Real number
Real Number :
Student Learning Outcomes:
After study this topic you will be able to learn the following
- Set of real number.
- Depict real number on number line. Demonstrate a number with terminating and non terminating recurring decimals on the number line.
- Give decimal represention of rational and irrational number.
- Explain the concept of radical and radicands
- Different between radical from and exponential form of an expression.
- Apply the exponential law and simplify expression with real exponents.Introduction is base of mathematics We us different kind of number in daily life. Real number is base of pure math.
Definition of real number:
The union rational and irrational is called real number and is debited by R
Natural Number:
The number 1,2,3,...........which is use for counting is called natural number or positive number and denoted by N . N={1,2,3,........} Integers :
The set integers consist of positive integers,0 and negative and is denoted by Z Z={-3,-2,-1,0,1,2,3,............}.
Rational Number
All number in form p/q where p,q are integers and q is not zero are called rational numbers and denoted by Q.
Irrational Number:
The numbers which cannot in the form p/q where p,q are integers and q is not zero are called irrational numbers and is denoted by Q`
Depiction of Real number on number Line
The Real numbers are represention geometrically by point on a number line l we first choose an arbitrary point O (the origin) on horizontal line l and associate with it the real number 0 The right of the origin are positive numbers and left of the origin are negative numbers Assign the number 1to the point A so that the line segment OA represents one unit of length.
(a) Rational Number:
The decimal representations of rational numbers are of two types , terminating and recurring.(اختتام پزیراور تکراری) Terminating Decimal Fraction: (اختتام پذیر اعشاری ناطق اعداد)
The decimal fraction in which they are finite number of digits in its decimal part is called a terminating decimal fraction for example 3/5 =0.6 and 3/8 = 0.375
Recurring and Non terminating Decimal Fraction: ( غیر اختتام پذیرائی تکراری اعشاریہ اعداد).
The decimal fraction (non terminating) (کسر اعشاریہ) in which some digits are repeated again and again in the same order in its decimal part(کسری حصہ) is called a recurring decimal fraction. For example ,2/9=0.2222...... and 4/11=0.363636.......
(b) Irrational Number: (غیر ناطق اعداد).
It may be noted that the decimal representations for irrational numbers (غیر ناطق اعداد) are neither terminating nor repeated in blocks. The decimal from of an irrational number would continue forever and never Begin to repeat the same block of digits. For example √2= 1.4142135..........and π= 3.1415926........ and e=2.718281829........
Example:
Express the following decimal in the form p/q where p,q belong to Z and q doesnot equal to zero a 0.3`= 0.3333333......
Solution:
Let x=0.3` which can be written as x= 0.3333333.......eq(1) . Multiply both side 10 we get then we will 10x=3.33333......eq(2) Subtracting eq1 from 2 we get 10x-x=3.33333.....0.33333...... =9x= 3 or x= 3/9 or x=1/3 Hence 0.3`=1/3.
Example 2:
0.15`= .151515........
Solution:
let x = 0.15` or x= .151515.....eq(1) Multiply eq1 by 100 we get 100x= 15.151515...... eq(2) Subtracting eq1 from eq2 we get 100x-x = 15.151515.........-0.151515......... =99x=15 or x=15/99 is rational Number.
Example 3: . 0 33`
Solution:
let x =0.333333...... eq(1) Multiply 100 by ep 1 we get 100x= 33.3333..... eq2 Subtracting eq1 from eq2 we have 100x-x=33.3333.....-0.333333...... 99x= 33 or x=33/99; whch is a rational Number
Representation of Rational and Irrational numbers on number line:
(نمبر لائن پر ناطق اور غیر ناطق اعداد کو ظاھر کرنا )
in order to locate a number with terminating and non terminating recurring decimal on the number line, (اختتام پذیراعشاری ناطق اعداد اور غیر اختتام پذیر کو ناطق اعداد کو نمر لائن پر ظاہر کرنے کی خاطر ھم نمبر لائن پر ) the point associated with rational number m/n and -m/n where m,n are positive integers we subdivide each unit length into n equal parts Examples Represent the following numbers on the number line
1. -3/5 2. 15/7 3. 2/5
Solution
سب سے پہلے ایک افقی لائن کھنچے گٕے اس کے بإیں جانب منفی والے نمبر اور دايں طرف منفی والے نمبر لکھیں گٕے ۔سب سے پہلے رونڈ دیکھیں گے کہ وہ مثبت کا ہے یا منفی کا۔ اس کے بعد جو بٹے کے نیچے والا نمبر اس رونڈ میں اتنی ہی لائن لگاے گے بٹے کے اوپر والے نمبر کے لحاظ سے نقطہ لگاے گے
Radicals and Radicands: (ریڈیکلز اور ریڈیکنڈز)
Concept of Radical and Radicand (ریڈیکل اور ریڈیکنڈ کا تصور )
If n is a positive integer greater 1 and a is real number ,then any real number x such that x ke power n= a is called the nth root of a and in symbols is written as
The Real Number System
Some of the properties of the set R of real numbers which has the binary operations of ordinary addition and ordinary multiplication, are stated as follows
Properties of real numbers
(1) Clousure property( w.r.t addition and multiplication(خاصیت بندش بلحاظ جمع اور ضرب)
Sum of two real number is also a real number
For all a,b belongto R this imply that a+b belongto R
For example -3and 5 belongto R then -3+5=2 belongto R
And for all a,b belong to R this imply that ab belong to R
For example -3and 5 belong to R then -3*5=-15 belongto R
Associative property w.r.t addition and multiplication(خاصیت تلازم بلحاظ خمع اورضرب)
For all a,b,c belong to R this imply that
a+(b+c)=(a+b)+c And a(bc)=(ab)c
For example 5+(7+8)=(5+7)+8 and 5(7*8)=(5*7)8
Commutative property w.r.t addition and multiplication(خاصیت مبادلہ بلحاظ جمع اور ضرب)
For all a,b belong to R this imply that a+b=b+a and ab=ba
For example 5+7=7+5 and 5*7=7*5
Multiplicative Identity:(ضربی ذاتی عنصر )
There exits a unique real number 1 called the multiplicative Identity, such that a.1 =1.a forall a belong to R For example 1*7=7*1
Additive Identity:(جمعی ذاتی عنصر )
There exists a unique real number 0 called additive Identity ,such that
a+0=0+a for all a belong to R
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