Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: Find the absolute value. $$ |3-4 i| $$

Step-by-Step Solution

Answer

The absolute value is 5.

Step 1: Understanding the Problem

We need to find the absolute value of the complex number \(3 - 4i\). The absolute value of a complex number \(a + bi\) is calculated using the formula \(\sqrt{a^2 + b^2}\).

Step 2: Identify Components

In the complex number \(3 - 4i\), \(a = 3\) and \(b = -4\).

Step 3: Apply the Formula

Substitute \(a = 3\) and \(b = -4\) into the formula for the absolute value: \(\sqrt{3^2 + (-4)^2}\).

Step 4: Calculate Squared Terms

First, calculate \(3^2 = 9\) and \((-4)^2 = 16\).

Step 5: Add Squared Terms

Add the squared terms: \(9 + 16 = 25\).

Step 6: Calculate the Square Root

Find the square root of 25: \(\sqrt{25} = 5\).

Step 7: Conclusion

The absolute value of the complex number \(3 - 4i\) is 5.

Key Concepts

Absolute Value of a Complex Number

The absolute value of a complex number, also known as its magnitude, measures the distance between the point representing the complex number in the complex plane and the origin. For a complex number of the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, the absolute value can be found using the formula \(\sqrt{a^2 + b^2}\). This formula is similar to the Pythagorean theorem, as it involves finding the hypotenuse of a right triangle where \(a\) and \(b\) are the legs.

In this specific problem, we were tasked with finding the absolute value of \(3 - 4i\). By identifying \(a = 3\) and \(b = -4\), we compute \(\sqrt{3^2 + (-4)^2}\). After simplifying, this yields \(\sqrt{9 + 16} = \sqrt{25} = 5\). Therefore, the distance from the origin to the point \(3 - 4i\) is 5 units.

Understanding the Complex Plane

The complex plane, often referred to as the Argand plane, is a two-dimensional plane used to graph complex numbers. The horizontal axis represents the real part of the complex number, and the vertical axis represents the imaginary part. When you plot a complex number like \(3 - 4i\), you move 3 units along the real axis and -4 units along the imaginary axis.

This representation allows complex numbers to be visualized as vectors. The absolute value of a complex number, as discussed, is the length of this vector from the origin. Understanding how complex numbers are positioned on the complex plane helps in visualizing operations like addition, subtraction, and finding the absolute value, offering a geometric perspective to algebraic operations.

Grasping Imaginary Numbers

Imaginary numbers are a fundamental part of complex numbers, introduced to extend the real number system. A typical imaginary number is expressed as \(bi\), where \(b\) is a real number and \(i\) represents the imaginary unit \(\sqrt{-1}\). Imaginary numbers arise when solving equations that do not have solutions within the realm of real numbers, such as \(x^2 + 1 = 0\).

In this context, the term \(-4i\) in our complex number \(3 - 4i\) serves as the imaginary component. It influences how the complex number is positioned in the complex plane, directly affecting the calculation of its absolute value by contributing to the imaginary part \(b\) in our formula \(\sqrt{a^2 + b^2}\). Moreover, combining real and imaginary numbers into complex numbers enriches mathematical analysis beyond what's possible with real numbers alone.