MATHGUIDE E-BOOK

www.mathguide.org

MATHGUIDE E-BOOK

www.mathguide.org

Chapters

NUMBER SYSTEMS

1.           NATURAL NUMBERS

2.           WHOLE NUMBERS

3.           INTEGERS

4.           RATIONAL NUMBERS

5.           IRRATIONAL NUMBERS

    1. NATURAL NUMBERS (OR COUNTING NUMBERS) 

         {1,2,3,4…}

       E.G.

       A.  NUMBERS OF DAYS IN A WEEK.

       B.  NUMBER OF PAGES IN A BOOK.

       C.   NUMBER OF SCHOOLS IN A CITY.

    2. WHOLE NUMBERS

         {0,1,2,3, 4…}

    3.  INTEGER NUMBERS – NUMBER SYSTEM CONSISTING OF NATURAL NUMBERS, THEIR NEGATIVES AND NUMBER 0 IS CALLED INTEGER NUMBER.

                 {…-3, -2, -1,0,1,2, 3…}

    4. RATIONAL NUMBERS – A RATIONAL NUMBER IS OF THE FORM P/Q WHERE P AND Q ARE INTEGERS AND Q≠0. E.G. 6/5, 5/6, -8/9 ETC.

    5. IRRATIONAL NUMBERS – NUMBERS WHICH CANNOT BE WRITTEN IN THE FORM P/Q WHERE P, Q ARE INTEGERS AND Q≠0. THESE NUMBERS CAN BE WRITTEN AS DECIMALS BUT CANNOT BE EXPRESSED AS FRACTIONS I.E.  IRRATIONAL NUMBERS HAVE NON-TERMINATION AND NON-RECURRING DECIMAL. E.G. Π, √2, √5 ETC.

RATIO AND PROPORTION

1.              RATIO – WHEN WE COMPARE TWO QUANTITIES OF THE SAME KIND (WITH RESPECT TO THEIR MAGNITUDES) BY DIVISION, WE SAY THAT WE HAVE FORMED A RATIO.

POINTS TO REMEMBER –

    (I)     A RATIO HAS NO UNITS IN ITSELF I.E. A RATIO IS PURELY A NUMBER, IT HAS NO UNIT ATTACHED TO IT.

    (II)    THE NUMBERS A AND B IN A RATIO A:B ARE CALLED TERMS OF A RATIO.

2.           PROPORTION – EQUALITY OF TWO RATIOS IS CALLED PROPORTION. WE WRITE PROPORTION AS A:B :: C:D OR A/B = C/D.

 PERCENTAGE AND ITS APPLICATIONS

A FRACTION WITH DENOMINATOR 100 IS CALLED A PERCENT. LATIN WORD “PER CENTUM” MEANS ‘PER HUNDRED’.

  APPLICATIONS OF PERCENTAGE

     1.PROFIT AND LOSS

          PROFIT – THE MARGIN (EXTRA MONEY) EARNED AFTER SELLING A PRODUCT AT A HIGHER PRICE THAN ITS    COST IS CALLED PROFIT.

          LOSS – THE MONEY LOST ON SELLING A PRODUCT AT A PRICE LOWER THAN ITS COST.

     2. SIMPLE INTEREST

            PRINCIPAL (P) – THE AMOUNT OF MONEY BORROWED OR DEPOSITED

            RATE (R) –  A PERCENTAGE OF THE PRINCIPAL IS ADDED TO THE PRINCIPAL, MAKING MONEY GROW IN A GIVEN PERIOD OF TIME.

            TIME (T) – TIME FOR WHICH THE MONEY IS BORROWED OR DEPOSITED.

            INTEREST (I) – THE ADDITIONAL OR EXTRA MONEY RETURNED OR OBTAINED ON BORROWING OR DEPOSITING A MONEY FOR A GIVEN PERIOD OF TIME AT A GIVEN RATE OF INTEREST.

            AMOUNT (A) – THE TOTAL MONEY WHICH IS RETURNED OBTAINED AT THE END OF A SPECIFIED PERIOD OF TIME AT A GIVEN RATE OF INTEREST.

           SIMPLE INTEREST (S.I.) – IF THE PRINCIPAL REMAINS THE SAME FOR THE ENTIRE PERIOD OF TIME FOR WHICH THE MONEY IS BORROWED OR DEPOSITED, THE INTEREST IS CALLED SIMPLE INTEREST.

        3.  DISCOUNT

            MARKED PRICE – THE PRICE ATTACHED TO THE ARTICLE IS CALLED MARKED PRICE.

            RATE - THE PERCENTAGE REDUCTION IN THE PRICE OF THE ARTICLE IS CALLED RATE OF DISCOUNT. 

            DISCOUNT – THE REDUCTION ALLOWED ON THE PRICE OF THE ARTICLE IS CALLED DISCOUNT.

      4. Compound Interest

            COMPOUND INTEREST IS INTEREST ON INTEREST I.E. IN COMPOUND INTEREST THE FIRST INTEREST IS ADDED TO THE PRINCIPAL OF A DEPOSIT OR LOAN AND AMOUNT SO OBTAINED BECOMES THE NEW PRINCIPAL. THIS PROCESS IS REPEATED FOR THE GIVEN NUMBER OF TERMS (YEARS / HALF YEARS / QUARTER YEARS). THIS ADDITION OF INTEREST TO THE PRINCIPAL IS CALLED COMPOUNDING.

SETS

THEORY OF SET WAS DEVELOPED BY GERMAN MATHEMATICIAN GEORG CANTOR. SET IS A WELL-DEFINED COLLECTION OF OBJECTS SO THAT WE CAN DEFINITELY DECIDE WHETHER A GIVEN OBJECT BELONGS TO A GIVEN COLLECTION OR NOT.E.G. VOWELS IN ENGLISH LANGUAGE, SOLUTION OF A QUADRATIC EQUATION. SET IS USUALLY DENOTED BY CAPITAL ALPHABETS. ELEMENTS ARE DENOTED BY SMALL ALPHABETS.SETS CAN BE WRITTEN IN TWO WAYS :

1.  ROSTER FORM – E.G. {A,E,I,O,U}, {1,2,3,4}

2.  SET BUILDER FORM – E.G. {X : X IS A VOWEL}

LOGARITHMS

    JOHN NAPIER, A SCOTISH MATHEMATICIAN INVENTED LOGARITHMS IN 1614. LOGARITHM IS DERIVED FROM WORD ‘LOGOS’ MEANING RATIO AND ‘ARITHMOS’ MEANING NUMBER. HENRY BRIGGS INTRODUCED COMMON LOGARITHMS IN 1624.

    DEFINITION – FOR EACH POSITIVE REAL NUMBER A, A≠1, THE UNIQUE REAL NUMBER ‘M’ IS CALLED THE LOGARITHM OF ‘B’ TO THE BASE ‘A’. 

NATURAL LOGARITHM AND COMMON LOGARITHM

NATURAL LOGARITHM – LOG TO THE BASE ‘E’ (E = 2.71828 APPROX.)

COMMON LOGARITHM – LOG TO THE BASE 10

SEQUENCE AND SERIES

SEQUENCE – AN ARRANGEMENT OF NUMBERS IN A DEFINITE ORDER ACCORDING TO SOME RULES.

TERM – VARIOUS NUMBERS OCCURRING IN A SEQUENCE ARE CALLED ITS TERMS. 

WE DENOTE A SEQUENCE AS :

        A1, A2, A3, …, AN

IT IS A SEQUENCE HAVING N TERMS. A1 IS THE FIRST TERM AND AN IS THE NTH TERM. NTH TERM IS ALSO CALLED THE GENERAL TERM OF A SEQUENCE.

SEQUENCE HAVING FINITE NUMBER OF TERMS IS CALLED FINITE SEQUENCE AND SEQUENCE HAVING INFINITE NUMBER OF TERMS IS CALLED INFINITE SEQUENCE.

      1.      ARITHMETIC PROGRESSION (A.P)

              A SEQUENCE   A1, A2, A3, …, AN IS CALLED ARITHMETIC PROGRESSION IF AN+1 = AN + D, N Ε N

       2.    GEOMETRIC PROGRESSION (G.P.)

              A SEQUENCE   A1,A2,A3,…,AN IS CALLED GEOMETRIC PROGRESSION IF EACH TERM IS NON-ZERO AND, AK+1 / AK =R (CONSTANT) FOR ALL K ≥ 1

        A, AR, AR2, AR3,… IS A G.P. WHERE A IS THE FIRST TERM AND R IS THE COMMON RATIO.

 ALGEBRAIC EXPRESSIONS

      THE LETTERS WHICH ARE USED TO REPRESENT NUMBERS ARE OFTEN GIVEN THE NAME LITERAL NUMBERS OR LITERALS. AN ALGEBRAIC EXPRESSION IS A COMBINATION OF NUMBERS, LITERALS AND ARITHMETICAL OPERATIONS.

        MONOMIALS: AN ALGEBRAIC EXPRESSION CONSISTING OF ONLY ONE TERM E.G. 23, 3Y, XYZ ETC.

       BINOMIALS: AN ALGEBRAIC EXPRESSION CONSISTING OF TWO TERMS E.G. 3X-Y, 2X+1 ETC.

POLYNOMIALS

      A POLYNOMIAL IS AN ALGEBRAIC EXPRESSION CONSISTING OF SUM OF TERMS WITH EACH TERM CONTAINING A VARIABLE OR VARIABLES RAISED TO A POWER AND MULTIPLIED BY A COEFFICIENT. IN POLYNOMIAL, VARIABLES HAVE NON-NEGATIVE INTEGRAL EXPONENTS.

     A POLYNOMIAL THAT CONTAINS ONLY ONE VARIABLE, SAY X, IS KNOWN AS POLYNOMIAL IN ONE VARIABLE X.

     HIGHEST EXPONENT OF POLYNOMIAL IS CALLED DEGREE OF THE POLYNOMIAL.

LINEAR EQUATION

      A STATEMENT OF EQUALITY WHICH INVOLVES LITERALS / VARIABLES IS CALLED EQUATION.

      AN EQUATION IN WHICH THE HIGHEST DEGREE OF THE VARIABLE IS ONE IS CALLED LINEAR EQUATION.

LINEAR EQUATION IN ONE VARIABLE

      IF ONLY ONE VARIABLE IS INVOLVED, THEN IT IS CALLED LINEAR EQUATION IN ONE VARIABLE.

      AN EQUATION OF THE FORM AX+B=C, WHERE A,B,C ARE NUMBERS, A≠0 AND ‘X’ IS A VARIABLE

LINEAR EQUATION IN TWO VARIABLES

      A LINEAR EQUATION WITH TWO VARIABLES IS CALLED LINEAR EQUATION IN TWO VARIABLE SAY X AND Y BUT THE HIGHEST DEGREE OF EACH VARIABLE IS ONE. LINEAR EQUATION OF THE FORM AX+BY+C=0 IS CALLED LINEAR EQUATION IN TWO VARIABLES.

   

QUADRATIC EQUATION

      A POLYNOMIAL OF DEGREE TWO IS CALLED QUADRATIC POLYNOMIAL

      GENERAL FORM OF QUADRATIC POLYNOMIAL IS AX2+BX+C, A,B,C ARE REAL NUMBERS AND A≠0.

      IF FOR X= Α, Α Ε R, THE VALUE OF QUADRATIC POLYNOMIAL BECOMES ZERO, THEN Α IS CALLED ZERO OF THE QUADRATIC POLYNOMIAL.

      THE EQUATION AX2+BX+C=0, A≠0 IS CALLED QUADRATIC EQUATION.

      IF Α AND Β ARE TWO ZEROS OF  AX2+BX+C=0, THEN WE SAY THAT Α AND Β ARE ROOTS OF THE QUADRATIC EQUATION.