Absolute ValueMeaning, How to Calculate Absolute Value, Examples
A lot of people comprehend absolute value as the distance from zero to a number line. And that's not inaccurate, but it's by no means the entire story.
In mathematics, 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 look 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 number is at all times positive or zero (0). It is the magnitude of a real number irrespective to its sign. This refers that if you have a negative number, the absolute value of that figure is the number without the negative sign.
Definition of Absolute Value
The previous explanation refers that the absolute value is the distance of a figure from zero on a number line. So, if you think about that, the absolute value is the distance or length a figure has from zero. You can see it if you check out a real number line:
As you can see, the absolute value of a figure is the length of the figure is from zero on the number line. The absolute value of negative five is five due to the fact it is 5 units apart from zero on the number line.
Examples
If we plot -3 on a line, we can observe that it is three units apart from zero:
The absolute value of negative three is 3.
Presently, let's look at more absolute value example. Let's suppose we posses an absolute value of sin. We can plot this on a number line as well:
The absolute value of 6 is 6. Hence, what does this refer to? It shows us that absolute value is constantly positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Figure or Expression
You need to know a couple of things before working on how to do it. A few closely related characteristics will support you grasp how the expression within the absolute value symbol works. Luckily, here we have an meaning of the ensuing 4 fundamental features of absolute value.
Basic Properties 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 figure itself. Instead, the absolute value of a negative number is the non-negative value of that same number.
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 equal to the product of absolute values.
With above-mentioned four basic characteristics in mind, let's look at two more useful 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 difference among two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Now that we learned these properties, we can in the end initiate learning how to do it!
Steps to Find the Absolute Value of a Figure
You have to follow few steps to find the absolute value. These steps are:
Step 1: Write down the figure whose 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 applicable to the absolute value equations.
Step 5: The absolute value of the number is the figure you have after steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on both side of a expression or number, similar to this: |x|.
Example 1
To start out, let's assume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To work 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 given the equation |x+5| = 20, and we must find the absolute value within the equation to find x.
Step 2: By using the basic characteristics, we learn that the absolute value of the sum of these two expressions is the 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 get rid of 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 right.
Example 2
Now let's work on one more absolute value example. We'll use the absolute value function to solve a new equation, similar to |x*3| = 6. To do this, we again have to observe the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We are required to find the value of x, so we'll initiate 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: Therefore, the initial equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can include a lot of complicated values 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, this states it is varied at any given point. The ensuing formula provides 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 distance is all positive real numbers. The absolute value function rises 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 differentiable at 0 reason being the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is given by:
I'm →0−(|x|/x)
The right-hand limit is given by:
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 differentiable at zero (0).
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