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Chapter 1: Problem 87

The absolute value of any real number is always positive.

Short Answer

Expert verified
The absolute value of any real number is always nonegative, as it represents the distance from zero on the number line. So, for a positive real number or zero, the absolute value is the number itself; while for a negative real number, the absolute value is the positive version of that number.

Step by step solution

01

Understand Absolute Value

First, the definition of absolute value needs to be understood. The absolute value of a number is its distance from 0 on the number line; it is always nonegative. This is denoted as \(|a|\), where \(a\) can be any real number.
02

Case of Positive Numbers

If a real number \(a\) is positive (i.e, greater than 0), then its absolute value is the same as the number itself. So, \(|a| = a\).
03

Case of Negative Numbers

If a real number \(a\) is negative (i.e, less than 0), then its absolute value is the positive version of that number or the opposite. So, \(|a| = -a\), because multiplying any negative number by (-1) changes it to its positive version.
04

Case of Zero

If the real number \(a\) is zero, then \(|a| = 0\). Zero is neither positive nor negative, but we can typically think of it as a nonnegative number in the context of absolute value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers
Real numbers are a broad category of numbers that include everything from everyday counting numbers to more abstract numbers. They encompass:
  • Whole numbers like 1, 2, 3...
  • Fractions and decimals such as 1/2, 0.75...
  • Integers, which cover both positives and negatives like -3, -2, 0, 1...
  • Irrational numbers such as \( \pi \) and \sqrt{2}\.

Real numbers form the continuum that we plot on the number line. They are the basis for performing arithmetic operations and solving equations in mathematics. Understanding real numbers is crucial as they serve as the foundation for more complex concepts like absolute value.
Number Line
Imagine a straight horizontal line with numbers placed at regular intervals along it. This is the number line, a visual representation of real numbers. On this line:
  • Zero is placed at the center.
  • Positive numbers extend to the right of zero.
  • Negative numbers stretch to the left of zero.

The number line helps us understand the size and direction of a number in relation to zero. For instance, the absolute value of a number is its distance from zero on this line. This visual tool is essential for grasping concepts such as positive and negative values, and understanding how they relate to each other.
Positive Numbers
Positive numbers are the set of all real numbers greater than zero. They can be whole numbers, fractions, or decimals: \( a > 0 \). Examples include 1, 3.5, and \frac{7}{2}\. These numbers are found to the right of zero on the number line.
  • Absolute value of a positive number is the number itself: \( |a| = a \).
  • These numbers quantify increases, gains, and values typically viewed as more or larger than zero.

Understanding positive numbers is vital in math problems involving counting, measuring, and determining magnitude in various contexts.
Negative Numbers
Negative numbers are real numbers that are less than zero: \( a < 0 \). These include integers like -4, and decimal fractions like -2.7. On the number line, they appear to the left of zero.
  • The absolute value of a negative number is its corresponding positive number: \( |a| = -a \).
  • Negative numbers represent values below a defined point, such as temperature drops or debts.

Understanding how negative numbers work is essential for solving problems that involve subtraction, differences, and shifts in value or direction.
Zero
Zero holds a unique place as it is neither positive nor negative. Located at the midpoint on the number line, it functions as the origin point. Zero serves several important roles in mathematics:
  • It is the base point for measuring distance in the concept of absolute value.
  • Zero is the additive identity; adding zero to any number leaves the number unchanged: \( a + 0 = a \).
  • It represents a null value, indicating nothingness, emptiness, or the absence of quantity.

In the context of absolute value, zero鈥檚 absolute value is zero itself, maintaining its property of neutrality. Understanding zero is crucial for foundational mathematics and beyond.

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