Absolute ValueMeaning, How to Calculate Absolute Value, Examples
Many perceive absolute value as the length from zero to a number line. And that's not wrong, but it's by no means the complete story.
In mathematics, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is all the time a positive zero or number (0). Let's look at what absolute value is, how to find absolute value, some examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure is always zero (0) or positive. It is the extent of a real number irrespective to its sign. This refers that if you hold a negative number, the absolute value of that number is the number without the negative sign.
Meaning of Absolute Value
The last explanation states that the absolute value is the length of a figure from zero on a number line. Therefore, if you consider it, the absolute value is the length or distance a number 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 how far away the number is from zero on the number line. The absolute value of -5 is 5 reason being it is five units away from zero on the number line.
Examples
If we plot -3 on a line, we can watch that it is 3 units apart from zero:
The absolute value of negative three is three.
Presently, let's look at another absolute value example. Let's suppose we posses an absolute value of sin. We can graph this on a number line as well:
The absolute value of six is 6. So, what does this mean? It shows 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 prior going into how to do it. A couple of closely related properties will assist you grasp how the figure within the absolute value symbol functions. Luckily, here we have an explanation of the ensuing 4 fundamental properties of absolute value.
Essential Properties of Absolute Values
Non-negativity: The absolute value of any real number is constantly positive or zero (0).
Identity: The absolute value of a positive number is the figure itself. Otherwise, the absolute value of a negative number is the non-negative value of that same figure.
Addition: The absolute value of a sum is less than or equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned four essential characteristics in mind, let's take a look at two other beneficial characteristics 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 among two real numbers is less than or equivalent to the absolute value of the sum of their absolute values.
Taking into account that we know these properties, we can ultimately start learning how to do it!
Steps to Find the Absolute Value of a Expression
You are required to observe a handful of steps to find the absolute value. These steps are:
Step 1: Write 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 make the number positive.
Step3: If the figure is positive, do not convert it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the figure is the number 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 expression, similar to this: |x|.
Example 1
To begin with, let's presume an absolute value equation, like |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To work this out, we have to calculate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned above:
Step 1: We are given the equation |x+5| = 20, and we must discover the absolute value inside the equation to solve x.
Step 2: By using the essential properties, we know that the absolute value of the sum of these two figures is the 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 see, x equals 15, so its length from zero will also be as same as 15, and the equation above is right.
Example 2
Now let's try another absolute value example. We'll use the absolute value function to get a new equation, similar to |x*3| = 6. To do this, we again have to obey the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We have to calculate the value x, so we'll initiate by dividing 3 from both side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two potential answers: x = 2 and x = -2.
Step 4: So, the original equation |x*3| = 6 also has two potential solutions, x=2 and x=-2.
Absolute value can include many complex numbers or rational numbers in mathematical settings; however, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, meaning it is distinguishable at any given point. The ensuing formula offers the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the domain is all real numbers except zero (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 constant at 0, so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 because the left-hand limit and the right-hand limit are not uniform. The left-hand limit is stated as:
I'm →0−(|x|/x)
The right-hand limit is provided 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 0.
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