Absolute ValueDefinition, How to Calculate Absolute Value, Examples
A lot of people perceive absolute value as the distance from zero to a number line. And that's not incorrect, but it's by no means the entire story.
In mathematics, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is at all time a positive number or zero (0). Let's look at what absolute value is, how to discover absolute value, some examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a number is constantly positive or zero (0). It is the magnitude of a real number irrespective to its sign. That means if you possess a negative number, the absolute value of that number is the number without the negative sign.
Definition of Absolute Value
The previous definition states that the absolute value is the length of a number from zero on a number line. Hence, if you consider it, the absolute value is the length or distance a figure has from zero. You can observe it if you take a look at a real number line:
As shown, the absolute value of a number is the length of the number is from zero on the number line. The absolute value of negative five is five reason being it is 5 units away from zero on the number line.
Examples
If we plot negative three on a line, we can see that it is 3 units away from zero:
The absolute value of -3 is three.
Well then, let's look at more absolute value example. Let's assume we hold an absolute value of sin. We can graph this on a number line as well:
The absolute value of six is 6. So, what does this refer to? It shows us that absolute value is constantly positive, regardless if the number itself is negative.
How to Calculate the Absolute Value of a Number or Figure
You should know a handful of things before going into how to do it. A couple of closely associated properties will assist you understand how the expression inside the absolute value symbol functions. Luckily, here we have an definition of the following 4 essential characteristics of absolute value.
Basic Properties of Absolute Values
Non-negativity: The absolute value of ever real number is always positive or zero (0).
Identity: The absolute value of a positive number is the expression itself. Instead, the absolute value of a negative number is the non-negative value of that same number.
Addition: The absolute value of a total is less than or equivalent to the sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned four fundamental characteristics in mind, let's look at two other useful properties 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 less than or equal to the absolute value of the sum of their absolute values.
Now that we know these properties, we can finally initiate learning how to do it!
Steps to Calculate the Absolute Value of a Number
You have to follow a couple of steps to calculate the absolute value. These steps are:
Step 1: Write down the expression whose absolute value you desire to find.
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 alter it.
Step 4: Apply all properties applicable to the absolute value equations.
Step 5: The absolute value of the figure is the number you get after steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on either side of a expression or number, like this: |x|.
Example 1
To set out, let's presume an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To figure this out, we have to locate the absolute value of the two numbers in the inequality. We can do this by following the steps above:
Step 1: We are given the equation |x+5| = 20, and we have to discover the absolute value within the equation to get x.
Step 2: By using the essential characteristics, we learn that the absolute value of the total of these two figures is as same as the sum of each absolute value: |x|+|5| = 20
Step 3: The absolute value of 5 is 5, and the x is unidentified, so let's get rid of the vertical bars: x+5 = 20
Step 4: Let's calculate for x: x = 20-5, x = 15
As we see, x equals 15, so its length from zero will also equal 15, and the equation above is true.
Example 2
Now let's check out one more absolute value example. We'll utilize the absolute value function to get a new equation, similar to |x*3| = 6. To get there, we again need to observe the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We are required to solve for x, so we'll begin by dividing 3 from each 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: Therefore, the original equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.
Absolute value can include a lot of intricate expressions or rational numbers in mathematical settings; still, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this states it is differentiable everywhere. The following formula offers the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the domain is all real numbers except zero (0), and the range is all positive real numbers. The absolute value function rises for all x<0 and all x>0. The absolute value function is consistent at zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinguishable at 0 reason being the left-hand limit and the right-hand limit are not equal. The left-hand limit is provided as:
I'm →0−(|x|/x)
The right-hand limit is offered as:
I'm →0+(|x|/x)
Since the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not differentiable at 0.
Grade Potential Can Guide You with Absolute Value
If the absolute value looks like a difficult topic, or if you're having problem with math, Grade Potential can help. We provide face-to-face tutoring by professional and qualified teachers. They can guide you with absolute value, derivatives, and any other concepts that are confusing you.
Contact us today to know more with regard to how we can help you succeed.
