Absolute ValueDefinition, How to Calculate Absolute Value, Examples
A lot of people comprehend absolute value as the length from zero to a number line. And that's not inaccurate, but it's not the whole story.
In mathematics, an absolute value is the extent of a real number without considering its sign. So the absolute value is at all time a positive zero or number (0). Let's look at what absolute value is, how to find absolute value, few examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a figure is at all times positive or zero (0). It is the extent of a real number without regard to its sign. This signifies if you hold a negative figure, the absolute value of that number is the number disregarding 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. So, if you think about it, the absolute value is the distance or length a figure has from zero. You can visualize it if you look at a real number line:
As you can see, the absolute value of a figure is how far away 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 see that it is three units away from zero:
The absolute value of negative three is three.
Presently, let's look at more absolute value example. Let's say we hold an absolute value of sin. We can plot this on a number line as well:
The absolute value of 6 is 6. So, what does this tell us? It shows us that absolute value is always positive, even if the number itself is negative.
How to Calculate the Absolute Value of a Expression or Figure
You should know a handful of things prior going into how to do it. A few closely linked features will assist you comprehend how the figure inside the absolute value symbol functions. Thankfully, what we have here is an explanation of the ensuing 4 fundamental features 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. Instead, the absolute value of a negative number is the non-negative value of that same expression.
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 equal to the product of absolute values.
With above-mentioned four fundamental properties in mind, let's take a look at two more useful 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 variance within two real numbers is less than or equal to the absolute value of the sum of their absolute values.
Taking into account that we went through these characteristics, we can ultimately initiate learning how to do it!
Steps to Calculate the Absolute Value of a Figure
You have to follow few steps to find the absolute value. These steps are:
Step 1: Write down the number of whom’s absolute value you want to calculate.
Step 2: If the number is negative, multiply it by -1. This will change it to a positive number.
Step3: If the figure is positive, do not change it.
Step 4: Apply all properties 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.
Remember that the absolute value symbol is two vertical bars on both side of a figure or number, like this: |x|.
Example 1
To begin with, let's consider an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we have to locate the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:
Step 1: We are given the equation |x+5| = 20, and we must find the absolute value within the equation to get x.
Step 2: By utilizing the fundamental characteristics, we know that the absolute value of the sum of these two figures 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 get rid of 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 distance from zero will also be as same as 15, and the equation above is right.
Example 2
Now let's work on another absolute value example. We'll utilize the absolute value function to find a new equation, such as |x*3| = 6. To make it, we again need to follow the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We have to find the value of x, so we'll begin by dividing 3 from both side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two possible results: x = 2 and x = -2.
Step 4: Hence, the first equation |x*3| = 6 also has two possible 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, this refers it is distinguishable at any given point. 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 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 zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not differentiable at 0 reason being the left-hand limit and the right-hand limit are not equal. The left-hand limit is stated 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 distinctable at zero (0).
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