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 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 all the time a positive number or zero (0). Let's check at what absolute value is, how to calculate absolute value, few examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a number is always positive or zero (0). It is the extent of a real number without considering its sign. This signifies if you hold a negative number, the absolute value of that number is the number overlooking the negative sign.
Definition of Absolute Value
The last explanation states that the absolute value is the length of a number from zero on a number line. Therefore, 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 demonstrated, 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 five due to the fact it is five units apart from zero on the number line.
Examples
If we graph negative three on a line, we can watch that it is 3 units apart from zero:
The absolute value of -3 is three.
Presently, let's look at another absolute value example. Let's say we posses an absolute value of sin. We can graph this on a number line as well:
The absolute value of 6 is 6. Hence, what does this tell us? It tells us that absolute value is at all times positive, regardless if the number itself is negative.
How to Locate the Absolute Value of a Number or Figure
You should know a couple of things before working on how to do it. A handful of closely related characteristics will assist you understand how the expression within the absolute value symbol functions. Thankfully, here we have an meaning of the following 4 fundamental characteristics of absolute value.
Fundamental Characteristics of Absolute Values
Non-negativity: The absolute value of all real number is at all time 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 equal to the product of absolute values.
With above-mentioned four fundamental properties in mind, let's take a look at two other helpful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is always zero (0) or positive.
Triangle inequality: The absolute value of the variance between two real numbers is less than or equivalent to the absolute value of the total of their absolute values.
Considering that we know these properties, we can in the end start learning how to do it!
Steps to Find the Absolute Value of a Expression
You are required to obey a handful of steps to find the absolute value. These steps are:
Step 1: Jot down the figure whose absolute value you desire to discover.
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 properties applicable to the absolute value equations.
Step 5: The absolute value of the figure is the number you obtain following steps 2, 3 or 4.
Bear in mind that the absolute value symbol is two vertical bars on either side of a number or expression, like this: |x|.
Example 1
To start out, let's presume an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To work this out, we need to find 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 have to find the absolute value within the equation to get x.
Step 2: By utilizing the basic characteristics, we know that the absolute value of the sum of these two numbers is as same as 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 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 be equivalent 15, and the equation above is true.
Example 2
Now let's try one more absolute value example. We'll utilize the absolute value function to get a new equation, similar to |x*3| = 6. To do this, we again have to follow the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We are required to find the value of x, so we'll begin by dividing 3 from each side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two possible results: x = 2 and x = -2.
Step 4: Hence, the original equation |x*3| = 6 also has two likely answers, x=2 and x=-2.
Absolute value can involve several complex values 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 varied at any given point. The following formula provides 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 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 differentiable 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)
Considering 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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