complex plane, which is a two-dimensional plane with real and imaginary axes. Complex numbers are also used to represent rotations and scaling in geometry and trigonometry.

History of Complex Numbers

The concept of complex numbers dates back to the 16th century, when Italian mathematician Girolamo Cardano first introduced them in his book "Ars Magna". However, it wasn't until the 19th century that complex numbers became a fundamental part of mathematics.

Conclusion

In conclusion, complex numbers are a powerful tool in mathematics and science. They can be used to represent points in the complex plane, rotations and scaling in geometry and trigonometry, and many other mathematical concepts. In this article, we evaluated the expression $(-4+9i)(11-11i)$ and wrote the result in the form $a+bi$. We used the distributive property and the fact that $i^2 = -1$ to simplify the expression. The result is $55 + 143i$.

Final Thoughts

Complex numbers are a fascinating topic in mathematics, and their applications are vast and varied. From representing points in the complex plane to solving equations in algebra, complex numbers are an essential tool in many mathematical and scientific fields. In this article, we have seen how to evaluate the expression $(-4+9i)(11-11i)$ and write the result in the form $a+bi$. We hope that this article has provided a useful introduction to complex numbers and their applications.

References

• Cardano, G. (1545). Ars Magna.
• Euler, L. (1749). Introductio in Analysin Infinitorum.
• Gauss, C. F. (1801). Disquisitiones Arithmeticae.

Further Reading

• Complex Numbers on Wikipedia
• Complex Numbers on MathWorld
• Complex Numbers on Wolfram MathWorld

Related Articles

• Evaluating the Expression $(3+4i)(2-5i)$
• Simplifying the Expression $(1+2i)(3-4i)$
• Solving the Equation $z^2 + 4z + 5 = 0$
  

Introduction

In this article, we will answer some frequently asked questions about complex numbers. Complex numbers are a fundamental concept in mathematics and have many applications in science and engineering. We will cover topics such as what complex numbers are, how to add and multiply complex numbers, and how to simplify complex expressions.

Q: What are complex numbers?

A: Complex numbers are numbers that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies $i^2 = -1$. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but with some additional rules.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if we have the complex numbers $3+4i$ and $2-5i$, we can add them as follows:

$$(3+4i) + (2-5i) = (3+2) + (4-5)i$$

$$= 5 - i$$

Q: How do I multiply complex numbers?

A: To multiply complex numbers, we use the distributive property, which states that for any complex numbers $a+bi$ and $c+di$, we have:

$$(a+bi)(c+di) = ac + adi + bci + bdi^2$$

Using the fact that $i^2 = -1$, we can simplify this expression to:

$$(a+bi)(c+di) = (ac - bd) + (ad + bc)i$$

Q: How do I simplify complex expressions?

A: To simplify complex expressions, we can use the fact that $i^2 = -1$ to eliminate any powers of $i$ greater than 1. For example, if we have the expression $i^3 + 2i^2 + 3i$, we can simplify it as follows:

$$i^3 + 2i^2 + 3i = i^2 \cdot i + 2i^2 + 3i$$

$$= -1 \cdot i + 2(-1) + 3i$$

$$= -i - 2 + 3i$$

$$= 2 + 2i$$

Q: What is the conjugate of a complex number?

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

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

A: To find the conjugate of a complex number, you simply change the sign of the imaginary part. For example, if we have the complex number $3+4i$, its conjugate is $3-4i$.

Q: What is the modulus of a complex number?

A: 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 $|a+bi|$.

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

A: To find the modulus of a complex number, you can use the formula:

$$|a+bi| = \sqrt{a^2 + b^2}$$

Q: What is the argument of a complex number?

A: 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 $\arg(a+bi)$.

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

A: To find the argument of a complex number, you can use the formula:

$$\arg(a+bi) = \tan^{-1}\left(\frac{b}{a}\right)$$

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

A: The polar form of a complex number $a+bi$ is the expression $r(\cos\theta + i\sin\theta)$, where $r$ is the modulus of the complex number and $\theta$ is the argument of the complex number.

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

A: To convert a complex number to polar form, you can use the formulas:

$$r = \sqrt{a^2 + b^2}$$

$$\theta = \tan^{-1}\left(\frac{b}{a}\right)$$

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

A: The exponential form of a complex number $a+bi$ is the expression $re^{i\theta}$, where $r$ is the modulus of the complex number and $\theta$ is the argument of the complex number.

Q: How do I convert a complex number to exponential form?

A: To convert a complex number to exponential form, you can use the formulas:

$$r = \sqrt{a^2 + b^2}$$

$$\theta = \tan^{-1}\left(\frac{b}{a}\right)$$

Conclusion

In this article, we have answered some frequently asked questions about complex numbers. We have covered topics such as what complex numbers are, how to add and multiply complex numbers, and how to simplify complex expressions. We have also discussed the conjugate, modulus, argument, polar form, and exponential form of complex numbers. We hope that this article has provided a useful introduction to complex numbers and their applications.

References

• Complex Numbers on Wikipedia
• Complex Numbers on MathWorld
• Complex Numbers on Wolfram MathWorld

Further Reading

• Complex Numbers on Khan Academy
• Complex Numbers on MIT OpenCourseWare
• Complex Numbers on Coursera

Related Articles

• Evaluating the Expression $(3+4i)(2-5i)$
• Simplifying the Expression $(1+2i)(3-4i)$
• Solving the Equation $z^2 + 4z + 5 = 0$