Period, Place & Place Value

Definition :

Place Value : The value of the position, or place  of a digit digit in a number or series .

Examples : 

897 : Here 7 is the place value of ones .

(a) . The place value is divided into two parts - 

1. International Place value chart .

2. Indian Place value chart .

2. Indian Place Value chart : 

Crores            Lakhs          Thousands     ones

Ten , one.      Ten,   one.   Ten,     one      H,T,o

Cr  ,    Cr          Lk,     Lk.     Th,     Th              

Here : 

Cr = crore 

Lk = Lakh

Th = Thousand 

H = Hundreds

T = Tens

O = Ones 

Examples : 

Numeral         Period          Place           Place 

                                                                    Value

71,38,291.       Lakhs.          Ten lakh   7000000

60,46,295.       Th.                Ten Th        40000

REAL NUMBER & property of real numbers

Real Number : 

The set of rational and irrational numbers are known as Real number .

1. It is represented by R .

R = Q U Q^c 

where 

Q represent rational number & Q^c irrational number .

2. Every real  number can be represented at a point on  number line or real number line .

Property of real numbers : 

(a) . Rational number + Rational number = Rational number .

3/4 + 1/2 = 5/4 .

(b) . Rational - Rational = Rational number 

3/4 - 1/2 = 1/4 .

(C) . Rational × Rational = Rational number 

 3/4 × 1/2  = 3/8 . 

(d) . Rational / Rational = Rational 

3/4 ÷ 1/2 = 3/2 .

(e) . Irrational + Irrational = Irrational or rational number .

(2+√3) + (√3 - 2 ) = 2√3 (irrational number)

(4+√5) + (4 - √5) = 8(Rational number)

(f) . irrational - irrational = irrational or rational number .

(√7+2) - (√7-2) = 4 (Rational number)

(√7+2) - (2 - √7) = 2√7(irrational number)

(g) . Irrational × Irrational = Irrational or Rational number . 

(2+√3) × (2-√3) = 1 . (Rational number ) 

(2+√3) × (2+√3) = 4+ 4√3+3 = 7+ 4√3 (Irrational number) . 

(h) . Irrational ÷ Irrational = x÷y 

(Where y  not equal to zero ) = Rational or Irrational number .

(2+√3) ÷ (2+√3) = 1 ( Rational number)

(2+√3) ÷ (√3) = - (2√3 + 3)/3 = 1+( 2/√3) .

(I) . Rational + Irrational = Irrational 

3 + √7 .

(J) . Rational - Irrational = Irrational number .

3 - √7 .

(K) . Irrational - Rational = Irrational number .

√7 - 3 .

(l) . Irrational × Rational = Irrational number . 

7×√3 = 7√3 .

(m) . Rational ÷ Irrational  = Irrational .

4/√5 .

(n) . Irrational ÷ Rational = Irrational number . 

{√3+2} ÷ 2 = 1 + (√3/2) .

Integer Number

Definition of Integer Number : 

An Integer number is a number which can be written without a fractional component .

It is commonly known as " Whole number ".

For examples -

21,45,-1, -2002,76,89,----- are Integer number , while 8.67,3.6,11/2,√2, are not Integer number .

a set of integer number is written as {-3,-2,-1,0,1,2,3} or {-5,-4,-3,-2,-1,0,1,2,3,4,5} e.t.c .

In these examples we see that the set of Integer consist of zero (0) the natural numbers (1,2,3,4,-----) and their additive inverse (the negative Integer ,i.e -1,-2,-3,-4,-----) .

This is denoted by Z or I . It is countably infinite .

Types of Integer number - 

1. Positive Integer ( I^+ or Z^+):

The positive natural number or the set of Positive numbers are known as positive Integer .

{ 1,2,3,---}

2. Negative Integer (I^- or z^-) : 

The additive inverse of natural numbers are known as negative Integer .

{ ------,-3,-2,-1 }

3. Non positive Integer : 

{ -----,-4,-3,-2,-1}

4. Non Negative Integer : 

{1,2,3,4,----------}

a. Even Integer : 

{0, +2,-2,+4,-4,+6,-6,-------}

Symbol : 2n , n belong to Integer (I)

b. Odd Integer : 

{ +1,-1,+3,-3,+5,-5,+7,-7,-------}

Symbol : 2n+1 or 2n-1, where n belong to Integer (I)

Here n is positive or negative number .