Absolute Value of Complex Numbers

Oct 13, 2025 by miodeya.com 141 views

Introduction

In mathematics, the absolute value of a complex number is a measure of its distance from the origin in the complex plane. It is also known as the modulus or magnitude of the complex number. The absolute value of a complex number $a+bi$ is denoted by $|a+bi|$ and is calculated using the formula $\sqrt{a^2+b^2}$ . In this article, we will explore the concept of absolute value of complex numbers and calculate the absolute value of the complex number $-4-\sqrt{2}i$ .

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, which satisfies the equation $i^2=-1$ . Complex numbers can be represented graphically in the complex plane, which is a two-dimensional plane with the real axis and the imaginary axis. The real axis is the horizontal axis, and the imaginary axis is the vertical axis.

Absolute Value of Complex Numbers

The absolute value of a complex number $a+bi$ is calculated using the formula $\sqrt{a^2+b^2}$ . This formula is derived from the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides. In the complex plane, the real axis and the imaginary axis form a right triangle with the complex number as the hypotenuse.

Calculating the Absolute Value of $-4-\sqrt{2}i$

To calculate the absolute value of the complex number $-4-\sqrt{2}i$ , we will use the formula $\sqrt{a^2+b^2}$ . In this case, $a=-4$ and $b=-\sqrt{2}$ . Plugging these values into the formula, we get:

\sqrt{(-4)^2+(-\sqrt{2})^2}

Simplifying the expression, we get:

\sqrt{16+2}

\sqrt{18}

\boxed{\sqrt{18}}

Simplifying the Absolute Value

The absolute value of the complex number $-4-\sqrt{2}i$ is $\sqrt{18}$ . We can simplify this expression by factoring out a perfect square:

\sqrt{18}

\sqrt{9 \cdot 2}

\sqrt{9} \cdot \sqrt{2}

3 \cdot \sqrt{2}

Conclusion

In conclusion, the absolute value of the complex number $-4-\sqrt{2}i$ is $3 \cdot \sqrt{2}$ . This is calculated using the formula $\sqrt{a^2+b^2}$ , where $a=-4$ and $b=-\sqrt{2}$ . The absolute value of a complex number is a measure of its distance from the origin in the complex plane, and it is an important concept in mathematics.

Frequently Asked Questions

What is the absolute value of a complex number?
How is the absolute value of a complex number calculated?
What is the formula for calculating the absolute value of a complex number?
How do you simplify the absolute value of a complex number?

Answers

The absolute value of a complex number is a measure of its distance from the origin in the complex plane.
The absolute value of a complex number is calculated using the formula $\sqrt{a^2+b^2}$ .
The formula for calculating the absolute value of a complex number is $\sqrt{a^2+b^2}$ .
To simplify the absolute value of a complex number, you can factor out a perfect square.

References

"Complex Numbers" by Math Open Reference
"Absolute Value of Complex Numbers" by Purplemath
"Complex Numbers and the Complex Plane" by Khan Academy

Introduction

In our previous article, we explored the concept of absolute value of complex numbers and calculated the absolute value of the complex number $-4-\sqrt{2}i$ . In this article, we will answer some frequently asked questions about absolute value of complex numbers.

Q&A

Q: What is the absolute value of a complex number?

A: The absolute value of a complex number is a measure of its distance from the origin in the complex plane. It is also known as the modulus or magnitude of the complex number.

Q: How is the absolute value of a complex number calculated?

A: The absolute value of a complex number is calculated using the formula $\sqrt{a^2+b^2}$ , where $a$ and $b$ are the real and imaginary parts of the complex number, respectively.

Q: What is the formula for calculating the absolute value of a complex number?

A: The formula for calculating the absolute value of a complex number is $\sqrt{a^2+b^2}$ .

Q: How do you simplify the absolute value of a complex number?

A: To simplify the absolute value of a complex number, you can factor out a perfect square. For example, $\sqrt{18}$ can be simplified to $3 \cdot \sqrt{2}$ .

Q: What is the absolute value of the complex number $3+4i$ ?

A: To calculate the absolute value of the complex number $3+4i$ , we will use the formula $\sqrt{a^2+b^2}$ . In this case, $a=3$ and $b=4$ . Plugging these values into the formula, we get:

\sqrt{3^2+4^2}

Simplifying the expression, we get:

\sqrt{9+16}

\sqrt{25}

5

Q: What is the absolute value of the complex number $-2-3i$ ?

A: To calculate the absolute value of the complex number $-2-3i$ , we will use the formula $\sqrt{a^2+b^2}$ . In this case, $a=-2$ and $b=-3$ . Plugging these values into the formula, we get:

\sqrt{(-2)^2+(-3)^2}

Simplifying the expression, we get:

\sqrt{4+9}

\sqrt{13}

Q: Can the absolute value of a complex number be negative?

A: No, the absolute value of a complex number cannot be negative. The absolute value of a complex number is always non-negative, since it is a measure of its distance from the origin in the complex plane.

Q: Can the absolute value of a complex number be zero?

A: Yes, the absolute value of a complex number can be zero. This occurs when the complex number is equal to zero, i.e., $a+bi=0$ .

Conclusion

In conclusion, the absolute value of a complex number is a measure of its distance from the origin in the complex plane. It is calculated using the formula $\sqrt{a^2+b^2}$ , where $a$ and $b$ are the real and imaginary parts of the complex number, respectively. We have answered some frequently asked questions about absolute value of complex numbers and provided examples of how to calculate the absolute value of complex numbers.

Frequently Asked Questions

What is the absolute value of a complex number?
How is the absolute value of a complex number calculated?
What is the formula for calculating the absolute value of a complex number?
How do you simplify the absolute value of a complex number?
What is the absolute value of the complex number $3+4i$ ?
What is the absolute value of the complex number $-2-3i$ ?
Can the absolute value of a complex number be negative?
Can the absolute value of a complex number be zero?

Answers

The absolute value of a complex number is a measure of its distance from the origin in the complex plane.
The absolute value of a complex number is calculated using the formula $\sqrt{a^2+b^2}$ .
The formula for calculating the absolute value of a complex number is $\sqrt{a^2+b^2}$ .
To simplify the absolute value of a complex number, you can factor out a perfect square.
The absolute value of the complex number $3+4i$ is 5.
The absolute value of the complex number $-2-3i$ is $\sqrt{13}$ .
No, the absolute value of a complex number cannot be negative.
Yes, the absolute value of a complex number can be zero.

References

"Complex Numbers" by Math Open Reference
"Absolute Value of Complex Numbers" by Purplemath
"Complex Numbers and the Complex Plane" by Khan Academy