Frequently Asked Questions About Complex Numbers

Oct 13, 2025 by windailys.com 118 views

Simplify $(-7-6i)^2$ and Write Your Answer in the Form $a + bi$

In mathematics, the concept of simplifying complex numbers is a crucial aspect of algebra and calculus. 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 the equation $i^2 = -1$ . In this article, we will focus on simplifying the expression $(-7-6i)^2$ and write the answer in the form $a + bi$ .

Before we dive into simplifying the expression, let's take a moment to understand complex numbers. 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 the complex number is $a$ , and the imaginary part is $b$ . For example, the complex number $3 + 4i$ has a real part of $3$ and an imaginary part of $4$ .

Now that we have a good understanding of complex numbers, let's focus on simplifying the expression $(-7-6i)^2$ . To simplify this expression, we can use the formula $(a + bi)^2 = a^2 + 2abi + (bi)^2$ . In this case, $a = -7$ and $b = -6$ .

import cmath

# Define the complex number
z = complex(-7, -6)

# Calculate the square of the complex number
z_squared = z ** 2

print(z_squared)

Using this formula, we can simplify the expression as follows:

$(-7-6i)^2 = (-7)^2 + 2(-7)(-6i) + (-6i)^2$

$= 49 + 84i - 36$

$= 13 + 84i$

Therefore, the simplified expression is $13 + 84i$ .

In this article, we simplified the expression $(-7-6i)^2$ and wrote the answer in the form $a + bi$ . We used the formula $(a + bi)^2 = a^2 + 2abi + (bi)^2$ to simplify the expression. The final answer is $13 + 84i$ . This is an important concept in mathematics, and it has many applications in fields such as engineering, physics, and computer science.

If you want to learn more about complex numbers and how to simplify expressions like $(-7-6i)^2$ , here are some additional resources:

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

Q: What is the imaginary unit $i$ ? A: The imaginary unit $i$ is a number that satisfies the equation $i^2 = -1$ .
Q: How do I simplify a complex number? A: To simplify a complex number, you can use the formula $(a + bi)^2 = a^2 + 2abi + (bi)^2$ .
Q: What is the difference between a real number and a complex number? A: A real number is a number that can be expressed in the form $a$ , where $a$ is a real 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.
Frequently Asked Questions About Complex Numbers =====================================================

Complex numbers are a fundamental concept in mathematics, and they have many applications in fields such as engineering, physics, and computer science. In this article, we will answer some frequently asked questions about complex numbers.

Q: What is the imaginary unit $i$ ?

A: The imaginary unit $i$ is a number that satisfies the equation $i^2 = -1$ . It is used to extend the real number system to the complex number system.

Q: How do I simplify a complex number?

A: To simplify a complex number, you can use the formula $(a + bi)^2 = a^2 + 2abi + (bi)^2$ . This formula allows you to expand the square of a complex number and simplify it.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed in the form $a$ , where $a$ is a real 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.

Q: How do I add complex numbers?

A: To add complex numbers, you can add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$ , you can add them as follows:

$(a + bi) + (c + di) = (a + c) + (b + d)i$

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you can subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$ , you can subtract them as follows:

$(a + bi) - (c + di) = (a - c) + (b - d)i$

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the formula $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$ . This formula allows you to multiply two complex numbers and simplify the result.

Q: How do I divide complex numbers?

A: To divide complex numbers, you can use the formula $\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ . This formula allows you to divide two complex numbers and simplify the result.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is $a - bi$ . The conjugate of a complex number is used to simplify expressions and to find the reciprocal of a complex number.

Q: What is the modulus of a complex number?

A: The modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$ . The modulus of a complex number is used to find the distance of a complex number from the origin.

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 complex number. The argument of a complex number is used to find the polar form of a complex number.

In this article, we answered some frequently asked questions about complex numbers. We covered topics such as the imaginary unit $i$ , simplifying complex numbers, adding and subtracting complex numbers, multiplying and dividing complex numbers, the conjugate of a complex number, the modulus of a complex number, and the argument of a complex number. We hope that this article has been helpful in answering your questions about complex numbers.

If you want to learn more about complex numbers and how to simplify expressions like $(-7-6i)^2$ , here are some additional resources:

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

Q: What is the difference between a real number and a complex number? A: A real number is a number that can be expressed in the form $a$ , where $a$ is a real 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.
Q: How do I simplify a complex number? A: To simplify a complex number, you can use the formula $(a + bi)^2 = a^2 + 2abi + (bi)^2$ .
Q: What is the imaginary unit $i$ ? A: The imaginary unit $i$ is a number that satisfies the equation $i^2 = -1$ .
Q: How do I add complex numbers? A: To add complex numbers, you can add the real parts and the imaginary parts separately.
Q: How do I subtract complex numbers? A: To subtract complex numbers, you can subtract the real parts and the imaginary parts separately.
Q: How do I multiply complex numbers? A: To multiply complex numbers, you can use the formula $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$ .
Q: How do I divide complex numbers? A: To divide complex numbers, you can use the formula $\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ .
Q: What is the conjugate of a complex number? A: The conjugate of a complex number $a + bi$ is $a - bi$ .
Q: What is the modulus of a complex number? A: The modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$ .
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 complex number.