Modulus Numbers - Original

8/2/2019 Modulus Numbers - Original

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Real Number

Complex Number


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yIntroduction To Find Modulus

Normally In Mathematics, the word modulus has

different meanings in mathematics with respect to

congruence and complex numbers and real numbers.

Modulus is used mainly in statistics. It is also used to

represent the complex numbers.


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y In mathematics, the absolute value (ormodulus)|a| ofa real number is a numerical value without

regard to its sign.

y So, forthe example, 3 is the absolute value ofboth 3

and -3.

|3| and |-3| = 3


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y an absolute value is alsodefinedfor the complexnumbers, the quaternion, fields andvectorspace.


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The graph of the absolute value function forreal

numbers.


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y Forany real numbera the absolute

value ormodulus ofa is denoted by |a|

(a vertical baron each side of the quantity) and

is defined as

yAs can be seen from the above definition, the

absolute value ofa is always

eitherpositive orzero, but nevernegative.T

hesame notation is used with sets to

denote coordinate; the meaning depends on

context.

Real number


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y Since the square-root notation without signrepresents the positive square root, it follows that

which is sometimes used as a definition ofabsolute

value.


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The absolute value has the following four

fundamental properties:

Non-negativity

Positive-

definiteness

Multiplication

Subadditivity


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Other important properties of the

absolute value include:Symmetry

Identity of indiscerribles

(equivalent to positive-

definiteness)

Preservation ofdivision

(equivalent to

multiplicativeness

Equivalent to subadditivity


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Ifb > 0, twootheruseful properties

concerning inequalities are:


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Modulus Functiony returns positive value of a variable or an expression.

y absolute value function.

y Interpretations of modulus can represent distance of a

point with respect to the reference point.

y Consists of:

y

Modulus and equalityy Modulus and inequality


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y Here, the initial values are calculated to draw the plotas:

y For x=2,y=|x|=x=(2)=2

y For x=1,y=|x|=x=(1)=1

y Forx=0,y=|x|=x=0

y For x=1,y=|x|=x=1

y For x=2,y=|x|=x=2

y The graph of the function is shown


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y The graph is differentiable at all points except at x=0.

y The graph also shows that modulus function is an evenfunction.

y Modulus function is invertible.

y Based on the graph, the range of modulus function isupper half of the real number set, including zero.


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Modulus and Equalityy For the sake of understanding, we consider a non-

negative number "2" equated to modulus of

independent variable "x" like |x|=2.y Then, the values of x satisfying this equation isx=2

y It is intuitive to note that values of "x" satisfying above

equation is actually the intersection of modulusfunction "y=[x]" and "y=2" plots as shown in thefigure-----------------------------------------------------------


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y We express these results in general form, using anexpression f(x) in place of "x" as :

y |f(x)|=a;a>0f(x)=a

y |f(x)|=a;a=0f(x)=0

y

|f(x)|=a;

a


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Modulus and Inequalityy Interpretation of inequality involving modulus

depends on the nature of number being compared

with modulus.y Case 1: a0


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Case 1: a


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Case 2: a>0y |x|


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Modulus ofComplex Numbery If we write Z in polar form as with ,

,then |Z|=r. It follows that the modulus is

a positive real number or zero. Alternatively, if isthe real part ofZ , and the imaginary part, then:


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Reference listy http://cnx.org/content/m15505/latest/

y http://myyn.org/m/article/modulus-of-complex-

number/y http://www.tutorvista.com/math/find-modulus

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