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Chapter 1: Problem 52

True or false: If \(a\) is negative, then \(|a|=-a\)

Short Answer

Expert verified
True

Step by step solution

01

- Understand the Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. Mathematically, it is denoted as \(|a|\).
02

- Definition of Absolute Value for Negative Numbers

For a negative number \(a\), such that \(a < 0\), the absolute value is defined as \(|a| = -a\). This means that you take the opposite (positive) value of \(a\) to get its absolute value.
03

- Conclusion

Since the original statement is: 'If \(a\) is negative, then \(|a| = -a\)', and based on the definition of absolute value for negative numbers, this statement is correct.

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Key Concepts

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

Negative Numbers
Negative numbers are numbers less than zero. They are typically written with a minus (-) sign preceding them.
For instance, -5 is a negative number. Negative numbers are found to the left of zero on the number line.
They represent values below zero and can denote a deficit, loss, or decrease in quantity.
Whenever we discuss the absolute value of a negative number, we are essentially talking about how far the number is from zero, irrespective of its direction. Simply put, the absolute value of any number is always positive or zero.
Number Line
A number line is a visual representation of numbers laid out in a straight line. The center of the number line is 0, called the origin.
Positive numbers are plotted to the right of zero, and negative numbers to the left.
The farther a number is from zero, the greater its absolute value. The number line helps in understanding concepts like absolute value and negative numbers, as it visually shows the distance from zero. When evaluating absolute values, this concept is crucial because the absolute distance of any number from zero is the same, regardless of whether the number is positive or negative.
Mathematical Definitions
In mathematics, definitions give us precise meanings of concepts. Here are some vital definitions relevant to our exercise:
  • Absolute Value: The absolute value of a number is its distance from zero on the number line, without considering its direction.
    It is always non-negative. For example, \(|a|\), where \(|a|\) denotes the absolute value of \(a\).
  • Negative Numbers: Numbers that are less than zero. Represented with a minus (-) sign.
  • Absolute Value for Negative Numbers: For any negative number \(a\), the absolute value of \(a\) is obtained by taking the positive counterpart of \(a\). Mathematically, for \(a < 0\), \(|a| = -a\). This happens because we negate the negative number to get a positive result.
Understanding these definitions helps students grasp various mathematical concepts more effectively.

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Most popular questions from this chapter

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 9 if the sum of its digits is divisible by \(9,\) and not otherwise. Show that (a) \(4,114,107\) is divisible by 9 and (b) \(2,287,321\) is not divisible by \(9 .\)

Simplify each expression. $$ 6(3 p-2)-(5 p+1) $$

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 8 if its last three digits form a numble by \(8,\) and not otherwise. Show that (a) \(2,923,296\) is divisible by 8 and (b) \(7,291,623\) is not divisible by 8 .

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ \left(\frac{1}{3} x-\frac{4}{5} y\right)\left(-\frac{1}{5} a\right) $$

Write a numerical expression for each phrase and simplify. The quotient of \(-12\) and the sum of \(-5\) and \(-1\)
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