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Magazine Chapter 1: Problem 62
Evaluate each absolute value expression. $$|\sqrt{3}-3|$$
Short Answer
Expert verified
1.268
Step by step solution
01
Understand Absolute Value
Absolute value refers to the non-negative value of a number, regardless of its sign. For a number \( x \), the absolute value is denoted by \( |x| \). It is equal to \( x \) if \( x \) is positive or zero, and \(-x\) if \( x \) is negative.
02
Calculate \( \sqrt{3} \)
First, we need to find the value of \( \sqrt{3} \). While \( \sqrt{3} \) is an irrational number, we know that it is approximately 1.732. This approximation helps us understand whether \( \sqrt{3} - 3 \) is positive or negative.
03
Subtract to Find Expression Inside Absolute Value
Calculate \( \sqrt{3} - 3 \). Since \( \sqrt{3} \approx 1.732 \), we have \( 1.732 - 3 = -1.268 \). So, \( \sqrt{3} - 3 \) is negative.
04
Apply Absolute Value
Since the expression inside the absolute value, \( \sqrt{3} - 3 \), is negative, the absolute value is its positive counterpart. Therefore, \(|\sqrt{3} - 3| = |1.732 - 3| = |-1.268| = 1.268\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Irrational Numbers
In mathematics, irrational numbers hold a special place as numbers that cannot be exactly written as a simple fraction. They have non-repeating, non-terminating decimal expansions.
For instance, the square root of 3, denoted as \( \sqrt{3} \), is an example of an irrational number. It cannot be expressed precisely as a fraction of two integers.
Approximations, like \( \sqrt{3} \approx 1.732 \), are widely used to give us a practical understanding of irrational numbers.
For instance, the square root of 3, denoted as \( \sqrt{3} \), is an example of an irrational number. It cannot be expressed precisely as a fraction of two integers.
Approximations, like \( \sqrt{3} \approx 1.732 \), are widely used to give us a practical understanding of irrational numbers.
- Irrational numbers can't be perfectly represented, so approximations are often necessary.
- Some well-known irrational numbers include \( \pi \), \( e \), and certain square roots like \( \sqrt{2} \) and \( \sqrt{3} \).
Numerical Approximation
Numerical approximation is a method used to find a close estimate of an irrational number or a more complex mathematical expression. While exact values might be impossible to pin down due to their infinite nature, approximations make calculations feasible.
For example, approximating \( \sqrt{3} \) to 1.732 enables easier arithmetic operations when solving expressions like \(|\sqrt{3} - 3|\).
For example, approximating \( \sqrt{3} \) to 1.732 enables easier arithmetic operations when solving expressions like \(|\sqrt{3} - 3|\).
- Approximation is crucial in practical applications like engineering or physics.
- Good approximations involve a trade-off between precision and ease of use.
Mathematical Expression
A mathematical expression is a combination of numbers, variables, and operations that define a particular value or relationship. Understanding these expressions is essential for problem-solving.
In this exercise, the expression \( \sqrt{3} - 3 \) requires simplification to evaluate its absolute value.
In this exercise, the expression \( \sqrt{3} - 3 \) requires simplification to evaluate its absolute value.
- Operations like subtraction or addition are used to simplify expressions.
- The absolute value operation requires special attention, especially when it involves negative results.
Step-by-Step Solution
Solving mathematical problems often requires a structured approach, and a step-by-step solution can guide someone through the process effectively.
Here's how the process was applied in this exercise:
Here's how the process was applied in this exercise:
- **Step 1:** Understanding the concept of absolute value helps in identifying the non-negative result that's needed.
- **Step 2:** Calculate numerical approximations for irrational numbers like \( \sqrt{3} \).
- **Step 3:** Perform arithmetic operations to simplify the expression within the absolute value.
- **Step 4:** Apply the absolute value rule to derive the final positive result.
