Absolute ValueMeaning, How to Calculate Absolute Value, Examples
A lot of people think of absolute value as the distance from zero to a number line. And that's not inaccurate, but it's not the entire 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 number or zero (0). Let's observe at what absolute value is, how to find absolute value, several 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 irrespective to its sign. That means if you possess a negative figure, the absolute value of that figure is the number without the negative sign.
Meaning of Absolute Value
The last definition means that the absolute value is the distance of a number 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 take a look at a real number line:
As shown, the absolute value of a number is the distance of the number is from zero on the number line. The absolute value of negative five is five because it is 5 units apart from zero on the number line.
Examples
If we plot -3 on a line, we can see that it is 3 units away from zero:
The absolute value of -3 is three.
Presently, let's look at more absolute value example. Let's say we have 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 tell us? It states that absolute value is at all times positive, regardless if the number itself is negative.
How to Calculate the Absolute Value of a Figure or Expression
You should know a couple of points prior going into how to do it. A few closely related characteristics will support you comprehend how the number inside the absolute value symbol functions. Luckily, what we have here is an meaning of the following four essential properties 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 expression itself. Instead, 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 equal to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned 4 fundamental properties in mind, let's take a look at two more beneficial properties of the absolute value:
Positive definiteness: The absolute value of any real number is constantly zero (0) or positive.
Triangle inequality: The absolute value of the variance between two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Now that we know these characteristics, we can ultimately initiate learning how to do it!
Steps to Find the Absolute Value of a Figure
You are required to obey few steps to calculate the absolute value. These steps are:
Step 1: Write down the figure of whom’s absolute value you desire to find.
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 alter it.
Step 4: Apply all characteristics relevant to the absolute value equations.
Step 5: The absolute value of the figure is the expression you obtain following steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on both side of a figure or expression, like this: |x|.
Example 1
To set 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 figure this out, we are required to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned priorly:
Step 1: We are given the equation |x+5| = 20, and we must calculate the absolute value within the equation to find x.
Step 2: By utilizing the basic characteristics, we know 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 unknown, so let's remove the vertical bars: x+5 = 20
Step 4: Let's calculate for x: x = 20-5, x = 15
As we see, x equals 15, so its distance from zero will also be equivalent 15, and the equation above is right.
Example 2
Now let's work on one more absolute value example. We'll utilize the absolute value function to get a new equation, like |x*3| = 6. To get there, we again need to observe the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We have to calculate the value x, so we'll begin by dividing 3 from both 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: So, the first equation |x*3| = 6 also has two likely answers, x=2 and x=-2.
Absolute value can involve several complicated values or rational numbers in mathematical settings; nevertheless, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a constant function, meaning it is differentiable everywhere. The following formula offers 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 increases 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 distinguishable at 0 reason being 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 provided as:
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 0.
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