Vikasana – Bridge Course 2012VIKASANA - VIGNANA PATHADEDEGE NIMMA NADIGE

Bridge Course Program for SSLC Students who want to take up Science in I PUC in 2012

NUMERALS ARE SYMBOLS USED TO WRITE NUMBERS

ARABICARABICHINDU-ARABICBABYLONIANBABYLONIANROMAN etc

Vikasana – Bridge Course 2012

HINDU ARABIC NUMERALS HAVE BEENHINDU ARABIC NUMERALS HAVE BEEN UNIVERSALY ACCEPTED AND ADOPTED

THEY ARE CUTE AND ADAPTIVETHEIR DIGITAL REPRESENTATION IS EASYTHESE NUMERALS ARE ALSO CALLED INTERNATIONAL STANDARD NUMERALS

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

Vikasana – Bridge Course 2012

THE INVENTION OF ‘0’ NUMERAL IS A REVOLUTION IN THE HISTORY OFREVOLUTION IN THE HISTORY OF NUMBERS

Vikasana – Bridge Course 2012

“NUMBERS” ALONG WITH APPROPRIATE UNITS REPRESENT QUANTITYREPRESENT QUANTITY

QUANTITY MAY BE OF SPACE, MASS, TIME, TEMPERATURE OR ANY THINGR UR OR G

Vikasana – Bridge Course 2012

EXAMPLES:‐3 METERS3 METERS,12 KMS7 SQ MTRS, 7 SQ MTRS, 30 CUBIC MTRS3 KG,3 KG,5 GMS, 1O MINUTES100 RS60 DEGREES Etc

Vikasana – Bridge Course 2012

‘0’ ZERO AS A NUMBER REPRESENTS “NOTHING” ORREPRESENTS “NOTHING” OR NO QUANTITYO QU

Vikasana – Bridge Course 2012

BASE ‘10’ NUMBERS ARE INTERNATIONALLY ACCEPTED AND USE NUMBERS

IN THIS SYSTEM OF NUMBERS EVERY PLACE IN THE NUMBERS HAS A PLACE VALUE WHICH IS AN ORDERED MULTIPLE OF 10

Vikasana – Bridge Course 2012

IN THE NUMBER 4536The place value of

6 is -- 10º-- one3 i 10¹ t3 is -- 10¹ -- ten5 is -- 10² -- hundred4 is – 10³ -- thousand4 is 10 thousand

The number is read as FOUR THOUSAND FIVE HUNDRED THIRTY SIXThe same is read as numeral FOUR FIVE THREE SIX

Vikasana – Bridge Course 2012

THE BASIC NUMERALS,THE BASIC NUMERALS, INTERNATIONALLY ACCEPTED AND USED AREACCEPTED AND USED ARE

0, 1, 2, 3, 4, 5, 6, 7, 8, & 9

Vikasana – Bridge Course 2012

THE HISTORY OF NUMBERS, BEGIN WITH THE HUMAN ACTIVITY OFWITH THE HUMAN ACTIVITY OF “COUNTING”

COUNTING, NATURALLY STARTS FROM 1 (ONE) CONTINUES IN ORDER WITH1 (ONE), CONTINUES IN ORDER, WITH UNIQUE CORRESPONDENCE WITH EVERY OBJECT IN A NON EMPTY SETEVERY OBJECT IN A NON-EMPTY SET OF OBJECTS, TILL THE OBJECTS IN THE SET ARE EXHAUSTEDVikasana – Bridge Course 2012THE SET ARE EXHAUSTED

WE MAY SAY

THE PROCESS OF COUNTING IS ESTABLISHING“ONE-TO-ONE” CORRESPONDENCE BETWEEN THE SET OFNATURAL NUMBERS AND SET OF OBJECTS,STARTING FROM 1 IN ORDERSTARTING FROM 1, IN ORDER

Vikasana – Bridge Course 2012

THE SET OF NATURAL NUMBERS N = { 1 2 3 4 }N = { 1,2,3,4,…………….}

THE SALIENT FEATURES OF NATURAL NUMBERS :-It starts with ‘1’ Successive natural number is obtained by addingSuccessive natural number is obtained by adding

1 to the numberEg : successor 4 is 4+1=5 and so ongEvery natural number has a successorThere is end to the array of natural numbers

Vikasana – Bridge Course 2012There is end to the array of natural numbers

INVENTION OF ‘0’ AND USING IT AS A NUMBER ENRICHED THE NUMBER SYSTEM

We have, SET OF WHOLE NUMBERSW= {0,1,2,3,4,…………..}

We see N is a subset of W WN

Vikasana – Bridge Course 2012

AS THE HUMAN CIVILISATION ADVANCED, THE ASSOCIATION OF ‘DIRECTION’ DIMENSION, TO THE NUMBERS EXPANDED THE UTILITY OF NUMBERS 

Eg:‐ Moving 1 step right wards ‐‐ +1Moving 1 step left wards  1Moving 1 step left wards ‐‐ ‐1Climbing up 1 step ‐‐‐ +1Climbing down 1 step ‐‐‐ ‐1Climbing down 1 step ‐‐‐ ‐1Moving 1 step forward ‐‐‐ +1 Moving 1 step backward ‐‐ ‐1

Vikasana – Bridge Course 2012Moving 1 step backward  1

THE NUMBER SYSTEM EXTENDED WITH NEGATIVE AND POSITIVE NUMBERS ALONG NEGATIVE AND POSITIVE NUMBERS ALONG WITH ‘0’ ‘0’ IS NEITHER POSITIVE NOR NEGATIVE

We have THE SET OF INTEGERSI = {……….-4, -3,-2,-1,0,+1,+2,+3,+4,……..}THE SET OF INTEGERS HAVE THE SAME FEATURES AS OF WHOLE NUMBERS BUT WITH OPEN ENDS 

Vikasana – Bridge Course 2012

AS THE HUMAN ACTIVITIES EXTENED AND CIVILISATION CONTINUED TO GROW, ‘SHARING’, ‘DIVINDING’ CONCEPTS STARTED

IT WAS THE BEGINNING OF FRACTIONAL NUMBERSNUMBERS

A FRACTION IS PART OF “WHOLE”A FRACTION IS PART OF WHOLE

Vikasana – Bridge Course 2012

EXAMPLE= ONE HALF-1/2

= 2 FIFTHS-2×1/5=2/5

= 2 & 1HALF = 2 1/2

OR FIVE HALVES = 5×1/2=5/2Vikasana – Bridge Course 2012

FRACTION ARE 2 TYPES PROPER FRACTIONSRO R R C ONSIMPROPER FRACTIONS

• IN PROPER FRACTIONS NUMERATOR IS LESS IN PROPER FRACTIONS NUMERATOR IS LESS THAN DENOMINATOR

1/2, 3/5, ‐7/9, 1/12, …………….1/2, 3/5,  7/9, 1/12, …………….• IN IMPROPER FRACTIONS THE DENOMINATOR IS LESS THAN THE NUMERATOR

3/2, 7/3, 1/5, 3 ½, 12/5…………..IMPROPER FRACTIONS ARE ALSO CALLED AS 

Vikasana – Bridge Course 2012IMPROPER FRACTIONS ARE ALSO CALLED AS 

MIXED FRACTIONS

ALL THESE NUMBERS CAN BE EXPRESSED IN THE FORM OF P/qTHE FORM OF P/q

Eg: - ½ p=1 q=22 1/3 – 7/3 p=7 q=32 1/3 – 7/3 p=7 q=3

• p and q both are integers• q is not zeroq is not zero• p,q do not have a common factor

(co-prime)(co prime)The set of all such numbers is called RATIONAL

NUMBERSVikasana – Bridge Course 2012

EVERY NATURAL NUMBER IS A RATIONAL NUMBER EVERY INTEGER IS A RATIONAL NUMBEREVERY INTEGER IS A RATIONAL NUMBER EVERY FRACTIONAL NUMBER IS A RATIONAL NUMBER

We see, SET OF RATIONAL NUMBER IS A SUPER SET OF ALL

PREVIOUS SETS OF NUMBERSPREVIOUS SETS OF NUMBERS

QIWN

Q

Vikasana – Bridge Course 2012

EVERY RATIONAL NUMBER CAN BE EXPRESSED AS A DECIMALBE EXPRESSED AS A DECIMAL FRACTION

AS A TERMINATING DECIMALAS A TERMINATING DECIMAL OR

A NON TERMINATING, RECURRING DECIMAL

Vikasana – Bridge Course 2012DECIMAL

Eg for terminating decimalsa) 1/2 = 0.5b) 1/5 = 0.2) 2 ¼ 2 25c) 2 ¼= 2.25

d) 7 = 7/1E f t i ti i d i lEg for non terminating recurring decimal

a) 1/3 = 0.33333333…….b) 1/7 = 0 142857142857b) 1/7 = 0.142857142857……………..c) 2/3 = 0.6666666666……..

Vikasana – Bridge Course 2012

AS WE PROCEED TO THE OPERATIONS ON REAL NUMBERS WE COME ACROSSREAL NUMBERS WE COME ACROSS STRANGE CASES WHERE THE RESULTS OF OPERATIONS ARE NOT RATIONAL NUMBERS

Eg:- √2, √7 etcThese numbers are called IRRATIONAL

NUMBERSThese numbers expressed in decimal

form are non terminating nonVikasana – Bridge Course 2012

form are non terminating, nonrecurring.

THE UNION OF SET OF RATIONAL NUMBERS AND SET OF IRRATIONAL NUMBERS FORM SET AND SET OF IRRATIONAL NUMBERS FORM SET OF REAL NUMBERS 

THE SET OF RATIONAL NUMBERS AND SET OF IRRATIONAL NUMBERS ARE DISJOINT SETSJ

Q RQ IR

Vikasana – Bridge Course 2012