Absolute ValueMeaning, How to Discover Absolute Value, Examples
A lot of people comprehend absolute value as the distance from zero to a number line. And that's not wrong, but it's by no means the entire story.
In math, an absolute value is the magnitude of a real number without regard to its sign. So the absolute value is at all time a positive zero or number (0). Let's check at what absolute value is, how to discover absolute value, several examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure is constantly positive or zero (0). It is the magnitude of a real number without considering its sign. This refers that if you have a negative figure, the absolute value of that number is the number without the negative sign.
Meaning of Absolute Value
The prior definition refers that the absolute value is the distance of a number from zero on a number line. Therefore, if you think about that, the absolute value is the distance or length a number 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 length of the number is from zero on the number line. The absolute value of negative five is 5 because it is five units away from zero on the number line.
Examples
If we graph negative three on a line, we can see that it is three units apart from zero:
The absolute value of -3 is three.
Now, let's look at another absolute value example. Let's suppose we hold an absolute value of sin. We can plot this on a number line as well:
The absolute value of six is 6. Therefore, what does this mean? 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 Expression
You need to know a handful of things prior working on how to do it. A handful of closely linked features will help you grasp how the expression within the absolute value symbol works. Luckily, what we have here is an definition of the ensuing 4 rudimental properties of absolute value.
Essential 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 figure itself. Alternatively, the absolute value of a negative number is the non-negative value of that same expression.
Addition: The absolute value of a total is less than or equal to the sum of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned four fundamental properties in mind, let's check out two more beneficial characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is at all times positive or zero (0).
Triangle inequality: The absolute value of the variance within two real numbers is less than or equal to the absolute value of the total of their absolute values.
Considering that we learned these properties, we can finally initiate learning how to do it!
Steps to Discover the Absolute Value of a Number
You have to follow a couple of steps to calculate 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 expression is negative, multiply it by -1. This will change it to a positive number.
Step3: If the number is positive, do not alter it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the number is the number you have following steps 2, 3 or 4.
Bear in mind that the absolute value sign is two vertical bars on both side of a expression or number, similar to this: |x|.
Example 1
To set 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 have to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps above:
Step 1: We are given the equation |x+5| = 20, and we are required to calculate the absolute value within the equation to get x.
Step 2: By utilizing the fundamental characteristics, we understand that the absolute value of the total 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 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 try one more 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 observe the steps:
Step 1: We use 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 potential answers, x=2 and x=-2.
Absolute value can include many complicated expressions or rational numbers in mathematical settings; nevertheless, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, meaning it is differentiable at any given point. 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 length 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 uniform. The left-hand limit is given by:
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 distinctable at 0.
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