Introduction to absolute value rules

 Introduction to absolute value rules

                The numerical value of a real number without considering it sign is called absolute value. For example we can consider the value |-5| absolute value of this is 5. We have to follow some rules for writing the absolute values. Which is called absolute value rules. Here we didn’t consider the value of the sign. If it is positive or negative the absolute value is positive. To find the absolute value of the  complex number is square root of the sum of the real and imaginary parts of the number. If we found like that we will get a real number there is no imaginary part in that.Absolute value rules are used to write the absolute values in the standard form.

The Rules Applied to Absolute Values

                Normally the absolute value is the distance between the number and the origin. It connects the absolute value of the complex number and magnitude of a vector.

Definition

                                          |a| =  `{(a if a>=0),(-a if a<0 span="span">

 It is the piecewise definition for absolute value.

Rule 1:

                A nonnegative that mean a positive  it must be multiplied times itself to equal a given number.

The square root of x can be written as square root (x) or  x^½. Totally we are having seven absolute value rules.

For example:

                Square root of (16) = 4  and  4²  = 16

Here there square root(x) never refers a negative value. Because (-4) * ( -4) is also +16 we can’t say

Square root (x) is negative if it is negative mean then it is imaginary.

Rule 2:

                |- a| = |a|

                |-a| = Square root ((-a) * ( -a) ) = a =|a|

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More Rules on Absolute Value

 Rule 3:

                |a| >0

                |-a| = |+a | =a  (|a| is always greater than 0)

Rule 4:

                Products of any two absolute value will also give a absolute value       

                                          |a b| = |a||b|

Rule 5:

                The result of dividing one absolute value by another is  equal.

                                |a/b| = |a| / |b|

Rule 6:

                Exponents the absolute values are equal.

                                                |aⁿ| = |a| ⁿ

Rule 7:

                Triangle Inequality:

                                |a +b | ≤ |a| + |b|

                 Alternative triangular property:

                                 |a – b| ≥ |a| – |b|

If we are going to write any absolute value we have to consider all the absolute value rules.