The absolute value of any complex number [tex]\(a + bi\)[/tex] is the Euclidean distance from [tex]\((a, b)\)[/tex] to [tex]\((0, 0)\)[/tex] in the complex plane.
To understand this, let's break it down:
1. Complex Number Representation: A complex number is often written as [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] is the real part and [tex]\(b\)[/tex] is the imaginary part.
2. Magnitude or Absolute Value: The magnitude (or absolute value) of a complex number [tex]\(a + bi\)[/tex] is denoted as [tex]\(|a + bi|\)[/tex]. This value represents the distance of the point [tex]\((a, b)\)[/tex] in the complex plane from the origin [tex]\((0, 0)\)[/tex].
3. Euclidean Distance Formula: This distance can be calculated using the Euclidean distance formula:
[tex]\[
\text{Distance} = \sqrt{(a - 0)^2 + (b - 0)^2} = \sqrt{a^2 + b^2}
\][/tex]
4. Conclusion: Thus, the absolute value of the complex number [tex]\(a + bi\)[/tex] is the Euclidean distance from the point [tex]\((a, b)\)[/tex] to the origin [tex]\((0, 0)\)[/tex] in the complex plane.
Putting it all together, the completed definition is:
The absolute value of any complex number [tex]\(a + bi\)[/tex] is the Euclidean distance from [tex]\((a, b)\)[/tex] to [tex]\((0, 0)\)[/tex] in the complex plane.
