UNDERSTANDING ENGINEERING MATHEMATICS
CONTENTS NUMBER AND ARITHMETIC REVIEW REVISION REINFORCEMENT APPLICATIONS ANSWERS TO REINFORCEMENT EXERCISES ALGEBRA FUNCTIONS AND SERIES EXPONENTIAL AND LOGARITHM FUNCTIONS GEOMETRY OF LINES, TRIANGLES AND CIRCLES TRIGONOMETRY COORDINATE GEOMETRY TECHNIQUES OF DIFFERENTIATION TECHNIQUES OF INTEGRATION APPLICATIONS OF DIFFERENTIATION AND INTEGRATION VECTORS INTRODUCTION – REPRESENTATION OF A VECTOR QUANTITY VECTORS AS ARROWS ADDITION AND SUBTRACTION OF VECTORS RECTANGULAR CARTESIAN COORDINATES IN THREE DIMENSIONS DISTANCE IN CARTESIAN COORDINATES DIRECTION COSINES AND RATIOS ANGLE BETWEEN TWO LINES THROUGH THE ORIGIN BASIS VECTORS PROPERTIES OF VECTORS THE SCALAR PRODUCT OF TWO VECTORS THE VECTOR PRODUCT OF TWO VECTORS VECTOR FUNCTIONS DIFFERENTIATION OF VECTOR FUNCTIONS COMPLEX NUMBERS WHAT ARE COMPLEX NUMBERS? THE ALGEBRA OF COMPLEX NUMBERS COMPLEX VARIABLES AND THE ARGAND PLANE MULTIPLICATION IN POLAR FORM DIVISION IN POLAR FORM EXPONENTIAL FORM OF A COMPLEX NUMBER DE MOIVRE’S THEOREM FOR INTEGER POWERS DE MOIVRE’S THEOREM FOR FRACTIONAL POWERS MATRICES AND DETERMINANTS AN OVERVIEW OF MATRICES AND DETERMINANTS DE?NITION OF A MATRIX AND ITS ELEMENTS ADDING AND MULTIPLYING MATRICES DETERMINANTS CRAMER’S RULE FOR SOLVING A SYSTEM OF LINEAR EQUATIONS THE INVERSE MATRIX EIGENVALUES AND EIGENVECTORS ANALYSIS FOR ENGINEERS – LIMITS, SEQUENCES, ITERATION, SERIES AND ALL THAT CONTINUITY AND IRRATIONAL NUMBERS LIMITS SOME IMPORTANT LIMITS CONTINUITY THE SLOPE OF A CURVE INTRODUCTION TO IN?NITE SERIES IN?NITE SEQUENCES ITERATION IN?NITE SERIES TESTS FOR CONVERGENCE IN?NITE POWER SERIES ORDINARY DIFFERENTIAL EQUATIONS INTRODUCTION DE?NITIONS FIRST ORDER EQUATIONS – DIRECT INTEGRATION AND SEPARATION OF VARIABLES LINEAR EQUATIONS AND INTEGRATING FACTORS SECOND ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS THE INHOMOGENEOUS EQUATION FUNCTIONS OF MORE THAN ONE VARIABLE – PARTIAL DIFFERENTIATION FUNCTION OF TWO VARIABLES PARTIAL DIFFERENTIATION HIGHER ORDER DERIVATIVES THE TOTAL DIFFERENTIAL AN APPRECIATION OF TRANSFORM METHODS THE LAPLACE TRANSFORM LAPLACE TRANSFORMS OF THE ELEMENTARY FUNCTIONS PROPERTIES OF THE LAPLACE TRANSFORM THE INVERSE LAPLACE TRANSFORM SOLUTION OF INITIAL VALUE PROBLEMS BY LAPLACE TRANSFORM LINEAR SYSTEMS AND THE PRINCIPLE OF SUPERPOSITION ORTHOGONALITY RELATIONS FOR TRIGONOMETRIC FUNCTIONS NUMBER AND ARITHMETIC IN THIS CHAPTER WE REVIEW THE KEY FEATURES OF ELEMENTARY NUMBERS AND ARITHMETIC THE TOPICS COVERED ARE THOSE FOUND TO BE MOST USEFUL LATER ON PREREQUISITES IT WILL BE HELPFUL IF YOU KNOW SOMETHING ABOUT: SIMPLE TYPES OF NUMBERS SUCH AS INTEGERS, FRACTIONS, NEGATIVE NUMBERS, DECIMALS THE CONCEPTS OF ‘GREATER THAN’ AND ‘LESS THAN’ ELEMENTARY ARITHMETIC: ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION POWERS AND INDICES NOTATION, 23 = 2 × 2 × 2, FOR EXAMPLE HOW TO CONVERT A SIMPLE FRACTION TO A DECIMAL AND VICE VERSA OBJECTIVES IN THIS CHAPTER YOU WILL ?ND: DIFFERENT TYPES OF NUMBERS AND THEIR PROPERTIES (PARTICULARLY ZERO) THE USE OF INEQUALITY SIGNS HIGHEST COMMON FACTORS AND LOWEST COMMON DENOMINATORS MANIPULATION OF NUMBERS (BODMAS) HANDLING FRACTIONS FACTORIAL (N!) AND COMBINATORIAL NCR OR N R NOTATION POWERS AND INDICES DECIMAL NOTATION ESTIMATION OF NUMERICAL EXPRESSIONS MOTIVATION YOU MAY NEED THE MATERIAL OF THIS CHAPTER FOR: NUMERICAL MANIPULATION AND CALCULATION IN ENGINEERING APPLICATIONS CHECKING AND USING SCIENTI?C FORMULAE ILLUSTRATING AND CHECKING RESULTS USED LATER IN MATHEMATICS STATISTICAL CALCULATIONS NUMERICAL ESTIMATION AND ‘BACK OF AN ENVELOPE’ CALCULATIONS 27 ?? A FOR EACH NUMBER CHOOSE ONE OR MORE DESCRIPTIONS FROM THE FOLLOWING: (A) INTEGER, (B) NEGATIVE, (C) RATIONAL NUMBER (FRACTION), (D) REAL, (E) IRRATIONAL, (F) DECIMAL, (G) PRIME (I) IS DONE AS AN EXAMPLE (I) ?1 (A, B, C, D) (II) (III) (IV) (V) (VIII) (IX) (X) ? (XII) ? B WHICH OF THE FOLLOWING DESCRIPTIONS APPLY TO THE EXPRESSIONS IN (I)–(X) BELOW? (A) IN?NITE (B) DOES NOT EXIST (C) NEGATIVE (D) ZERO (E) ?NITE (F) NON-ZERO (I) 0 × 1 (D, E) (II) 0 + (VII) (VIII) 3 × 0 + (X) 27 ?? EXPRESS SYMBOLICALLY: (I) X IS A POSITIVE, NON-ZERO, NUMBER (X > 0) (II) X LIES STRICTLY BETWEEN 1 AND (III) X LIES STRICTLY BETWEEN ?1 AND (IV) X IS EQUAL TO OR GREATER THAN ?2 AND IS LESS THAN (V) THE ABSOLUTE VALUE OF X IS LESS THAN 2 28 ?? A EXPRESS IN TERMS OF PRIME FACTORS (I) 15 (= 3 × 5) (II) (VI) B FIND THE HIGHEST COMMON FACTOR (HCF) OF EACH OF THE FOLLOWING SETS OF NUMBERS (I) 24, 30 (6) (II) 27, (III) 28, (IV) 12, 54, (V) 3, 6, 15, C FIND THE LOWEST COMMON MULTIPLE (LCM) OF EACH OF THE FOLLOWING SETS OF NUMBERS (I) 3, 7 (21) (II) 3, (III) 12, (IV) 3, 5, (V) 2, 4, 28 ?? EVALUATE (I) 2 + 3 ?7 (= ?2) (II) 4 × 3 ÷ (III) 3 + 2 × (IV) (3 + 2) × (V) 3 + (2 × 5) (VI) 18 ÷ 2 × 28 ?? A SIMPLIFY (I) (II) 3 × 5 × 4 ÷ 2 + (IX) 1 + 4 + B IF THE NUMBERS A AND B ARE IN THE RATIO A : B = 3 : 2 AND A = 6, WHAT IS B? (VII) (?AB2)3/A2B (VIII) 22 B EXPRESS IN TERMS OF SIMPLE SURDS SUCH AS ? 2, ? 3, ETC (I) ? 50 (= ? 2) (II) ? 72 ? ? (IV) ? (V) ? (VI) ? 3 + ? 3 ? ? 30 ?? A EXPRESS IN DECIMAL FORM (A) (C) (D) D WRITE THE NUMBERS IN C TO THREE SIGNI?CANT ?GURES 8 892 × 1 2 REVISION 1 2 1 TYPES OF NUMBERS ? 27 ? NUMBERS CAN BE CLASSI?ED INTO DIFFERENT TYPES: NATURAL NUMBERS ZERO DIRECTED NUMBERS INTEGERS RATIONAL NUMBERS (FRACTIONS) IRRATIONAL NUMBERS REAL NUMBERS COMPLEX NUMBERS (?CHAPTER 12) THE COUNTING NUMBERS 1, 2, 3, 4, ARE CALLED NATURAL NUMBERS ZERO, 0, IS REALLY IN A CLASS OF ITS OWN – WE ALWAYS HAVE TO BE CAREFUL WITH IT IT IS AN INTEGER AND ALSO, OF COURSE, A REAL NUMBER ESSENTIALLY, ZERO ENABLES US TO DE?NE NEGATIVE NUMBERS THUS, THE NEGATIVE OF 3 IS THE NUMBER DENOTED N = ?3 SATISFYING 3 + N = THIS ENABLES US TO ‘COUNT IN OPPOSITE DIRECTIONS’ USING DIRECTED OR NEGATIVE NUMBERS ?1, ?2, ?3, ?4, THE FULL SET OF NUMBERS { ?4, ?3, ?2, ?1, 0, 1, 2, 3, 4, } IS CALLED THE SET OF INTEGERS NUMBERS THAT CAN BE WRITTEN IN THE FORM: INTEGER NON-ZERO INTEGER E G INCLUDING INTEGERS, SUCH AS 6 = A ALL NUMBERS HERE ARE REAL, SO D APPLIES TO THEM ALL (I) ?1 IS A NEGATIVE NATURAL NUMBER, I E AN INTEGER; (A, B, C, D) (II) 2 IS A RATIO OF INTEGERS AND IS THEREFORE RATIONAL; (C, D) (III) 0 IS AN INTEGER – THE ONLY ONE THAT IS ITS OWN NEGATIVE; (A, C, D) (IV) 7 IS A NATURAL NUMBER AND AN INTEGER IT IS IN FACT ALSO A PRIME NUMBER – THAT IS, ONLY DIVISIBLE BY ITSELF OR 1 (SECTION 1 2 3) IT IS ALSO AN ODD NUMBER (CANNOT BE EXACTLY DIVIDED BY 2); (A, C, D, G) (V) 4 IS A RATIONAL NUMBER – A PROPER FRACTION (SECTION 1 2 5); (B, C, D) (VII) 0 73 IS ACTUALLY A DECIMAL REPRESENTATION OF A RATIONAL NUMBER 0 73 = SOMETIMES CALLED A DECIMAL FRACTION, BUT USUALLY SIMPLY A DECIMAL (SECTION 1 2 8); (C, D, F) (VIII) 11 IS A NATURAL NUMBER AND AN INTEGER – LIKE 7 IT IS ALSO PRIME, AND IS ALSO ODD AS ANY PRIME GREATER THAN 2 MUST BE; (A, C, D, G) (IX) 8 IS ANOTHER NATURAL NUMBER AND INTEGER – BUT IT IS NOT PRIME, SINCE IT CAN BE WRITTEN AS 2 × 2 × 2 = 23 (SECTION 1 2 7) IT IS ALSO AN EVEN NUMBER; (A, C, D) (X) THE SQUARE ROOT OF 2, ? 2, IS NOT A RATIONAL NUMBER THIS CAN BE SHOWN BY ASSUMING THAT ? 2 = M N U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S WHERE M AND N ARE TWO INTEGERS AND DERIVING A CONTRADICTION ? (B, C, D, F) (XII) ?, THE RATIO OF THE CIRCUMFERENCE OF A CIRCLE TO ITS DIAMETER, IS NOT A RATIONAL NUMBER – IT IS AN IRRATIONAL NUMBER THAT IS, IT CANNOT BE WRITTEN AS A FRACTION 22/7, FOR EXAMPLE, IS JUST AN APPROXIMATION TO ?; (D, E) B (I) 0 × 1 = 0, I E ZERO – WHICH OF COURSE IS ALSO ?NITE (D, E) (II) 0 + 1 = 1, ?NITE, NON-ZERO (E, F) (III) 0 DOES NOT EXIST – IT IS NOT IN?NITE, NEGATIVE, ZERO, ?NITE OR NON- ZERO – IT JUST DOES NOT EXIST (B) (IV) 2 ?0 = 2, ?NITE AND NON-ZERO (E, F) (V) 02 = 0 × 0 = 0, ZERO AND ?NITE (D, E) (VI) 0 ?1 = ?1, NEGATIVE, ?NITE, NON-ZERO (C, E, F) (VII) 0 DOES NOT EXIST (YOU CAN’T ‘CANCEL’ THE ZEROS!) IT IS NOT IN?NITE, NEGATIVE, ZERO, ?NITE OR NON-ZERO – IT JUST DOES NOT EXIST (B) (VIII) BECAUSE OF THE 0 THE EXPRESSION 3 × 0 + 0 DOES NOT EXIST (B) (IX) 0 AGAIN, DOES NOT EXIST (B) (X) 2 = 1 – NO PROBLEM HERE, ?NITE AND NON-ZERO (E, F) NOTE THAT NONE OF THE NUMBERS IN B IS REFERRED TO AS ‘IN?NITE’ 1 2 2 USE OF INEQUALITY SIGNS ? 27 ? THE REAL NUMBERS ARE ORDERED THAT IS, WE CAN ALWAYS SAY WHETHER ONE NUMBER A IS LESS THAN, EQUAL TO, OR GREATER THAN ANOTHER GIVEN NUMBER B TO DENOTE THIS WE USE THE ‘COMPARATOR’ SYMBOLS OR INEQUALITIES, < AND ?, > AND ? A > B MEANS A IS GREATER THAN B; A < B MEANS A IS LESS THAN B THUS 6 > 5, 4 < 5 A ?B MEANS A IS GREATER THAN OR EQUAL TO B, AND SIMILARLY A ?B MEANS A IS LESS THAN OR EQUAL TO B BE VERY CAREFUL TO DISTINGUISH BETWEEN, FOR EXAMPLE A > B AND A ?B SOMETIMES IT IS ALSO USEFUL TO USE THE ‘NOT EQUAL TO’ SYMBOL, ?= CARE IS NEEDED WHEN CHANGING SIGNS AND FORMING RECIPROCALS WITH INEQUALITIES FOR EXAMPLE, IF A > B > 0, THEN ?A < ?B AND A < B HOWEVER, IF A > 0 > B THEN ?A < ?B IS STILL TRUE, BUT A > B TRY A FEW NUMERICAL EXAMPLES TO CHECK THESE STATEMENTS MOST U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S OF US ?ND INEQUALITIES DIF?CULT TO HANDLE AND THEY REQUIRE A LOT OF PRACTICE HOWEVER, IN THIS BOOK WE WILL NEED ONLY THE BASIC PROPERTIES OF INEQUALITIES WE WILL SAY MORE ABOUT INEQUALITIES IN SECTION 3 2 6 OFTEN WE WISH TO REFER TO THE POSITIVE OR ABSOLUTE VALUE OF A NUMBER X (FOR EXAMPLE IN A RECTI?ED SINE WAVE) WE DENOTE THIS BY THE MODULUS OF X, |X| FOR EXAMPLE | ?4| = BY DE?NITION |X| IS NEVER NEGATIVE, SO |X| ?0 ALSO, NOTE THAT |X| < A MEANS ?A < X < A FOR EXAMPLE: |X| < MEANS ?3 < X < (I) ‘X IS A POSITIVE NON-ZERO NUMBER’ IS EXPRESSED BY X > (II) ‘X LIES STRICTLY BETWEEN 1 AND 2’ IS EXPRESSED BY 1 < X < (III) ‘X LIES STRICTLY BETWEEN ?1 AND 3’ IS EXPRESSED BY ?1 < X < (IV) ‘X IS EQUAL TO OR GREATER THAN ?2 AND IS LESS THAN 2’ IS EXPRESSED BY ?2 ?X < (V) IF THE ABSOLUTE VALUE OF X IS LESS THAN 2 THEN THIS MEANS THAT IF X IS POSITIVE THEN 0 ?X < 2, BUT IF X IS NEGATIVE THEN WE MUST HAVE ?2 < X ?0 SO, COMBINING THESE WE MUST HAVE ?2 < X < 2 THIS CAN ALSO BE EXPRESSED IN TERMS OF THE MODULUS AS |X| < 2 1 2 3 HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE ? 28 ? A PRIME NUMBER IS A POSITIVE INTEGER WHICH CANNOT BE EXPRESSED AS A PRODUCT OF TWO OR MORE SMALLER DISTINCT POSITIVE INTEGERS THAT IS, A PRIME NUMBER CANNOT BE DIVIDED EXACTLY BY ANY INTEGER OTHER THAN 1 OR ITSELF FROM THE DE?NITION, 1 IS NOT A PRIME NUMBER 6, FOR EXAMPLE, IS NOT A PRIME, SINCE IT CAN BE WRITTEN AS 2 × 3 THE NUMBERS 2 AND 3 ARE CALLED ITS (PRIME) FACTORS ANOTHER WAY OF DE?NING A PRIME NUMBER IS TO SAY THAT IT IS HAS NO INTEGER FACTORS OTHER THAN 1 AND ITSELF THERE ARE AN IN?NITE NUMBER OF PRIME NUMBERS: 2, 3, 5, 7, 11, 13, BUT NO FORMULA FOR THE NTH PRIME HAS BEEN DISCOVERED PRIME NUMBERS ARE VERY IMPORTANT IN THE THEORY OF CODES AND CRYPTOGRAPHY THEY ARE ALSO THE ‘BUILDING BLOCKS’ OF NUMBERS, SINCE ANY GIVEN INTEGER CAN BE WRITTEN UNIQUELY AS A PRODUCT OF PRIMES: 12 = 2 × 2 × 3 = THIS IS CALLED FACTORISING THE INTEGER INTO ITS PRIME FACTORS IT IS AN IMPORTANT OPERATION, FOR EXAMPLE, IN COMBINING FRACTIONS A (I) 15 = 3 × (II) 21 = 3 × (III) 60 = 3 × 20 = 3 × 4 × 5 = 2 × 2 × 3 × 5 = 22 × 3 × (IV) 121 = 11 × (V) 405 = 5 × 81 = 5 × 9 × 9 = 5 × 3 × 3 × 3 × 3 = 34 × (VI) 1024 = 4 × 256 = 4 × 16 × = 4 × 4 × 4 × 4 × 4 = 2 × 2 × 2 × 2 × 2 × 2 × × 2 × 2 × = 210 (OR, ANTICIPATING THE RULES OF INDICES = 45 = (22)5 = 210) (VII) 221 = 13 × NOTICE THAT THERE MAY BE MORE THAN ONE WAY OF FACTORISING, BUT THAT THE ?NAL RESULT IS ALWAYS THE SAME YOU MAY ALSO HAVE NOTICED THAT IT GETS INCREASINGLY DIF?CULT TO FACTORISE, COMPARED TO MULTIPLYING – THUS IN (VII), IT IS SO MUCH EASIER TO MULTIPLY 13 × 17 THAN TO DISCOVER THESE FACTORS FROM 221 THIS FACT IS ACTUALLY THE KEY IDEA BEHIND MANY POWERFUL CODING SYSTEMS – THE TRAP-DOOR PRINCIPLE – IN SOME CASES IT IS MUCH EASIER DOING A MATHEMATICAL OPERATION THAN UNDOING IT! B (I) 24, 30 IN THIS CASE IT IS CLEAR THAT THE LARGEST INTEGER THAT EXACTLY DIVIDES THESE TWO IS 6 AND SO THE HCF OF 24 AND 30 IS 6 (II) 27, 99 AGAIN THE FAIRLY OBVIOUS ANSWER HERE IS 9 (III) 28, 98 PERHAPS NOT SO OBVIOUS, SO SPLIT EACH INTO PRIME FACTORS: 28 = 4 × 7 = 2 × 2 × 98 = 2 × 49 = 2 × 7 × FROM WHICH WE SEE THAT THE HCF IS 2 × 7 = 14 (IV) 12, 54, 78 SPLITTING INTO PRIME FACTORS WILL AGAIN GIVE THE ANSWER HERE, BUT NOTICE A SHORT CUT: 2 CLEARLY DIVIDES THEM ALL, LEAVING 6, 27, 39 3 DIVIDES ALL OF THESE LEAVING 2, 9, 13 THESE CLEARLY HAVE NO FACTORS IN COMMON (EXCEPT 1) AND SO WE ARE DONE, AND THE HCF IS 2 × 3 = 6 U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S (V) 3, 6, 15, 27 HERE 3 IS THE ONLY NUMBER THAT DIVIDES THEM ALL AND IS THE HCF IN THIS CASE C (I) SINCE 3, 7 ARE BOTH PRIMES, THE LCM IS SIMPLY THEIR PRODUCT, 21 (II) 9 = 3 × 3, SO 3 AND 9 BOTH DIVIDE 9 AND THERE IS NO SMALLER NUMBER THAT DOES SO THE LCM IS THUS 9 (III) SINCE 3 × 12 = 36 AND 2 × 18 = 36, WE SEE DIRECTLY THAT THE LCM IS 36 (IV) WE HAVE TO DEAL WITH 3, 5, AND 9 = 32 THE LCM MUST CONTAIN AT LEAST TWO FACTORS OF 3 AND ONE OF 5 SO THE LCM IS 5 × 32 = 45 (V) 2, 4, 6 2 DIVIDES 4, SO WE MUST HAVE A FACTOR OF 4 IN THE LCM ALSO, 4 AND 6 BOTH DIVIDE 12, BUT NO SMALLER NUMBER AND SO THE LCM IS 12 1 2 4 MANIPULATION OF NUMBERS ? 28 ? MUCH OF ARITHMETIC IS BASED ON JUST A FEW OPERATIONS: ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION, SATISFYING A SMALL NUMBER OF RULES THE EXTENSION OF THESE RULES TO INCLUDE SYMBOLS AS WELL AS NUMBERS LEADS US ON TO ALGEBRA (CHAPTER 2) ADDITION, DENOTED +, PRODUCES THE SUM OF TWO NUMBERS: 6 + 3 = 9 = 3 + (ADDITION IS ‘COMMUTATIVE’) SUBTRACTION, DENOTED ?, PRODUCES THE DIFFERENCE OF TWO NUMBERS: 6 ?3 = 3 = ?(3 ?6) (MINUS SIGN CHANGES SIGNS IN BRACKETS) MULTIPLICATION, DENOTED BY A × B OR SIMPLY AS AB IN ALGEBRA, PRODUCES THE PRODUCT OF TWO NUMBERS: 6 × 3 = (6)(3) = 18 = 3 × (MULTIPLICATION IS COMMUTATIVE) A B IS SOMETIMES USED TO DENOTE THE PRODUCT BUT CAN BE CONFUSED WITH DECIMAL NOTATION IN ARITHMETIC DIVISION, DENOTED BY A ÷ B OR A/B OR, BETTER, A B , PRODUCES THE QUOTIENT OF TWO NUMBERS: 6 ÷ 3 = 6/3 = 3 = (OF COURSE, 6 ÷ 3 ?= 3 ÷ 6!) NOTE THAT ÷ AND / ARE VERY RARELY USED IN WRITTEN CALCULATIONS, WHERE WE USE THE FORM 3 UNLESS WE NEED TO CALL INTO PLAY ÷ OR / BECAUSE WE HAVE A LARGE NUMBER OF DIVISIONS ALSO NOTICE HOW WE HAVE SIMPLI?ED THE QUOTIENT TO 2 WE ALWAYS SIMPLIFY SUCH FRACTIONS TO LOWEST FORM WHENEVER WE CAN (?12) THE WAY THE ABOVE AND OTHER ARITHMETIC OPERATIONS ARE COMBINED IS ACCORDING TO A SET OF CONVENTIONAL PRECEDENCES – THE RULES OF ARITHMETIC THUS WE ALWAYS PERFORM MULTI- PLICATION BEFORE ADDITION, SO: 2 × 3 + 5 = 6 + 5 = U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S BRACKETS CAN BE USED IF WE WANT TO OVERRIDE SUCH RULES FOR EXAMPLE: 2 × (3 + 5) = 2 × 8 = FOR EXAMPLE (?2)(?3) = (?1)(?1)6 = (II) 4 × 3 ÷ 2 = 12 ÷ 2 = (III) 3 + 2 × 5 = 3 + 10 = (IV) (3 + 2) × 5 = 5 × 5 = (V) 3 + (2 × 5) = 3 + 10 = 13 IN THIS CASE THE BRACKETS ARE ACTUALLY UNNECESSARY, SINCE THE BODMAS RULES TELL US TO EVALUATE THE MULTI- PLICATION ?RST (VI) 18 ÷ 2 × 3 = 9 × 3 = 27 FOLLOWING THE CONVENTION OF WORKING FROM LEFT TO RIGHT (VII) 18 ÷ (2 × 3) = 18 ÷ 6 = 3 BECAUSE THE BRACKETS OVERRIDE THE LEFT TO RIGHT RULE U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S (VIII) ?2 ?(4 ?5) = ?2 ?(?1) = ?2 + (IX) (4 ÷ (?2)) × 3 ?4 = NOTICE THE CARE TAKEN IN THESE EXAMPLES, SPELLING OUT EACH STEP THIS MAY SEEM TO BE A BIT LABOURED, BUT I WOULD ENCOURAGE YOU TO TAKE SIMILAR CARE, PARTICULARLY WHEN WE COME TO ALGEBRA SLIPS WITH BRACKETS AND SIGNS CROP UP FREQUENTLY IN MOST PEOPLE’S CALCULA- TIONS (MINE INCLUDED!) WHEREAS THIS MAY ONLY LOSE YOU ONE OR TWO MARKS IN AN EXAM, IN REAL LIFE, AN ERROR IN SIGN CAN CONVERT A STABLE CONTROL SYSTEM INTO AN UNSTABLE ONE, OR A HEALTHY BANK BALANCE INTO AN OVERDRAFT 1 2 5 HANDLING FRACTIONS ? 28 ? A FRACTION OR RATIONAL NUMBER IS ANY QUANTITY OF THE FORM M N N ?= WHERE M, N ARE INTEGERS BUT N IS NOT EQUAL TO 0 IT IS OF COURSE ESSENTIAL THAT N ?= 0, BECAUSE AS NOTED IN SECTION 1 2 1 DIVISION BY ZERO IS NOT DE?NED M IS CALLED THE NUMERATOR N IS THE DENOMINATOR IF M ?N THE FRACTION IS SAID TO BE IMPROPER, AND IF M < N IT IS PROPER A NUMBER EXPRESSED IN THE FORM 2 2 (MEANING 2 + 2) IS CALLED A MIXED FRACTION IN MATHEMATICAL EXPRESSIONS IT IS BEST TO AVOID THIS FORM ALTOGETHER AND WRITE IT AS A VULGAR FRACTION, 2, INSTEAD, OTHERWISE IT MIGHT BE MISTAKEN FOR ‘2 × 2 = 1’, AND IT IS ALSO MORE DIF?CULT TO DO CALCULATIONS SUCH AS MULTIPLICATION AND DIVISION USING MIXED FRACTIONS THE NUMERATOR AND DENOMINATOR OF A FRACTION MAY HAVE COMMON FACTORS THESE MAY BE CANCELLED TO REDUCE THE FRACTION TO ITS SIMPLEST OR ‘LOWEST’ FORM 12 = 2 × 3 × 4 = 4 = 1 × 2 × 2 = EACH OF THESE FORMS ARE EQUIVALENT FRACTIONS, BUT CLEARLY THE LAST ONE IS THE SIMPLEST HOWEVER, SOMETIMES ONE OF THE OTHER FORMS MAY BE CONVENIENT FOR PARTICULAR PURPOSES, SUCH AS ADDING FRACTIONS A VERY COMMON FRACTION WHERE WE TEND NOT TO CANCEL DOWN IN U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S THIS WAY IS THE PERCENTAGE THUS WE USUALLY EXPRESS 32/100 AS ‘32 PERCENT’ RATHER THAN AS ITS EQUIVALENT, ‘8 OUT OF 25’! FRACTIONS ARE MULTIPLIED ‘TOP BY TOP AND BOTTOM BY BOTTOM’ AS YOU MIGHT EXPECT: M N × P Q = MP NQ (N, Q ?= 0) E G 2 × 11 = WITH P AND Q ALSO ANY INTEGERS THERE MAY, OF COURSE, BE COMMON FACTORS TO CANCEL DOWN, AS FOR EXAMPLE IN 7 = THE INVERSE OR RECIPROCAL OF A FRACTION IS OBTAINED BY TURNING IT UPSIDE DOWN M N = N M E G WHERE BOTH M AND N MUST BE NON-ZERO SO DIVIDING BY A VULGAR FRACTION IS DONE BY INVERTING IT AND MULTIPLYING: P Q ÷ M N = P Q M N = P Q × N M = NP MQ E G 2 ÷ 6 = 14 = MULTIPLICATION AND DIVISION OF FRACTIONS ARE THEREFORE QUITE SIMPLE ADDITION AND SUBTRAC- TION ARE LESS SO TWO FRACTIONS WITH THE SAME DENOMINATOR ARE EASILY ADDED OR SUBTRACTED: M N ± P N = M ± P N E G 2 = SO TO ADD AND SUBTRACT FRACTIONS IN GENERAL WE REWRITE THEM ALL WITH THE SAME COMMON DENOMINATOR, WHICH IS THE LOWEST COMMON MULTIPLE OF ALL THE DENOMINATORS FOR EXAMPLE 3 = 3 × ?4 × IN THE EXAMPLE, 12 IS THE LCM OF 3 AND 4 AN ELECTRICAL EXAMPLE – RESISTANCES IN PARALLEL THREE RESISTANCES R1, R2, R3 CONNECTED IN PARALLEL ARE EQUIVALENT TO A SINGLE RESISTANCE R GIVEN BY R = SO, FOR EXAMPLE IF R1 = 2, R2 = 2, R3 = 2 THEN 2 + 2 + 3 (UNITS OF INVERSE OHMS) U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S OR, WITH 6 THE LCM OF 2 AND 6 + 6 (?1) AND SO THE EQUIVALENT RESISTANCE IS R = FINALLY, ON FRACTIONS, RECALL THE IDEAS OF RATIO AND PROPORTION THESE ARE MET EARLY IN OUR MATHEMATICAL EDUCATION, YET OFTEN CONTINUE TO CONFUSE US LATER IN LIFE SPECI?CALLY, IT IS NOT UNCOMMON TO SEE SOMEONE MAKE ERRORS SUCH AS: ‘ A B = 3 MEANS A = 1 AND B = 3’ SO IT IS WORTH HAVING A QUICK REVIEW OF THIS TOPIC THE NOTATION A : B IS USED TO INDICATE THAT THE NUMBERS A AND B ARE IN A CERTAIN RATIO OR PROPORTIONALITY TO EACH OTHER A : B = 1 SIMPLY MEANS THAT A B = AND THIS MOST CERTAINLY DOES NOT MEAN A = 1 AND B = 3 FOR EXAMPLE 3 : 9 = 2 : 6 = 7 : 21 = 1 ALL A : B = 1 : 3 MEANS IS THAT A = B A (I) 6 = 2 × 2 × 3 = 9 = 9 × U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S (III) (CANCELLING 7 FROM TOP AND BOTTOM, AS IN (III)) = 1 × 5 × 2 = 5 = 4 × 4 = 2 × 3 + ON MULTIPLYING TOP AND BOTTOM APPROPRIATELY TO GET THE COMMON DENOMINATOR IN BOTH FRACTIONS, = 6 = 3 + 15 ?5 × HERE WE FOUND THE LCM OF 2 AND 3 (6) AND PUT EVERYTHING OVER THIS, INCLUDING THE 1 (X) WE WANT THE LCM OF 3, 4, 8 THIS IS 24, SO 8 = 2 × ?3 × = 16 ?18 + U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S B IF A : B = 3 : 2 THEN A B = 2 SO B = 2A 3 SO, IF A = 6 THEN B = 2 × = 4 1 2 6 FACTORIAL AND COMBINATORIAL NOTATION – PERMUTATIONS AND COMBINATIONS ? 29 ? THE FACTORIAL NOTATION IS A SHORTHAND FOR A COMMONLY-OCCURRING EXPRESSION INVOLVING POSI- TIVE INTEGERS IT PROVIDES SOME NICE PRACTICE IN MANIPULATION OF NUMBERS AND FRACTIONS, AND GENTLY INTRODUCES ALGEBRAIC IDEAS IF N IS SOME POSITIVE INTEGER ?1 THEN WE WRITE N! = N(N ?1)(N ?2) 2 × READ AS ‘N FACTORIAL’ FOR EXAMPLE 5! = 5 × 4 × 3 × 2 × 1 = NOTICE THAT THE FACTORIAL EXPRESSION YIELDS LARGE VALUES VERY QUICKLY, THAT IS N! INCREASES RAPIDLY WITH N IN CALCULATIONS INVOLVING FACTORIALS IT IS OFTEN USEFUL TO REMEMBER SUCH RESULTS AS 10! = 10 × 9 × 8 × 7! I E WE CAN PICK OUT A LOWER FACTORIAL IF THIS IS CONVENIENT, AND THIS OFTEN HELPS WITH CANCELLATIONS IN EXPRESSIONS CONTAINING FACTORIALS NOTE THAT 1! = 1 ALSO, WHILE THE ABOVE DE?NITION DOES NOT DE?NE 0!, THE CONVENTION IS ADOPTED THAT 0! = THE FACTORIAL NOTATION IS USEFUL IN THE BINOMIAL THEOREM (?71) AND IN STATISTICS IT CAN BE USED TO COUNT THE NUMBER PERMUTATIONS OF N OBJECTS, I E THE NUMBER OF WAYS OF ARRANGING N OBJECTS IN A GIVEN ORDER: FIRST OBJECT CAN BE CHOSEN IN N WAYS SECOND OBJECT CAN BE CHOSEN IN (N ?1) WAYS THIRD OBJECT CAN BE CHOSEN IN (N ?2) WAYS LAST OBJECT CAN ONLY BE CHOSEN IN 1 WAY SO THE TOTAL NUMBER OF PERMUTATIONS OF N OBJECTS IS N × (N ?1) × (N ?2) 2 × 1 = N! NOTE THAT N! = N × (N ?1)! FOR 3 OBJECTS A, B, C, FOR EXAMPLE, THERE ARE 3! = 6 PERMUTATIONS, WHICH ARE: ABC, ACB, BAC, BCA, CAB, CBA U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S EACH OF THESE IS THE SAME COMBINATION OF THE OBJECTS A, B, C – THAT IS A SELECTION OF THREE OBJECTS IN WHICH ORDER IS NOT IMPORTANT NOW SUPPOSE WE SELECT JUST R OBJECTS FROM THE N EACH SUCH SELECTION IS A DIFFERENT COMBINATION OF R OBJECTS FROM N AN OBVIOUS QUESTION IS HOW MANY DIFFERENT PERMUTATIONS OF R OBJECTS CHOSEN FROM N CAN BE FORMED IN THIS WAY? THIS NUMBER IS DENOTED BY NPR IT MAY BE EVALUATED BY REPEATING THE PREVIOUS COUNTING PROCEDURE, BUT ONLY UNTIL WE HAVE CHOSEN R OBJECTS: THE ?RST MAY BE CHOSEN IN N WAYS THE SECOND MAY BE CHOSEN IN (N ?1) WAYS THE THIRD MAY BE CHOSEN IN (N ?2) WAYS THE RTH MAY BE CHOSEN IN (N ?(R ?1)) WAYS SO THE TOTAL NUMBER OF PERMUTATIONS WILL BE NPR = N × (N ?1) × (N ?2) × × (N ?R + 1) = N(N ?1)(N ?2) (N ?R + 1)(N ?R)(N ?R ?1) 2 × (N ?R)(N ?R ?1) 2 × = N! (N ?R)! FOR EXAMPLE THE NUMBER OF WAYS THAT WE CAN PERMUTE 3 OBJECTS CHOSEN FROM 5 DISTINCT OBJECTS IS 5P3 = 5 × 4 × 3 = 5 × 4 × 3 × 2 × = 5! (5 ?3)! = SINCE THE ORDER DOES NOT MATTER IN A COMBINATION, NPR WILL INCLUDE R! PERMUTATIONS OF THE SAME COMBINATIONS OF R DIFFERENT OBJECTS SO THE NUMBER OF COMBINATIONS OF R OBJECTS CHOSEN FROM N IS R! NPR = N! (N ?R)!R! THIS IS USUALLY DENOTED BY NCR (CALLED THE ‘N ?C ?R’ NOTATION) OR N R – ‘CHOOSE R OBJECTS FROM N’: NCR = N R = N! (N ?R)!R! E G 5C3 = = 5! (5 ?3)!3! = WHICH IS VERY USEFUL IN BINOMIAL EXPANSIONS (SEE SECTION 2 2 13) AND OTHER AREAS, SIMPLY AS A NOTATION, REGARDLESS OF ITS ‘COUNTING’ SIGNI?CANCE A (I) 3! = 3(3 ?1)(3 ?2) = 3 × 2 × 1 = (II) 6! = 6 × 5 × 4 × 3 × 2 × 1 = (III) 24! 23! = 24 × 23! 23! = (IV) 12! 9!3! = 12 × 11 × 10 × 9! 9!3! = 12 × 11 × B (I) (A) 3C2 = 3! (3 ?2)!2! = 3! 1!2! = (B) 6C4 = 6! (6 ?4)!4! = 6 × 5 × 4! 2!4! = (C) 6P3 = 6! 3! = 6 × 5 × 4 = (II) TWO LETTERS CAN BE CHOSEN FROM ABCD IN 4C2 = 4! 2!2! = 6 WAYS (III) THERE ARE 5! = 120 PERMUTATIONS OF ?VE DIFFERENT LETTERS 1 2 7 POWERS AND INDICES ? 29 ? POWERS, OR INDICES, PROVIDE, IN THE ?RST INSTANCE, A SHORTHAND NOTATION FOR MULTIPLYING A NUMBER BY ITSELF A GIVEN NUMBER OF TIMES: 2 × 2 = 2 × 2 × 2 = 2 × 2 × 2 × 2 = ETC FOR A GIVEN NUMBER A WE HAVE AN = A × A × A × × A N TIMES A IS CALLED THE BASE, N THE POWER OR INDEX A1 IS SIMPLY A BY CONVENTION WE TAKE A0 = (A ?= 0) WE INTRODUCE A?1 TO DENOTE THE RECIPROCAL A , SINCE THEN 1 = A × A = A1 × A = A1 × A?1 = A1?1 = A0 FOLLOWS IN GENERAL, A?N = AN FROM THESE DE?NITIONS WE CAN DERIVE THE RULES OF INDICES: AM × AN = AM+N U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S AM AN = AM?N (AM)N = AMN (AB)N = ANBN NOTE THAT FOR ANY INDEX N, 1N = 1 EXAMPLES 23 × 24 = 23+4 = 35 × 30 = (52)3 = (22)0 = 20 = 43/22 = (22)3/22 = 26/22 = A SQUARE ROOT OF A POSITIVE NUMBER A, IS ANY NUMBER THAT, WHEN SQUARED, YIELDS THE NUMBER A WE USE ?A TO DENOTE THE POSITIVE VALUE OF THE SQUARE ROOT (ALTHOUGH THE NOTATION HAS TO BE STRETCHED WHEN WE GET TO COMPLEX NUMBERS) FOR EXAMPLE 2 = ? SINCE 22 = SINCE ?2 = ? ? 4 ALSO SATIS?ES (?2)2 = 4, ? ? 4 IS ALSO A SQUARE ROOT OF 4 SO THE SQUARE ROOTS OF 4 ARE ± ? 4 = ±2 WE CAN SIMILARLY HAVE CUBE ROOTS OF A NUMBER A, WHICH YIELD A WHEN THEY ARE CUBED IF A IS POSITIVE THEN 3?A DENOTES THE POSITIVE VALUE OF THE CUBE ROOT FOR EXAMPLE 2 = 3? BECAUSE 23 = ? ) THIS IS NOT JUST A MATHEMATICAL NICETY ? 2 FOR EXAMPLE, IS THE DIAGONAL OF THE UNIT SQUARE, AND YET BECAUSE IT IS IRRATIONAL, IT CAN NEVER BE WRITTEN DOWN EXACTLY AS A RATIONAL NUMBER OR FRACTION ( ? 2 = 1 4142 IS, FOR EXAMPLE, ONLY AN APPROXIMATION TO ? 2 TO FOUR DECIMAL PLACES) IN TERMS OF INDICES, ROOTS ARE REPRESENTED BY FRACTIONAL INDICES, FOR EXAMPLE: ?A = A U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S AND IN GENERAL N?A = A N THIS ?TS IN WITH THE RULES OF INDICES, SINCE A N N = A N ×N = A1 = A FRACTIONAL POWERS SATISFY THE SAME RULES OF INDICES AS INTEGER POWERS – BUT THERE ARE SOME NEW FEATURES: MULTIPLICITY OF ROOTS: 22 = (?2)2 = NON-EXISTENCE OF CERTAIN ROOTS OF NEGATIVE NUMBERS: ??1 IS NOT A REAL NUMBER IRRATIONAL VALUES FOR ROOTS OF PRIMES AND THEIR MULTIPLES: ? 2 CANNOT BE EXPRESSED AS A FRACTION QUANTITIES SUCH AS ? 2, ? 3, CONTAINING SQUARE ROOTS OF PRIMES, ARE CALLED SURDS THE TERM ORIGINATES FROM THE GREEK WORD FOR MUTE, REFERRING TO A NUMBER THAT CANNOT ‘SPEAK’ ITS VALUE – BECAUSE ITS DECIMAL PART NEVER ENDS (SEE SECTION 1 2 8) IN MATHEMATICAL MANIPU- LATION SURDS ARE ALWAYS BEST LEFT AS THEY ARE – RETAINING THE ROOT SIGN ANY DECIMAL FORM FOR THEM WILL SIMPLY BE AN APPROXIMATION AS NOTED FOR ? 2 ABOVE USUALLY WE TRY TO MANIPULATE SURDS SO THAT THE RESULT IS THE SIMPLEST FORM, AND NONE REMAIN IN DENOMINATORS (ALTHOUGH WE WOULD NORMALLY WRITE, FOR EXAMPLE, SIN 45° = A (I) 2324 = 23+4 = 27 (LEAVE IT AS A POWER, LIKE THIS) (II) 33 = 34?3 = 31 = (III) (52)3 = 52×3 = (9 × 23) = 44 = = (?1)262(?1)3 (VIII) 22 = 2223 = THE STEPS TO WATCH OUT FOR IN SUCH PROBLEMS ARE THE HANDLING OF THE MINUS SIGNS AND BRACKETS, AND DEALING WITH THE NEGATIVE POWERS AND RECIP- ROCALS DON’T FORGET THAT AN, N ?0, IS NOT DE?NED FOR A = 0 B WE CAN GET A LONG WAY SIMPLY BY USING ? AB = ?A ? B (I) ? 50 = ? 25 × 2 = ? 52? 2 = (II) ? 72 ? ? 8 = ?36 × 2 ??4 × 2 = ? 2 = ? 3)3 = 33( ? 3)3 = ? 3 = 34? 3 = 4 × 2 = ? 4 × (VI) TO SIMPLIFY ? 3 + (SEE SECTION 2 2 1) AND THE REMOVAL OF SURDS BY SQUARING FOR A ? 3 ? ? 2 ON THE BOTTOM WE MULTIPLY TOP AND BOTTOM BY ? 3 + ? 2, USING: ( ? 3 ? ? 2)( ? 3 + ? 2) = ( ? 3)2 ?( ? 2)2 = 3 ?2 = U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S THUS: ? 3 + × ? 3 + ? ? 3 + ? = ? 3 + ? 3 + = 3 + = 7 + THIS TOPIC, POWERS AND INDICES, OFTEN GIVES BEGINNERS A LOT OF TROUBLE IF YOU ARE STILL THE SLIGHTEST BIT UNSURE, GO TO THE REINFORCEMENT EXERCISES FOR MORE PRACTICE – THERE IS NO OTHER WAY THIS IS, LITERALLY, POWER TRAINING! 1 2 8 DECIMAL NOTATION ? 30 ? YOU PROBABLY KNOW THAT 2 MAY BE REPRESENTED BY THE DECIMAL 0 5, 4 BY 0 25 AND SO ON IN FACT ANY REAL NUMBER, A, 0 ?A < 1, HAS A DECIMAL REPRESENTATION, WRITTEN A = 0 D1D2D3 WHERE EACH DI IS ONE OF THE DIGITS 0, 1, 2, , 9, AND THE SEQUENCE MAY NOT TERMINATE (SEE BELOW) THE TERM DECIMAL ACTUALLY REFERS TO THE BASE 10 AND REPRESENTS THE FACT THAT: A = D1 × 10?1 + D2 × 10?2 + D3 × 10?3 NOTE THE IMPORTANCE OF ‘PLACE VALUE’ HERE – THE VALUE OF EACH OF THE DIGITS DEPENDS ON ITS PLACE IN THE DECIMAL ANY REAL NUMBER CAN BE REPRESENTED BY AN INTEGER PART AND SUCH A DECIMAL PART IF, FROM SOME POINT ON THE DECIMAL CONSISTS OF A REPEATING STRING OF ONE OR MORE DIGITS, THEN THE DECIMAL IS SAID TO BE A REPEATING OR RECURRING DECIMAL ALL RATIONAL NUMBERS CAN BE REPRESENTED BY A ?NITE DECIMAL REPRESENTATION OR A RECURRING ONE IRRATIONAL NUMBERS CANNOT BE REPRESENTED IN THIS WAY AS A TERMINATING OR RECURRING DECIMAL – THUS THE DECIMAL (4 SF), 500 00 (5 SF) SOMETIMES NUMBERS ARE APPROXIMATED BY TERMINATING THE DIGITS AFTER A GIVEN NUMBER OF DIGITS AND REPLACING THEM WITH ZEROS IF THIS IS DONE WITH NO REGARD TO THE SIZE OF THE REMOVED DIGITS, THEN WE SAY THE NUMBER HAS BEEN ‘CHOPPED’ OR ‘TRUNCATED’ FOR EXAMPLE 324829 1 CHOPPED TO 3 SIGNI?CANT ?GURES IS 324000 ANOTHER, MORE ACCURATE, METHOD OF APPROXIMATION IS ‘ROUNDING’, IN WHICH WE TAKE ACCOUNT OF THE SIZE OF THE REMOVED DIGITS U N D E R S T A N D I N G E N G I N E E R I N G M A T H E M A T I C S WHEN WE ‘ROUND’ A NUMBER WE CHANGE THE LAST NON-Z