Problem 3
Question
Determine each a bsolute value. $$ |12| $$
Step-by-Step Solution
Verified Answer
The absolute value of 12 is 12.
1Step 1: Understand Absolute Value
The absolute value of a number is its distance from 0 on the number line, regardless of direction. For any real number \( x \), the absolute value is denoted as \( |x| \) and is always non-negative.
2Step 2: Evaluate the Absolute Value
Since 12 is a positive number, the absolute value is the number itself. Thus, \(|12| = 12\).
3Step 3: Verify the Absolute Value
Check your result by considering that the number 12 is 12 units away from 0 on the number line, confirming that the absolute value is correct.
Key Concepts
Understanding the Number LineExploring Real NumbersThe Concept of Non-Negative Values
Understanding the Number Line
Imagine a simple line marked at regular intervals. This is a number line. It represents all real numbers and extends infinitely in both directions. The center of the number line is zero, where positive numbers lie to its right and negative numbers to its left.
The number line helps us visualize the concept of absolute value effectively. When you look at the number line, think of each point as a distance from zero. This is vital because it means absolute value is all about distance. And distance, by its nature, cannot be negative. So when you hear "absolute value," it's like saying "how far away is this number from zero," no matter in which direction it lies.
The number line helps us visualize the concept of absolute value effectively. When you look at the number line, think of each point as a distance from zero. This is vital because it means absolute value is all about distance. And distance, by its nature, cannot be negative. So when you hear "absolute value," it's like saying "how far away is this number from zero," no matter in which direction it lies.
- Zero is the central reference point.
- Positive numbers are to the right of zero.
- Negative numbers are to the left of zero.
Exploring Real Numbers
Real numbers include all the numbers you can think of: whole numbers, fractions, decimals, positive numbers, and negatives. They comprise both rational numbers, like \frac{1}{2}, and irrational numbers, such as \pi.
Real numbers fill up every possible space on the number line. This means between any two real numbers there's always another real number—it extends infinitely. When you're calculating absolute value, you're working within the realm of real numbers. The notation \(|x|\) indicates the absolute value of \(x\), and it’s a way to find out how far \(x\) is from zero on this grand number line, ignoring any negative signs.
Real numbers fill up every possible space on the number line. This means between any two real numbers there's always another real number—it extends infinitely. When you're calculating absolute value, you're working within the realm of real numbers. The notation \(|x|\) indicates the absolute value of \(x\), and it’s a way to find out how far \(x\) is from zero on this grand number line, ignoring any negative signs.
- Rational numbers can be fractions or whole numbers.
- Irrational numbers cannot be expressed as simple fractions.
- Real numbers cover every point on the number line.
The Concept of Non-Negative Values
Whenever you calculate the absolute value, you are ensuring the result is a non-negative number. This means your answer will be zero or greater, never less. It’s essential because in mathematics, taking the magnitude, or size, of something shouldn't involve direction.
Whether you start with a positive or negative number, the distance or absolute value from zero does not change. For instance, both \-3\ and \3\ have an absolute value of \3\, as both are three spaces away from zero on the number line. The purpose of absolute value is to negate any negativity, focusing purely on magnitude.
Whether you start with a positive or negative number, the distance or absolute value from zero does not change. For instance, both \-3\ and \3\ have an absolute value of \3\, as both are three spaces away from zero on the number line. The purpose of absolute value is to negate any negativity, focusing purely on magnitude.
- Absolute values are always non-negative.
- Zero is its own absolute value, i.e., \(|0|=0\).
- Removing the direction leaves only the size or magnitude.