Can the Absolute Value of a Number Be Negative

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 whatsoever integer 'a'. The absolute value of any integer, whether positive or negative, volition exist the real numbers, regardless of which sign it has.  It is represented by two vertical lines |a|, which is known equally the modulus of a.

For example: five is the accented value for both 5 and -v.

|-v| = +5 and |+ five| = +v

In this article, we volition learn what is the accented value of a number, symbol, examples, accented value in number line, absolute value of existent numbers, absolute value of complex numbers in detail.

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 ascertain the absolute values similar the following:

{ a if a ≥ 0 }

|a| = { -a if a < 0 }

Notation: There is no absolute value for 0 because the accented value changes the sign of the numbers into positive and zero has no sign.

If the number is positive then it will consequence in a positive number just. And if the number is negative, so the modulus of this number will besides be a positive number. It is denoted as |due north|, where north is an integer.

Absolute Value Symbol

The symbol of absolute value is represented by the modulus symbol, '| |', with the numbers betwixt it. For case, the absolute value of 9 is denoted as |9|.

The altitude of whatsoever number from the origin on the number line is the absolute value of that number. It also shows the polarity of the number whether information technology is positive or negative. It can be negative e'er a s it shows the distance and the altitude can't be negative. So, information technology is always positive.

'| |' – Accented value symbols, '| | pipes' are used to stand for the accented values.

For case: '|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 accented value of a number.

• |-1| = one
• |-fourteen| = 14
• |1| = 1
• |0| = 0
• |7| = vii
• |7-2| = |5| = 5
• |2+three| = |v| = 5
• |-three×v| = |-xv| = xv

Related Articles:

• Accented Value formula
• Integers
• Addition and Subtraction Integers

Absolute Value Part

The absolute value function is given by f(ten) = |x| such that:

• |x| = +x for x > 0
• |ten| = -x for 10 < 0

Absolute Value Properties

If x and y are real numbers and so the absolute values are satisfying the following properties:

Belongings	Expression
Non-negativity	| x | ≥ 0
Positive-definiteness	| x | = 0 ↔ a = 0
Multiplicativeness	| x × y| = |x| × |y|
Subadditivity	| 10 + y| ≤ | ten | + | y |
Symmetry	|-10| = |x|
Identity of indiscernible (equivalent to positive definiteness)	| 10 – y | = 0 ↔ a = b
Triangle inequality (equivalent to subadditivity)	| x – y | ≤ | ten – z | + | z – x |
Preservation of division (equivalent to multiplicativeness)	| x / y| = | 10 | / | y |
Equivalent to subadditivity	| x – y | ≥ | | 10 | – | y | |

Accented Value of a Real Number

If a real number ten, the absolute value will satisfy the following conditions.

| 10 | = x, if x ≥ 0

| 10 | = – x, if x < 0

Let's expect at the absolute value of two in the number line given below. Here, |2| is the distance of 2 from 0(zero). So, both +2 and -2 is the altitude of 2 from the origin. But it would be taken as 2 considering altitude is never measured in negative.

Absolute Value of Circuitous Number

Complex numbers consist of real numbers and imaginary numbers. Hence, unlike integers, it is difficult to detect the absolute value for them. Suppose, x+iy is the given circuitous number.

z = x+iy

The absolute value of z will be;

|z|= √[Re(z)ⁱⁱ+Im(z)²]

|z| = √(10²+y²)

Where x and y are the real numbers.

Absolute Value in Number Line

The graph of accented values is called the absolute value graph. Equally we know the absolute value of any real number is positive, then 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 accented value of -9 will wait like following

Problems and Solutions

Let's understand this topic amend with the assist of examples.

Question one:

Arrange the given numbers in ascending social club.

-|-14|, |12|, |7|, |-91|, |-five|, |-8|, |-65|, |6|

Solution:

Initially, find the absolute values of the given numbers

-xiv, 12, seven, 91, 5, eight, 65, 6

Now, arrange the numbers in ascending gild (smallest to the largest number)

-fourteen, 5, 6, 7, eight, 12, 65, 91

Question 2:

Adapt the given numbers in ascending order.

-|-24|, |21|, 17|, |-109|, |-fifteen|, |-19|, |-75|, |xvi|

Solution:

Find the accented values of the given numbers

-24, 21, 17, 109, 15, xix, 75, 16

At present, arrange the numbers in ascending gild (smallest to the largest number)

-24, 15, 16, 17,19, 21, 75, 109.

Example 3:

Detect the absolute value of a number -12/5

Solution:

To observe: |-12/5|

|-12/v| = 12/5

Hence, the absolute value of a number -12/5 is 12/five

Instance four:

Find the absolute value for the following numbers:

(a) |-i/2|

(b) |72|

(c) |3/four|

Solution:

The absolute value of the numbers are:

(a) |-1/2| = 1/2

(b) |72| = 72

(c) |three/four| = 3/4

Practice Questions

i. What is the absolute value of -13?

2. Find the absolute value of 100.

3. Adapt the following in ascending order.

           |three|, |-ix|, |1|, |-2|, |-5|

4. Plot the absolute value of -7 on the number line.

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Often Asked Questions on Absolute Values

What is an absolute value of a number?

The absolute value of a number represents the altitude of the number from null in a number line.

What is the accented value of iv?

The absolute value of 4 is 4. |4| = 4

What is the absolute value of -12?

The absolute value of -12 is 12. Since the altitude betwixt -12 and 0 in a number line is 12.

What is the symbol for absolute value?

The symbol to represent absolute value is |x|, where x is an integer.

What is the accented value of -seven?

The accented value of -7 is seven. It is represented by |-7| = 7.

Is accented value negative?

Absolute value of whatever integer is e'er positive and never negative.

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