The absolute value of a number refers to the distance of a number from the origin of a number line.It is represented as |a|, which defines the magnitude of any integer ‘a’. The absolute value of any integer, whether positive or negative, will be the real numbers, regardless of which sign it has.It is represented by two vertical lines |a|, which is known as the modulus of a.
For example: 5 is the absolute value for both 5 and -5.
|-5| = +5 and |+ 5| = +5
In this article, we will learn what is the absolute value of a number, symbol, examples, absolute value in number line, absolute value of real numbers, absolute value of complex numbers in detail.
- Definition
- Symbol
- Examples
- Absolute value function
- Properties
- Absolute value of real number
- Absolute value of complex number
- Absolute value in Number line
- Solved Problems
- Practice Questions
- FAQs
What is the Absolute Value of a Number?
The absolute value of a number or integer is the actual distance of the integer from zero, in a number line. Therefore, the absolute value is always a positive value and not a negative number.
We can define the absolute values like the following:
{ a if a ≥ 0 }
|a| = { -a if a < 0 }
Note: There is no absolute value for 0 because the absolute value changes the sign of the numbers into positive and zero has no sign.
If the number is positive then it will result in a positive number only. And if the number is negative, then the modulus of this number will also be a positive number. It is denoted as |n|, where n is an integer.
Absolute Value Symbol
The symbol of absolute value is represented by the modulus symbol, ‘| |’, with the numbers between it. For example, the absolute value of 9 is denoted as |9|.
The distance of any number from the origin on the number line is the absolute value of that number. It also shows the polarity of the number whether it is positive or negative. It can be negative ever a s it shows the distance and the distance can’t be negative. So, it is always positive.
For example: ‘|a|’, where “a” is the number whose absolute value has to be determined.
Absolute value of a Number Examples
Let us see some examples of the absolute value of a number.
- |-1| = 1
- |-14| = 14
- |1| = 1
- |0| = 0
- |7| = 7
- |7-2| = |5| = 5
- |2+3| = |5| = 5
- |-3×5| = |-15| = 15
Related Articles:
- Absolute Value formula
- Integers
- Addition and Subtraction Integers
Absolute Value Function
The absolute value function is given by f(x) = |x| such that:
- |x| = +x for x > 0
- |x| = -x for x < 0
Absolute Value Properties
If x and y are real numbers and then the absolute values are satisfying the following properties:
| Property | Expression |
| Non-negativity | | x | ≥ 0 |
| Positive-definiteness | | x | = 0 ↔ a = 0 |
| Multiplicativeness | | x × y| = |x| × |y| |
| Subadditivity | | x + y| ≤ | x | + | y | |
| Symmetry | |-x| = |x| |
| Identity of indiscernible (equivalent to positive definiteness) | | x – y | = 0 ↔ a = b |
| Triangle inequality (equivalent to subadditivity) | | x – y | ≤ | x – z | + | z – x | |
| Preservation of division (equivalent to multiplicativeness) | | x / y| = | x | / | y | |
| Equivalent to subadditivity | | x – y | ≥ | | x | – | y | | |
Absolute Value of a Real Number
If a real number x, the absolute value will satisfy the following conditions.
| x | = x, if x ≥ 0
| x | = – x, if x < 0
Let’s look at the absolute value of 2 in the number line given below. Here, |2| is the distance of 2 from 0(zero). So, both +2 and -2 is the distance of 2 from the origin. But it would be taken as 2 because distance is never measured in negative.
Absolute Value of Complex Number
Complex numbers consist of real numbers and imaginary numbers. Hence, unlike integers, it is difficult to find the absolute value for them. Suppose, x+iy is the given complex number.
z = x+iy
The absolute value of z will be;
|z|= √[Re(z)2+Im(z)2]
|z| =√(x2+y2)
Where x and y are the real numbers.
Absolute Value in Number Line
The graph of absolute values is called the absolute value graph. As we know the absolute value of any real number is positive, so the absolute value of any number or function graph will lie on the positive side only.
Example: Graph the absolute value of the number -9.
Solution:Absolute value of |-9| is +9.
So the graph for the absolute value of -9 will look like following
Problems and Solutions
Let’s understand this topic better with the help of examples.
Question 1:
Arrange the given numbers in ascending order.
-|-14|, |12|, |7|, |-91|, |-5|, |-8|, |-65|, |6|
Solution:
Initially, find the absolute values of the given numbers
-14, 12, 7, 91, 5, 8, 65, 6
Now, arrange the numbers in ascending order (smallest to the largest number)
-14, 5, 6, 7, 8, 12, 65, 91
Question 2:
Arrange the given numbers in ascending order.
-|-24|, |21|, 17|, |-109|, |-15|, |-19|, |-75|, |16|
Solution:
Find the absolute values of the given numbers
-24, 21, 17, 109, 15, 19, 75, 16
Now, arrange the numbers in ascending order (smallest to the largest number)
-24, 15, 16, 17,19, 21, 75, 109.
Example 3:
Find the absolute value of a number -12/5
Solution:
To find: |-12/5|
|-12/5| = 12/5
Hence, the absolute value of a number -12/5 is 12/5
Example 4:
Find the absolute value for the following numbers:
(a) |-1/2|
(b) |72|
(c) |3/4|
Solution:
The absolute value of the numbers are:
(a) |-1/2| = 1/2
(b) |72| = 72
Practice Questions
1. What is the absolute value of -13?
2. Find the absolute value of 100.
3. Arrange the following in ascending order.
|3|, |-9|, |1|, |-2|, |-5|
4. Plot the absolute value of -7 on the number line.
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Frequently Asked Questions on Absolute Values
Q1
What is an absolute value of a number?
The absolute value of a number represents the distance of the number from zero in a number line.
Q2
What is the absolute value of 4?
The absolute value of 4 is 4. |4| = 4
Q3
What is the absolute value of -12?
The absolute value of -12 is 12. Since the distance between -12 and 0 in a number line is 12.
Q4
What is the symbol for absolute value?
The symbol to represent absolute value is |x|, where x is an integer.
Q5
What is the absolute value of -7?
The absolute value of -7 is 7. It is represented by |-7| = 7.
Q6
Is absolute value negative?
Absolute value of any integer is always positive and never negative.
FAQs
Absolute Value of a Number - Definition, Symbol and Examples? ›
Definitions: The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.
What is the symbol for absolute value of a number? ›The most common way to represent the absolute value of a number or expression is to surround it with the absolute value symbol: two vertical straight lines. |6| = 6 means “the absolute value of 6 is 6.” |–6| = 6 means “the absolute value of –6 is 6.”
What is the absolute value of |- 2 |? *? ›For example, |-2| = 2. Irrespective of the sign of the numeric value, the absolute value is always non-negative.
What is the absolute value of a number on a number line example? ›The absolute value of a number is its distance from 0 on the number line. The absolute value of -7 is 7, because it is 7 units away from 0. The absolute value of 5 is 5, because it is 5 units away from 0.
What is the definition of absolute value of a number quizlet? ›The definition of absolute value is "the distance of x from zero on the number line". Since distance cannot be negative, this means that an absolute value is always non-negative ≥0.
What is an example of an absolute value? ›The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.
What is the absolute value of 2? ›For example,|2| represents the absolute value of 2.
What's the absolute value of 3? ›For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
What is the absolute value of 29? ›The absolute value of any positive number is the number itself, so 29 has 29 as an absolute value.
What is the absolute value of 14? ›14 is a positive number, so the absolute value of 14 is simply 14.
What is the absolute value of |- 9 |? A? ›
The absolute value of a number tells only one thing: 1. distance from 0. The absolute value of 9 is 9.
What is the absolute value of |- 1? ›FAQ. What is an absolute value? An absolute value is the positive value of any number. For example, the absolute value of -1 is 1.
What is the absolute value of |- 4? ›Therefore, the absolute value of -4 is 4.
How do you find the definition of absolute value? ›The absolute value of the number is defined as its distance from the origin. For example, to find the absolute value of 7, locate 7 on the real line and then find its distance from the origin. To find the absolute value of −7, locate −7 on the real line and then find its distance from the origin.
What is absolute mean value? ›Absolute Values. The absolute value of a number refers to the distance of a number from the origin of a number line.
What is the meaning of absolute value function and example? ›The absolute value function is defined as an algebraic expression in absolute bar symbols. Such functions are commonly used to find distance between two points. Some of the examples of absolute value functions are: f(x) = |x|
What is the absolute value 8th grade? ›An absolute value is the numerical distance from zero and can be used in equations and graphed as dilations or reflections.
What is the absolute value of 6? ›The absolute value of a number is its distance from 0 on a number line. Let's use the number 6 as an example. Using a number line, we can see that 6 is six units away from 0. Therefore, the absolute value of 6 is 6.
What is the absolute value of 25? ›The distance from 25 to 0 is 5 units . That means the absolute value of 25 is 5 .
What is the absolute value of 50? ›So we would say the absolute value of negative 50 is 50, and the absolute value of five is five.
What is the absolute value of 10? ›
Answer and Explanation: The absolute value of 10 is 10.
What is the absolute value of 13? ›The absolute value of 13 is 13.
What is the absolute value of 24? ›The number line also shows that |24| is the distance between 24 and 0 is 4. Thus, |24| 5 4. The absolute value of a positive number or zero is the number itself. The absolute value of a negative number is its opposite.
What is the absolute value of -(- 11? ›The distance for −11 will also equal 11 (because |−11 − 0| = |−11| = 11), and the absolute value of −11 is 11. Thus, the absolute value of any real number is equal to the absolute value of its distance from 0 on the number line.
What is the absolute value of [- 21? ›Therefore, the absolute value of -21 is 21.
What numbers have an absolute value of 7? ›The absolute value of 7 is 7. In fact, the absolute value of any positive number is equal to that number.
What is the absolute value of 16 answer? ›Hence, the answer is 16.
What is the absolute value of 12 answer? ›It's easy to understand that the absolute value of 12 and -12 are identical: they're both 12, since each number is 12 units away from zero. Absolute value is always positive — to get the absolute value of a negative number, just take away the minus sign.
What is the absolute value of 23? ›The absolute value of a number is the distance of that number from zero. The absolute value of 23 is 23 because it is 23 units from zero.
What's the absolute value of 18? ›1 Answer. The absolute value of |−18|=18 .
What is the absolute value of 36? ›
The absolute value of the number is represented by neglecting the negative sign. Hence, for negative numbers, the absolute value is the magnitude of the number without the sign. Hence, the absolute value of -36 is 36.
What is the absolute value of 15? ›Algebra Examples
The absolute value is the distance between a number and zero. The distance between −15 and 0 is 15 .
22 is 22 units from zero on the number line. This means that the absolute value of 22 is 22. Notice that our sign didn't change. The absolute value of a number will always be positive.
What is the absolute value of 17? ›The absolute value of 17 is "17" because the absolute value of 17 is seventeen units away from zero on the number line. In another way, |17| indicates that the absolute value of 17 is 17.
What is 100 in absolute value? ›The absolute value of the number 100 is 100, in mathematical notation |100| = 100. The absolute value of zero is zero, |0| = 0, the absolute value of −6.3 is 6.3, | − 6.3| = 6.3.
What has an absolute value of 9? ›The absolute value of a number is its distance from 0 on a number line. It is the magnitude of that number without considering its sign. The absolute value of 9 is written as |9| and is equal to 9. Thus, the absolute value of 9 = |9| = 9.
How do you write an absolute value equation? ›More generally, the form of the equation for an absolute value function is y=a| x−h |+k.
What is an example of a value in math? ›In math, value is a number signifying the result of a calculation or function. So, in the example above, you could tell your teacher that the value of 5 x 6 is 30 or the value of x + y if x = 6 and y = 3 is 9. Value can also refer to a variable or constant.
Why do we use absolute value? ›We often use absolute values when talking about distance because there is no such thing as negative distance. For example, just because the store you're going to is a block behind you doesn't mean it's negative one blocks away.
Can an absolute value be negative? ›Since the absolute value of any number other than zero is positive, it is not permissible to set an absolute value expression equal to a negative number. So, if your absolute value expression is set equal to a negative number, then you will have no solution.
How do you find the value of a number? ›
The value refers to the worth of each digit depending on where it lies in the number. We calculate it by multiplying the place value and face value of the digit.
What is the absolute value of |- 3? ›Negative 3 is three units away from zero, which means the absolute value of -3 is 3.
What is the absolute value of 4? ›The absolute value of 4 is 4 and –3 is 3. Subtract the smaller number from the larger and you get 4 – 3 = 1. The larger absolute value in the equation was 4 or a positive number so you give the result a positive result.
What is the absolute value of 9? ›The absolute value of a number is its distance from 0 on a number line. It is the magnitude of that number without considering its sign. The absolute value of 9 is written as |9| and is equal to 9. Thus, the absolute value of 9 = |9| = 9.
What is the absolute value of 12? ›It's easy to understand that the absolute value of 12 and -12 are identical: they're both 12, since each number is 12 units away from zero. Absolute value is always positive — to get the absolute value of a negative number, just take away the minus sign.
What is the absolute value of 11? ›The absolute value of any positive number is the number itself, so 11 has 11 as an absolute value.
What's the absolute value of 7? ›The absolute value of 7 is 7. In fact, the absolute value of any positive number is equal to that number.
What's the absolute value of 8? ›So, the absolute value of is, 8 = 8.
What is the absolute value of -(- 6? ›Explanation: The absolute value is defined as the distance from a number to 0 , and so it is always positive. So, the absolute value of −6 will be the distance from −6 to 0 on the number line, and it is 6 .
What is 42 absolute value? ›The absolute value is the distance between a number and zero. The distance between −42 and 0 is 42 .