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Chapter 1: Problem 88

The absolute value of a number is equal to the absolute value of its opposite.

Short Answer

Expert verified
The proof shows that the absolute value of a number is indeed equal to the absolute value of its opposite.

Step by step solution

01

Understand the Concept

The first step is understanding the nature of absolute value. Absolute value is the magnitude or size of a real number without its sign. It is the distance of that number from zero on a number line. For example, \( |5| = 5 \) and \( |-5| = 5 \)
02

Define the Variables

Next, we will use the variables 'a' and 'b' to represent any number and its opposite, respectively. So, if 'a' is any real number, 'b' would be '-a', which is the opposite of 'a'.
03

Apply the Definition of Absolute Value

Using the definitions of 'a' and 'b', we will show that \( |a| = |b| \). By the definition of absolute value, we have \( |a| = a \) when \( a \geq 0 \), and \( |a| = -a \) when \( a < 0 \). Likewise, \( |b| = b \) when \( b \geq 0 \), or using the definition of 'b', \( |b| = -[(-a)] \) when \( (-a) \geq 0 \). But this simplifies to \( |b| = a \), which shows that \( |a| = |b| \).

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Key Concepts

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

Properties of Absolute Value
The absolute value is a unique concept in mathematics, focusing on the size or magnitude of a number without considering its sign. This means the absolute value of a number is always non-negative, and it represents how far a number is from zero on the real number line.
\[ \text{For example, } |3| = 3 \text{ and } |-3| = 3. \] Here, both positive and negative 3 are equidistant from zero, hence their absolute values are equal.
The following are some core properties of absolute value:
  • The absolute value of zero is zero: \( |0| = 0 \).
  • The absolute value of any positive number is the number itself: \( |a| = a \), if \( a \geq 0 \).
  • The absolute value of any negative number is its opposite (positive): \( |a| = -a \), if \( a < 0 \).
  • For any two real numbers, \( a \) and \( b \), the property \( |a \, b| = |a| \, |b| \) holds true.

These properties make absolute value a powerful tool in both pure and applied mathematics, aiding our understanding from basic calculations to advanced equations.
Real Number Line
The real number line is an essential concept when dealing with absolute value, as it helps visualize numbers and their absolute values.
Imagine a long line stretching from negative infinity to positive infinity, with each point on the line representing a real number. Zero is placed at the center, acting as the reference point.
Absolute value represents the distance any number is from zero on this line.
  • If a number is to the right of zero, like 4, its distance is its absolute value: \( |4| = 4 \).
  • If a number is to the left of zero, like -4, its distance from zero (ignoring direction) is also the absolute value: \( |-4| = 4 \).

This visualization helps to understand that, although they have different signs, \(|4|\) and \(|-4|\) both measure 4 units from zero.
Magnitude and Sign
Magnitude and sign are two distinct concepts when exploring the absolute value of a number. Magnitude indicates the size of a number, while sign tells us its direction on the real number line鈥攅ither positive or negative.
Absolute value, denoted by vertical bars around a number like \(|x|\), provides the magnitude without regard to sign.
For instance:
  • If \( x = 7 \), then \(|x| = 7\), because 7 is seven units to the right of zero.
  • If \( x = -7 \), then \(|x| = 7\), because -7 is seven units to the left of zero.

This reiterates how absolute value strips away the direction (sign) and focuses solely on distance (magnitude), making it useful in various mathematical contexts and problem-solving scenarios.

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Most popular questions from this chapter

Determining Order of Operations Using the established order of operations, which of the following expressions has a value of 72 ? For those that do not, decide whether you can insert parentheses into the expression so that its value is 72 . (a) \(4+2^{3}-7\) (b) \(4+8 \cdot 6\) (c) \(93-25-4\) (d) \(70+10 \div 5\) (e) \(60+20 \div 2+32\) (f) \(35 \cdot 2+2\)

\(7-\frac{4+6}{2^{2}+1}+5\)

\(4\left(-\frac{2}{3}+\frac{4}{3}\right)\)

Writing Are \(2 \cdot 5^{2}\) and \(10^{2}\) equal? Explain.

Commutative Property of Multiplication: \(10(-3)=\)
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