Absolute ValueDefinition, How to Discover Absolute Value, Examples
Many think of absolute value as the length from zero to a number line. And that's not wrong, but it's not the complete story.
In mathematics, an absolute value is the magnitude of a real number without considering its sign. So the absolute value is all the time a positive number or zero (0). Let's check at what absolute value is, how to find absolute value, few examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a figure is at all times positive or zero (0). It is the magnitude of a real number without considering its sign. This refers that if you possess a negative number, the absolute value of that number is the number without the negative sign.
Meaning of Absolute Value
The previous explanation means that the absolute value is the length of a figure from zero on a number line. Hence, if you think about that, the absolute value is the distance or length a figure has from zero. You can see it if you take a look at a real number line:
As shown, the absolute value of a number is how far away the figure is from zero on the number line. The absolute value of -5 is 5 because it is 5 units away from zero on the number line.
Examples
If we plot -3 on a line, we can see that it is three units apart from zero:
The absolute value of negative three is 3.
Presently, let's look at one 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 six is 6. So, what does this refer to? It states that absolute value is constantly positive, regardless if the number itself is negative.
How to Calculate the Absolute Value of a Number or Expression
You should be aware of few points before going into how to do it. A few closely associated properties will help you comprehend how the expression inside the absolute value symbol works. Thankfully, what we have here is an explanation of the following four fundamental features of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is at all time 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 expression.
Addition: The absolute value of a sum is lower than or equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With these 4 essential characteristics in mind, let's look at two other helpful 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 among two real numbers is lower than or equivalent to the absolute value of the sum of their absolute values.
Considering that we know these characteristics, we can in the end begin learning how to do it!
Steps to Calculate the Absolute Value of a Figure
You have to follow a handful of steps to find the absolute value. These steps are:
Step 1: Jot down the number of whom’s absolute value you desire to calculate.
Step 2: If the number is negative, multiply it by -1. This will change it to a positive number.
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 number is the expression you get after steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on both side of a figure or number, like this: |x|.
Example 1
To set out, let's presume an absolute value equation, like |x + 5| = 20. As we can see, 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 observing the steps mentioned above:
Step 1: We are provided with the equation |x+5| = 20, and we are required to calculate the absolute value within the equation to solve x.
Step 2: By utilizing the essential characteristics, we know that the absolute value of the addition of these two numbers 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 unidentified, so let's eliminate 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 equivalent 15, and the equation above is genuine.
Example 2
Now let's work on another absolute value example. We'll utilize the absolute value function to get a new equation, such as |x*3| = 6. To make it, we again have to observe the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We need to find the value of x, so we'll start by dividing 3 from each side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two possible answers: x = 2 and x = -2.
Step 4: Therefore, the first equation |x*3| = 6 also has two possible answers, x=2 and x=-2.
Absolute value can contain many complex values or rational numbers in mathematical settings; nevertheless, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this refers it is differentiable everywhere. The following formula gives 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 distance is all positive real numbers. The absolute value function increases 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 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 distinguishable at 0.
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