Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The absolute value of \( 1-i \) is \( \sqrt{2} \).

Step 1: Understand the Absolute Value of a Complex Number

The absolute value (or modulus) of a complex number \( z = a + bi \), where \( a \) and \( b \) are real numbers, is given by \( |z| = \sqrt{a^2 + b^2} \). In our exercise, the complex number is \( 1 - i \).

Step 2: Identify Real and Imaginary Components

In the complex number \( 1 - i \), the real part \( a \) is 1 and the imaginary part \( b \) is -1. We need these values for our formula.

Step 3: Substitute Values into Formula

Substitute \( a = 1 \) and \( b = -1 \) into the modulus formula: \[ |1 - i| = \sqrt{1^2 + (-1)^2} \]

Step 4: Calculate Square of Components

Calculate \( 1^2 = 1 \) and \( (-1)^2 = 1 \). Now, substitute these into the formula: \[ |1 - i| = \sqrt{1 + 1} \]

Step 5: Simplify the Expression

Now, simplify the expression inside the square root: \( 1 + 1 = 2 \). Thus the expression becomes: \[ |1 - i| = \sqrt{2} \]

Step 6: Finalize the Answer

The absolute value of the complex number \( 1 - i \) is \( \sqrt{2} \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are usually written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Complex numbers are a significant part of mathematics because they extend the number system from the familiar real numbers. They are particularly useful in fields such as engineering and physics.

• Real numbers include values like 1, 0.5, and -3, which can be found on the number line.
• Imaginary numbers, like \( i \), arise from the square root of negative numbers. For instance, \( i \) is the square root of -1.

When you combine these components, you can represent any number as a complex number, providing more avenues for solutions in mathematical equations.

Modulus Formula

The modulus of a complex number measures its "size" or "distance" from the origin in the complex plane. It's an essential concept because it gives insight into the magnitude of the complex number, similar to how absolute value gives the magnitude of a real number. For a complex number \( z = a + bi \), the modulus is calculated using the formula: \( |z| = \sqrt{a^2 + b^2} \).

• The square of the real part \( a \) is \( a^2 \).
• The square of the imaginary part \( b \) is \( b^2 \).
• The modulus then becomes the square root of the sum of these squares.

Thus, the modulus reveals how far \( z \) is from the origin \((0,0)\) in the complex plane and is a useful tool for comparing and handling complex numbers.

Real and Imaginary Components

Every complex number \( z = a + bi \) has two main components: the real and the imaginary. Identifying these parts is crucial for operations such as addition, subtraction, and finding the modulus. In the exercise, with the complex number \( 1 - i \):

• The real component \( a \) is 1.
• The imaginary component \( b \) is -1 (note the minus sign).

These values are vital when using the modulus formula or performing other complex number calculations, ensuring you can manipulate and interpret these numbers accurately.

Square Root Calculation

In finding the modulus, calculating the square root is a key step. After summing the squares of the real and imaginary components, you determine the modulus by computing the square root of that sum. This process follows several straightforward steps:

• Square each component: for \( 1 - i \), calculate \( 1^2 = 1 \) and \((-1)^2 = 1\).
• Add the results: \( 1 + 1 = 2 \).
• Take the square root: \( \sqrt{2} \).

The square root represents the geometric "length" or "distance" in the complex plane, which is the modulus or absolute value of the complex number. Accurate square root calculations are fundamental to precise modulus evaluations.