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Magazine Chapter 1: Problem 96
Is it always true that \(|-a|=a\) for any number \(a\) ? Explain why or why not.
Short Answer
Expert verified
No, \(|-a| = a\) only holds for non-negative numbers; it's not true for negative \(a\).
Step by step solution
01
Understanding Absolute Value
The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always non-negative. By definition, \[\begin{cases} \text{If } x \geq 0, & |x| = x \\text{If } x < 0, & |x| = -x \end{cases}\]This implies that the absolute value symbol will effectively strip away any negative sign.
02
Considering Positive Numbers
If \( a \) is a positive number, then \( -a \) is negative. By taking its absolute value, we have:\[ |-a| = -(-a) = a \]In this case, \(|-a| = a\) is true.
03
Considering Zero
If \( a = 0 \), then \( -a \) is also 0. The absolute value of zero is zero, thus:\[ |-a| = |0| = 0 = a \]Here, \(|-a| = a\) holds true as well.
04
Considering Negative Numbers
If \( a \) is a negative number, then \( -a \) is positive. The absolute value of a positive number is the number itself, therefore:\[ |-a| = -a \]In such cases, \(|-a| = a\) is not correct since \(-a eq a\) when \(a\) is negative.
05
Conclusion
Based on the definitions and cases we considered, \(|-a| = a\) is not always true. It holds for positive numbers and zero, but not for negative numbers, where it becomes \(|-a| = -a\), the positive counterpart of the negative \(a\).
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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 represented with a minus sign, and they sit to the left of zero on the number line. For example, -1, -5, and -20 are all negative numbers. Negative numbers are crucial when examining real-world situations like temperatures below freezing or bank account overdrafts.
They are the opposite of positive numbers and illustrate a deficit or lack. For instance, if you owe someone money, the amount you owe can be seen as a negative number on your balance sheet.
Understanding how negative numbers interact with operations is key. For example, when a negative number is multiplied or divided by another negative number, the result is positive. This is because two negative signs cancel each other out. Similarly, subtracting a negative number is the same as adding the corresponding positive number, because the two negatives negate each other. However, the absolute value of a negative number, such as \(-a\), will be positive, because it measures the distance from zero without considering direction.
They are the opposite of positive numbers and illustrate a deficit or lack. For instance, if you owe someone money, the amount you owe can be seen as a negative number on your balance sheet.
Understanding how negative numbers interact with operations is key. For example, when a negative number is multiplied or divided by another negative number, the result is positive. This is because two negative signs cancel each other out. Similarly, subtracting a negative number is the same as adding the corresponding positive number, because the two negatives negate each other. However, the absolute value of a negative number, such as \(-a\), will be positive, because it measures the distance from zero without considering direction.
Number Line
The number line is a straight, horizontal line that represents all real numbers. Zero is typically at the center of a number line, with positive numbers extending to the right and negative numbers extending to the left. The number line is a vital tool for visualizing numbers' sizes and their relationships.
When you move to the right on the number line, numbers become larger. Conversely, moving to the left decreases number values. This line provides a simple way to understand operations like addition and subtraction. By moving forward or backward, you can easily calculate sums and differences.
Additionally, the number line helps in understanding the concept of absolute value, which is the distance a number is from zero on the line. Regardless of direction, distances are always positive. For instance, both -3 and 3 have an absolute value of 3 because they are both three units away from zero.
When you move to the right on the number line, numbers become larger. Conversely, moving to the left decreases number values. This line provides a simple way to understand operations like addition and subtraction. By moving forward or backward, you can easily calculate sums and differences.
Additionally, the number line helps in understanding the concept of absolute value, which is the distance a number is from zero on the line. Regardless of direction, distances are always positive. For instance, both -3 and 3 have an absolute value of 3 because they are both three units away from zero.
Non-Negative Numbers
Non-negative numbers are numbers that are either positive or zero. They do not include any negative values. This category includes numbers like 0, 1, 2, and so forth.
Mathematically, non-negative numbers are represented by the expression \( x \geq 0 \). This means that the numbers are either equal to zero or larger. They are commonly used in scenarios where measuring deficits doesn't make sense or is not possible, such as the size of a group or amount of resources.
The concept of non-negative numbers is important in functions and real-world applications such as measuring heights, weights, and counting objects. Their characteristic is that they never decrease below zero, which aligns with our everyday perception of countable quantities. Understanding that absolute values are always non-negative is crucial because the absolute value of any number, positive or negative, results in a non-negative number.
Mathematically, non-negative numbers are represented by the expression \( x \geq 0 \). This means that the numbers are either equal to zero or larger. They are commonly used in scenarios where measuring deficits doesn't make sense or is not possible, such as the size of a group or amount of resources.
The concept of non-negative numbers is important in functions and real-world applications such as measuring heights, weights, and counting objects. Their characteristic is that they never decrease below zero, which aligns with our everyday perception of countable quantities. Understanding that absolute values are always non-negative is crucial because the absolute value of any number, positive or negative, results in a non-negative number.
