As an expert in the field of mathematics, I'm here to provide a comprehensive explanation of the concept of absolute value. The absolute value of a number is a fundamental concept that is widely used in various branches of mathematics, including algebra, geometry, and calculus. It is a measure of the magnitude of a number without considering its sign. This means that the absolute value of a number is always a non-negative number.

Let's delve into the definition and properties of absolute values.

Definition:
The absolute value of a real number \( x \) is denoted by \( |x| \) and is defined as follows:
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0 \), then \( |x| = -x \).

In essence, the absolute value function removes any negative sign from a number, making it a positive or zero value.

Properties:

1. Non-Negativity: The absolute value of any real number is always non-negative. This is evident from the definition as it either takes the number itself (if it's non-negative) or its opposite (if it's negative).


2. Identity: The absolute value of zero is zero, i.e., \( |0| = 0 \).


3. Positivity: For any non-zero real number \( x \), \( |x| > 0 \).


4. Symmetry: The absolute value is symmetric with respect to the origin, meaning \( |x| = |-x| \).


5. Triangle Inequality: For any real numbers \( x \) and \( y \), \( |x + y| \leq |x| + |y| \).


6. Multiplication: For any real numbers \( x \) and \( y \), \( |xy| = |x||y| \).

7.
Addition: The absolute value of the sum of two numbers is not necessarily equal to the sum of their absolute values, i.e., \( |x + y| \neq |x| + |y| \) in general.

Now, let's apply these concepts to find the absolute value of 9.

Absolute Value of 9:
Given the definition of absolute value, since 9 is a positive number, its absolute value is the number itself. Therefore, \( |9| = 9 \). This is because 9 is 9 units away from 0 on the number line, and the absolute value function is designed to capture this distance without considering direction.

Absolute Value of -9:
Similarly, for the number -9, despite being negative, its absolute value is also 9. This is because the absolute value function disregards the negative sign and considers only the distance from zero. Hence, \( |-9| = 9 \) as well.

In conclusion, the absolute value of a number is a measure of its distance from zero on the number line, and it is always a non-negative number. It is a crucial concept that underlies many mathematical operations and theorems.

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Or, it's the number's distance from 0 on a number line. For example, the number 9 is 9 units away from 0. Therefore its absolute value is 9. The number -9 is the same distance from zero, so its absolute value is also just 9.read more >>