Absolute ValueDefinition, How to Find Absolute Value, Examples
Many think of absolute value as the distance from zero to a number line. And that's not inaccurate, but it's by no means the whole story.
In mathematics, an absolute value is the extent 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 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 zero (0) or positive. It is the magnitude of a real number without considering its sign. This signifies if you have a negative figure, the absolute value of that number is the number disregarding the negative sign.
Definition of Absolute Value
The previous explanation means that the absolute value is the distance of a figure from zero on a number line. So, if you consider it, the absolute value is the distance or length a figure has from zero. You can see it if you look at a real number line:
As you can see, the absolute value of a figure is the length of the figure is from zero on the number line. The absolute value of -5 is 5 reason being it is 5 units apart 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 negative three is 3.
Now, let's look at one more absolute value example. Let's suppose we posses an absolute value of sin. We can plot this on a number line as well:
The absolute value of six is 6. So, what does this mean? It shows us that absolute value is always positive, even if the number itself is negative.
How to Calculate the Absolute Value of a Figure or Expression
You should be aware of a couple of things before going into how to do it. A few closely associated features will support you grasp how the expression within the absolute value symbol functions. Thankfully, what we have here is an definition of the ensuing four essential characteristics of absolute value.
Basic Properties of Absolute Values
Non-negativity: The absolute value of ever real number is constantly zero (0) or positive.
Identity: The absolute value of a positive number is the number itself. Otherwise, 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 equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these 4 fundamental properties in mind, let's check out two other beneficial characteristics 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 difference among two real numbers is less than or equivalent to the absolute value of the total of their absolute values.
Considering that we went through these characteristics, we can finally start learning how to do it!
Steps to Find the Absolute Value of a Figure
You are required to observe a couple of steps to find the absolute value. These steps are:
Step 1: Jot down the number of whom’s absolute value you want to calculate.
Step 2: If the expression is negative, multiply it by -1. This will convert the number to positive.
Step3: If the figure is positive, do not change it.
Step 4: Apply all characteristics significant to the absolute value equations.
Step 5: The absolute value of the figure is the number you have following steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on either side of a figure or number, like this: |x|.
Example 1
To start out, let's assume an absolute value equation, like |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To solve this, we are required 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 have the equation |x+5| = 20, and we must calculate the absolute value inside the equation to find x.
Step 2: By using the basic characteristics, we learn that the absolute value of the addition of these two expressions is as 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 calculate for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length from zero will also equal 15, and the equation above is true.
Example 2
Now let's try another absolute value example. We'll use the absolute value function to find a new equation, similar to |x*3| = 6. To get there, we again need to obey the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We have to calculate the value 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 potential solutions: x = 2 and x = -2.
Step 4: So, the initial equation |x*3| = 6 also has two possible answers, x=2 and x=-2.
Absolute value can contain a lot of complex numbers 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 continuous function, this states it is varied everywhere. The ensuing formula gives the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area is all real numbers except 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 0, so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 due to the 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 given by:
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 distinguishable at zero (0).
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