Absolute ValueDefinition, How to Calculate Absolute Value, Examples
Many comprehend absolute value as the length from zero to a number line. And that's not incorrect, but it's not the complete story.
In mathematics, an absolute value is the magnitude of a real number without regard to its sign. So the absolute value is always a positive number or zero (0). Let's check at what absolute value is, how to find absolute value, some examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a number is at all times positive or zero (0). It is the magnitude of a real number without regard to its sign. This refers that if you have a negative number, the absolute value of that number is the number without the negative sign.
Definition of Absolute Value
The previous definition refers that the absolute value is the length of a figure from zero on a number line. Therefore, if you think about it, the absolute value is the distance or length a figure has from zero. You can observe it if you check out a real number line:
As demonstrated, the absolute value of a number is how far away the number is from zero on the number line. The absolute value of negative five is 5 because it is 5 units away from zero on the number line.
Examples
If we plot -3 on a line, we can observe that it is three units apart from zero:
The absolute value of -3 is three.
Well then, let's look at one more absolute value example. Let's say we have an absolute value of sin. We can graph this on a number line as well:
The absolute value of 6 is 6. Therefore, what does this refer to? It shows us that absolute value is constantly positive, even though the number itself is negative.
How to Locate the Absolute Value of a Number or Figure
You should know few points prior going into how to do it. A couple of closely associated characteristics will support you understand how the figure inside the absolute value symbol functions. Fortunately, here we have an explanation of the ensuing four fundamental properties of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of ever real number is always zero (0) or positive.
Identity: The absolute value of a positive number is the figure itself. Otherwise, the absolute value of a negative number is the non-negative value of that same expression.
Addition: The absolute value of a sum is less than or equal to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these 4 essential properties in mind, let's check out two other helpful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is always zero (0) or positive.
Triangle inequality: The absolute value of the difference between two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Now that we know these characteristics, we can ultimately begin learning how to do it!
Steps to Find the Absolute Value of a Number
You are required to obey a handful of steps to calculate the absolute value. These steps are:
Step 1: Note down the number whose absolute value you desire to calculate.
Step 2: If the figure is negative, multiply it by -1. This will make the number positive.
Step3: If the expression is positive, do not change it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the number is the number you have subsequently steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on both side of a expression or number, similar to this: |x|.
Example 1
To start out, let's consider an absolute value equation, like |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we have to find the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned priorly:
Step 1: We are provided with the equation |x+5| = 20, and we must discover the absolute value within the equation to find x.
Step 2: By using the essential properties, we understand that the absolute value of the addition of these two figures is equivalent to the sum of each absolute value: |x|+|5| = 20
Step 3: The absolute value of 5 is 5, and the x is unknown, so let's remove the vertical bars: x+5 = 20
Step 4: Let's solve for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's check out one more absolute value example. We'll use the absolute value function to find a new equation, similar to |x*3| = 6. To make it, we again need to follow the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We need to find the value of x, so we'll start by dividing 3 from both side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two possible answers: x = 2 and x = -2.
Step 4: Hence, the original equation |x*3| = 6 also has two likely results, x=2 and x=-2.
Absolute value can involve several complex expressions or rational numbers in mathematical settings; still, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, meaning it is distinguishable everywhere. The following formula provides the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area is all real numbers except zero (0), and the length is all positive real numbers. The absolute value function increases for all x<0 and all x>0. The absolute value function is constant at zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinguishable at 0 due to the the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is stated as:
I'm →0−(|x|/x)
The right-hand limit is offered as:
I'm →0+(|x|/x)
Considering the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not differentiable at zero (0).
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