Absolute ValueDefinition, How to Find Absolute Value, Examples
A lot of people think of absolute value as the distance from zero to a number line. And that's not wrong, but it's nowhere chose to the entire story.
In math, an absolute value is the extent of a real number without regard to its sign. So the absolute value is at all time a positive number or zero (0). Let's check at what absolute value is, how to calculate absolute value, some examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a figure 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 figure is the number without the negative sign.
Meaning of Absolute Value
The previous explanation refers that the absolute value is the distance of a figure from zero on a number line. Hence, if you think about it, the absolute value is the distance or length a figure has from zero. You can visualize it if you take a look at a real number line:
As you can see, the absolute value of a figure is the distance of the number is from zero on the number line. The absolute value of negative five is five because it is five units away from zero on the number line.
Examples
If we plot -3 on a line, we can observe that it is 3 units away from zero:
The absolute value of -3 is 3.
Well then, let's look at another absolute value example. Let's suppose we have an absolute value of sin. We can plot this on a number line as well:
The absolute value of six is 6. Hence, what does this mean? It states that absolute value is constantly positive, even if the number itself is negative.
How to Calculate the Absolute Value of a Number or Figure
You should be aware of a couple of points prior going into how to do it. A handful of closely associated properties will help you understand how the expression within the absolute value symbol works. Luckily, here we have an definition of the following four essential properties of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of all 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 lower 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 properties of the absolute value:
Positive definiteness: The absolute value of any real number is at all times zero (0) or positive.
Triangle inequality: The absolute value of the variance within 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 finally initiate learning how to do it!
Steps to Discover 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 find.
Step 2: If the number is negative, multiply it by -1. This will make the number positive.
Step3: If the figure is positive, do not alter it.
Step 4: Apply all characteristics relevant to the absolute value equations.
Step 5: The absolute value of the expression is the expression you get after steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on either side of a figure or expression, similar to this: |x|.
Example 1
To set out, let's assume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To work this out, we need to calculate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned priorly:
Step 1: We are provided with the equation |x+5| = 20, and we must calculate the absolute value inside the equation to find x.
Step 2: By using the fundamental characteristics, we learn that the absolute value of the addition of these two numbers is the same as the total of each absolute value: |x|+|5| = 20
Step 3: The absolute value of 5 is 5, and the x is unidentified, 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 distance from zero will also be as same as 15, and the equation above is true.
Example 2
Now let's work on another absolute value example. We'll utilize the absolute value function to find a new equation, like |x*3| = 6. To make it, we again have to obey the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We are required to find the value of x, so we'll initiate by dividing 3 from each side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two potential results: x = 2 and x = -2.
Step 4: So, the first equation |x*3| = 6 also has two potential answers, x=2 and x=-2.
Absolute value can involve several complex figures or rational numbers in mathematical settings; however, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, meaning it is varied everywhere. The ensuing formula offers the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the domain is all real numbers except 0, and the length is all positive real numbers. The absolute value function rises for all x<0 and all x>0. The absolute value function is constant at 0, so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 because the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is provided as:
I'm →0−(|x|/x)
The right-hand limit is provided as:
I'm →0+(|x|/x)
Because the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not differentiable at 0.
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