Vedic Math Presentation - MathMaverick
Vedic MathPresentationJohn L. k.comcheck out my mental math apps for android devices athttp://www.mathmaverick.com/viewMMIndex.php?vid 997practice and master mental math – the Vedic way
Vedic MathVeda means KnowledgeVedic Maths over 2000 years oldRediscovered in 20th century by Bharati KrishnaComprised of Sutras and sub-Sutraswhich are aphoristic formulasA system of Mental MathematicsRecommended Reference BookVedic Mathematics – Teacher’s Manual - Elementary LevelKenneth R. WilliamsISBN: 81-208-2774-0
Vedic MathCompleting the WholelessonsThe Ten Point CircleUsing Subtraction to Simply AdditionUsing Addition to Simplify SubtractionSimplifying Addition by Groups of 10sutrasBy Completion or Non-CompletionBy the Deficiency
The Ten Point Circle2010199181711182 127361651541413
The Ten Point Circlefive number pairs1 92 83 74 65 58By Completion orNon-CompletionUse number pairs tomake groups of1 “10” when addingnumbers109237465example:24 26 50
Below a Multiple of TenBy the DeficiencyView a number as to how close it is to the next multiple of ten49 is close to 50 and is 1 short38 is close to 40 and is 2 shortIt’s easy toadd zeros!Make useof this!59 4 6359 is close to 60 and is 1 below itSo, 59 4 is 1 below 60 459 4 60 4 – 1 64 – 1 6338 24 6238 is close to 40 and is 2 below itSo, 38 24 is 2 below 40 2438 24 40 24 – 2 64 – 2 62PracticeThisProcessMentally!
Sum to TenThe Ten Point Circle illustrates the pairs of numberswhose sum is 10There are eight unique groups of three numbers that sum to 101 2 7 10 is an example1 2 7 10 10 10 10 10 10 10 10Can you find the other seven groups of three numbers summing to 10?
Adding a List of NumbersBy Completion orNon-CompletionLook for number pairs that make a multiple of 107 6 3 4The list can be sequentially added as follows7 6 is 13,13 3 is 16And 16 4 is 20ORYou could look for number pairs that makemultiples of 107 3 is 10,6 4 is 1010 10 is 2048 16 61 3210101078923531237 9591010
Subtracting Near a BaseBy Completion orNon-CompletionWhen subtracting a number close to a multiple of 10,Just subtract from the multiple of 10 and correct the answeraccordinglymentally (53 – 30) 153 – 29 (23) 129 is close to 30, just 1 lower, So subtract 30 from 53 making 2324Then add 1 to make 2453 – 29 53 – (30 – 1) 53 – 30 1 23 1 24This process can be done mentally45 – 1845 – 18 45 – (20 – 2) 45 - 20 2 25 2 27
Vedic MathDoubling and HalvinglessonsMentally Multiplying and Dividing by 2Mentally Multiplying and Dividing by 4 and 8Mentally Multiplying and Dividing by 5, 50 and 25Using Number Relationships to Simplify a ProblemsutrasProportionately
DoublingProportionatelyAdding a number to itself is called Doubling23 232 2 46 3 3Mentally, we can double each column and then combine results58 58 double 50 and add double 8116 100 16 116Grouping columns may simply the problem263 263526 double 260 and add double 3 520 6 526Practice Doubling Mentally
DoublingProportionatelyPractice how to approach a problem736 7361472Mentally, the problem can be “broken” into two problems736 700 36double each of these and combine1400 72 1472Practice Doubling Mentally
Multiplying by 4 and 8Doubling can be used to multiply by 4just double the number twice35 x 4 (35 x 2 ) x 2so,35 x 4 70 x 2 140similarly,163 x 4 326 x 2 652Doubling can be used to multiply by 8just double the number three times35 x 8 ( (35 x 2 ) x 2 ) x 2so,35 x 8 70 x 4 140 x 2 280similarly,163 x 8 326 x 4 652 x 2 1304Proportionately
HalvingProportionatelyHalving is the opposite of DoublingHalf of 42 is 21just half each columnHalf of 56 is 28just half 50 and half 6 then addMentally, the problem can be “broken” into two problems736 700 36half each of these and combine350 18 368Practice Halving Mentally
Dividing by 4 and 8Halving can be used to divide by 4just half the number twice72 4 (72 2 ) 2so,72 4 36 2 18similarly,164 4 82 2 41Halving can be used to divide by 8just half the number three times72 8 ( (72 2 ) 2 ) 2so,72 8 (36 2) 2 18 2 9similarly,504 8 252 4 126 4 63Proportionately
Multiplying by 5, 50 and 25ProportionatelyMultiply by 5 by multiplying by 10 and halving the result26 x 5 ( 26 x 10 ) 2 260 2 130It’s easy to multiply by 10 and 100! Make use of this!Multiply by 50 by multiplying by 100 and half the result43 x 50 ( 43 x 100 ) 2 4300 2 2150Multiply by 25 by multiplying by 100 and half the result twice68 x 25 ( 68 x 100 ) 4 6800 4 3400 2 1700
Dividing by 5, 50 and 25ProportionatelyDivide by 5 by doubling and dividing the result by 10320 5 ( 2 x 320 ) 10 640 10 64It’s easy to divide by 10 and 100! Make use of this!Divide by 50 by doubling and dividing the result by 100850 50 ( 850 x 2 ) 100 1700 100 17Divide by 25 by doubling twice and dividing the result by 100325 25 ( 325 x 4 ) 100 1300 100 13
ProportionatelyProportionatelyWe know certain number facts well, such as 8 x 7 56But given the problem 16 x 7,we may use long multiplication,Instead, proportionately allows us to useour number facts along with halving and doubling16 x 7 2 x (8 x 7) 2 x (56) 112all of which can be done mentally!similarly,18 x 14 (2x9) x (2x7) 4 x (9x7) 4 x 63 2 x 126 252
Vedic MathDigit SumslessonsDefinition of Digit SumNine Point Circle“Casting Out” 9’sChecking with Digit SumssutrasWhen the Samuccaya is the Same it is Zero
Digit SumsA Digit Sum is the sum of all of the digits of a numberand is found by adding all of the digits of a numberThe Digit Sum of 35 is 3 5 8The Digit Sum of 142 is 1 4 2 7If the sum of the digits is greater than 9, thensum the digits of the result again until the result is less than 10 9, so sum the digits againThe Digit Sum of 57 is 5 7 12 1 2 3So the Digit Sum of 57 is 3The Digit Sum of 687 is 6 8 7 21 2 1 3So the Digit Sum of 687 is 3Keep finding the Digit Sum of the result until it’s less than 100 and 9 are equivalent!
Nine Point Circle262516241727189Numbers at each19 point on thecircle have the10same digit sum18271563514234132212112120
Casting Out 9’sWhen finding the Digit Sum of a number,9’s can be “cast out”The Digit Sum of 94993 is 4 3 7“cast out” the 9’sWhen finding the Digit Sum of a number,Group of numbers that sum to 9 can be “cast out”The Digit Sum of 549673 is 7“cast out” the 5 4, 9 and 6 3, leaving just 7By Casting Out 9’s,Finding a Digit Sum can be done more quicklyand mentally!
Checking with Digit SumsBoth Addition and Multiplication preserve Digit Sums21 1435 3 53 5 826 35 8 88 8 16 861 7If the sum of the Digit Sums does NOT equalThe Digit Sum of the sum, then there’s a problem! 821x14 294 63 3x5 15 6526x35 910 18 88x8 64 1If the product of the Digit Sums does NOT equalThe Digit Sum of the product, then there’s a problem!
Digit Sums of Perfect SquaresAll Perfect Squaresend in 1, 4, 5, 6, 9 or 0anddigit sums are 1, 4, 7 or 94539ends in 9digit sum is 3Therefore, 4539 is not a perfect square5776ends in 6digit sum is 7Therefore, 5776 may be a perfect square
Vedic MathLeft to Right
Vedic MathAll from 9 and the last from 10lessonsSubtracting Number from 10nApplications with MoneysutrasAll From 9 and the Last From 10
All From 9 and the Last From 10All from 9 and theLast from 10When subtracting a number from a power of 10Subtract all digits from 9 and last from 101000-276724from9from10276 724If the number ends in zero,use the last non-zero number as the last number10000-42505750from9from104250 5750
All From 9 and the Last From 10All from 9 and theLast from 10If the number is less digits, then append zeros to the startfrom910000-425from100425 95759575When subtracting from a multiple of a power of 10,Just decrement the first digit by 1, then subtract remaining digitsfrom94000- 257from10257 37434–1 3743
All from 9 and theLast from 10MoneyA great application of “All from 9 and last from 10” is money.Change can be calculated by applying this sutra mentally! 1 0 .0 0- 4 .2 5 5 .7 5from9from10 4 .2 5 5 .7 5It is often the case the payment is made with bills only,these are multiples of “100”THINK MENTALLY!
Vedic MathNumber SplittinglessonsSplitting Number to Simplify ProblemsutrasProportionately
Number SplittingQuick mental calculations can be performed more easily if thethe numbers are “split” into more manageable partsThis sum can looksomewhat daunting“split” into two moremanageable sums36 42 24 393642 243960 81The “split” allows oneto add “36 24” and“42 39”, both ofwhich can be donementallyThink about where to place the “split” line.It’s often best to avoid number “carries” over the line342 5873 42 5 8734 2 58 79 2992 9a carry of “1” overthe line is requiredNo carry is requiredcarry 1
Number SplittingThe same can be done for subtraction also!3642- 243936 42- 24 39The “split” allows oneto subtract “36-24”and “42-39”, both ofwhich can be donementally12 03The same can be done for multiplication also!263x226 3x2 x2526526The same can be done for division also!6234 262 34 2 231 173117
Number SplittingThe “split” may require more “parts”30155 5530 15 5 5 56 03244506 3163124 45 06 3 3 38 15281502
Vedic MathBase MultiplicationlessonsMultiplying Numbers Just Above or Below 10nUsing Number Relationships to Simplify ProblemsMultiplying Numbers Near Different BasesSquaring Numbers Near a BasesutrasVertically and CrosswiseProportionately
Multiplying Numbers Just Above 10Vertically andCrosswiseTraditionally, the multiplication of 2-digit numbers requiresFour single digit multiplies and series of summationsto combine the resultsTraditionalMethodVedicMethod12x 1312212 is 2 above 10361215613313 is 3 above 1015615 12 3 or 13 2think “crosswise”1566 2x3think “vertically”The Vedic Method requires an addition (crosswise), a single digitMultiplication (vertically) and possibly a carry
Multiplying Numbers Just Above 10Vertically andCrosswiseA carry may be required when combining the crosswise andvertical resultsTraditionalMethod16x 134816208VedicMethod16616 is 6 above 1013313 is 3 above 101918carry “1” andadd to 19208
Multiplying Numbers Just Below 100Vertically andCrosswiseTraditionally, the multiplication of 2-digit numbers requiresFour single digit multiplies and series of summationsto combine the resultsTraditionalMethod98x 887847848624VedicMethod9828812862486 98–12 or 88-2think “crosswise”98 is 2 below 10088 is 12 below 100862424 2x12think “vertically”The Vedic Method requires a subtraction (crosswise), anda single digit multiplication (vertically)
Multiplying Numbers Just Above 100Vertically andCrosswiseTraditionally, the multiplication of 3-digit numbers requiresSix single digit multiplies and series of summationsto combine the resultsTraditionalMethod102x 11220410210211424VedicMethod10221121211424114 112 2 or 102 12think “crosswise”102 is 2 above 100112 is 12 above 1001142424 2x12think “vertically”The Vedic Method requires an addition (crosswise), anda multiplication (vertically)
Multiplying Numbers on Either Side of 100Vertically andCrosswiseMultiplying Numbers Close to 100but on either sideTraditionalMethodVedicMethod112x 988961008112 129811010976110 112-2 or 98 12-2112 is 12 above 10098 is 2 below100-24 11000 – 24 - 10976-24 -2x12All from 9 and theLast from 10So, what’s 103 x 96?9900 – 12 9888
Multiplying Larger NumbersVertically andCrosswiseTraditionally, the multiplication of larger numbers requiresnumerous single digit multiplies and series of summationsto combine the resultsTraditionalMethod672x 998537660486048670656VedicMethod6723289982670 656670 672-2think “crosswise”672 is 328 below 1000“all from 9, the last from 10”998 is 2 below 1000670656656 2x328think “vertically”The Vedic Method requires an addition (crosswise), anda multiplication (vertically)
Vertically and CrosswiseProportionatelyProportionatelyOften problems can be simplified in order to be performedmentally more easilyTraditionalMethod212x 10484821222048VedicMethod212 x 1042 x (106 x 104)2 x (106 x 104) 2 x 11024 ther example46 x 192 can be rewritten as 92 x 96 by doubling and halving46 x 192 92 x 96 8832vertically/crosswise
Multiplying Numbers Near DifferentVertically andBases CrosswiseMultiplying numbers near different basesuses the same techniques94 x 9997999739997 is 3 below 1000094694 is 6 below 10093979397 9997-6x100think “crosswise”10006 x 100210026 10006 20think “crosswise”1893971818 3x6think “vertically”10006610006 is 6 above 10000100221002 is 2 above 1000100261210026012
Squaring Numbers Near a BaseVertically andCrosswiseTo square numbers near a base, just apply the same techniques94294694 is 6 below 100946883688 94-610052883636 62100551005 is 5 above 1000100551010251010025
Vedic MathBar NumberslessonsDefine Bar NumbersBar Number ArithmeticUsing Bar NumberssutrasAll From 9 and the Last From 10
Bar Numbers29 is close to the number 30Let’s rewrite 29 as 3131 means 30 – 1 or 29Bar numbers are analogous to timeThe time 5:45 can be“45 minutes past 5”OR“15 minutes before 6”52 means 50 – 2 or 48Note the distinctionbetween412 392412 means 400 – 10 2 392and413 means 400 – 13 387412 38863 means 60 – 3 or 57
Subtraction using Bar Numbers435-276159Traditionally, subtraction isperformed on columns right to left“borrowing” from the next leftcolumn when necessaryHowever, subtracting each column independentlygives the following:4–2 23 – 7 -45 – 6 -1Negative numbers can be replaced with their bar number equivalent , so4–2 23–7 45–6 1So,435-2762 4 1 241 159
Arithmetic of Bar Numbers29 48 31 5277The original problem can be rewritten three different waysusing bar numbers31 4829 5277777 8-131 5283 777 9-2In each case, the result is the same, 77!3 2 1
Arithmetic of Bar NumbersAddition:28 43 71Subtraction:Multiplication:63-3771 2628x 3 32x 387 3711 2 1 2-163-432684Division:31 426 3-3 3 36 2x39 6 84 93 33 1 291 3 3
Vedic MathSpecial MultiplicationlessonsMultiply by 11Multiply ab x ac where b c 10Multiply ba x ca where b c 10Multiply by 99, 999, etcUsing the Average to MultiplysutrasVertically and CrosswiseBy One More than The One BeforeBy One Less than The One BeforeThe First by the First, The Last by the LastSpecific General
Multiplying by 11Multiplying by 11 can be performed easilyTraditionalMethodVedicMethod45x 1145sum9445455 495495Carries may be required when the sum exceeds 9575sum12carry the 1and add to 57 627
Multiplying by 11TraditionalMethodVedicMethod243x 112432432432 4264 373 26732673Carries may be required when a center sum exceeds 956155 611carry the 1and add to 56 171 6171
Special MultiplicationBy One More ThanThe One BeforeWhen 2-digit numbers start with the same digitand their last digits sum to 10,The product of the two numbers can be easily computedboth numbersstart with 332x3812 1632 x 38the sum ofthe last digitsis 10As the numbers getfurther from 35,their product gets furtherfrom 352 1225by a factor of the differencesquared121612 3x416 2x8“By one morethan the one before”35 x 3534 x 3633 x 3732 x 3831 x 39 1225 1225 – 01224 1225 – 1 1225 - 121221 1225 – 4 1225 – 221216 1225 – 9 1225 – 321209 1225 – 16 1225 - 42
Special MultiplicationThe First by the FirstThe Last by the LastWhen 2-digit numbers end with the same digitand their first digits sum to 10,The product of the two numbers can be easily computedboth numbersEnd with 424 x 84the sum ofthe first digitsis 1036x7627 3624x8420 1620 2x8 4201616 4x462x42273626 042604
Special MultiplicationBy One Less ThanThe One BeforeWhen multiplying by a number with all digits equal to “9”,The product of the two numbers can be easily computedin two parts32x9931 6831 32-1“By one lessthan the one before”568x 999567 432316868 100-32“All from 9,the last from 10”1285x 99995674321284 871512848715
Special MultiplicationSpecificGeneralWhen multiplying numbers, the average can sometimes be usedto determine their product31 x 29their average is 30square this and subtract 1 to determine the product31 x 29 302 – 12 900 – 1 89930 (31 29)/21 (31-30)238 x 42 402 – 22 1600 – 4 1596“all from 9, the last from 10”47 x 53 502 – 32 2500 – 9 2491“all from 9, the last from 10”
Vedic MathGeneral MultiplicationlessonsMultiply 2-Digit NumbersMultiply 3-Digit NumberssutrasVertically and Crosswise
Vertically andCrosswiseMultiplying 2-Digit NumbersTraditionalMethodIntermediateMethod24x 4324x 437296126016080010321032VedicMethod 3x4 3x20 40x4 40x2088 2x4upperhalf“vertical”24431661216 4x46 2x3“crosswise”12 4x3lowerhalf“vertical”812 220 103246 x 52 2012 380 2392
Multiplying 3-Digit NumbersFind 504 x 321TraditionalMethod504x 32150410081512161784VedicMethod50432115, 10, 17, 8, 4161784Mentally, we think 15; 160; 1617; 16178; 161784Find 123 x 3213, 8, 14, 8, 339483Mentally, we think 3; 38; 394; 3948; 39483
Multiplying 3-Digit NumbersFind 617 x 893VedicMethod61789348, 62, 83, 66, 21550981Mentally, we think 48;542; 5503; 55096; 550981Vertically andCrosswise
Vedic MathSquaringlessonsSquaring Numbers Ending in 5Squaring Numbers Near 50General Squaring using the Duplex of a NumbersutrasBy One More Than the One BeforeProportionatelyVertically andCrosswise
Squaring Numbers that End in 5By One More Than the One Before352 3 4 / 25 12 / 25 1225The result is comprised of two “parts”The bottom “part” is always 25The top “part” is n(n 1)752 7 8 / 25 56 / 25 5625452 4 5 / 25 20 / 25 2025952 9 10 / 25 90 / 25 90251152 11 12 / 25 132 / 25 13225By One More ThanThe One Before
Squaring Numbers Near 50542 52 4 / 42 29 / 16 2916For numbers greater than 50The result is comprised of two “parts”The bottom “part” is always the bottom digit squaredThe top “part” is the top digit squared plus the bottom digit482 52-2 / 22 23 / 4 2304For numbers less than 50The result is comprised of two “parts”The bottom “part” is always the bar of the bottom digit squaredThe top “part” is the top digit squared minusthe bar of the bottom digit532 25 3 / 9 28 / 9 2809462 25-4 / 16 21 / 16 2116
General SquaringThe Duplex, D, of a number1 digit – D(n) n2e.g. D(5) 252 digits – D(n) twice the product of the digitse.g. D(26) 2(2)(6) 243 digits – D(n) twice the product of the outer digits the square of the middle digite.g. D(137) 2(1)(7) 32 14 9 23The square of a number is the “total” of its Duplexes342 1156D(3) 9, D(34) 24, D(4) 169, 24, 16 1156562 3136D(5) 25, D(56) 60, D(6) 3625, 60, 36 3136
General Squaring472 1156D(4) 16, D(47) 56, D(4) 4916, 56, 49 2209Number Splitting1232 - Split 123 into two parts 12 / 3D(12) 144, D(123) 72, D(3) 9144, 72, 9 151294122 - Split 412 into two parts 4 / 12D(4) 16, D(412) 96, D(12) 14416, 96, 144 16974416 495622 09
General Squaring (3 and 4-digit numbers)3412 116281D(3) 9, D(34) 24, D(341) 22, D(41) 8, D(1) 19, 24, 22, 8, 1 1162812632 D(2) 4, D(26) 24, D(263) 48, D(63) 36, D(3) 94, 24, 48, 36, 9 6916943322 D(4) 16, D(43) 24, D(433) 33, D(4332) 34,D(332) 21, D(32) 12, D(2) 416, 24, 33, 34, 21, 12, 4 18766224
Vedic MathSpecial DivisionlessonsDivision by 9Division Below and Above 10nsutrasNeed to Determine This
Division by 932 99)323 r5 where 5 3 252 99)525 r7 where 7 5 275 99)757 r12 where 12 7 5 remainder 9 8 r33102 99)31023 4 4 r6312 99 ) 3123 4 r6When dividing by 9,The remainder is always thedigit sum of the original number
Division by 9 with Carries3172 99 ) 3 17 23 4 11 r13 351 r13 352 r45555 99)5555 10 r15 61 r6Short Cut3172 99)31723 5 2 r46153 99)61536 8 3 r6Check to see if next sum is 9, if so add 1
Division Below a Base Number235 88211 7588 ) 23 51 22425 976 ) 21 12 5521121123 89891.2.3.4.Drop 2 (the first digit)Create 12 88Multiple 12 by 2 24Add 35 and2406 18989 ) 1 1 2 1 1 2 3101110 1 12 0 2 24 0 441 2 4 6 4 8 7
Division Above a Base Number1498 124 124 ) 1 42 429 8441 281 01.2.3.4.5.6.Drop 1 (the first digit)Create 24Multiply 24 by 1 24Add 4 2 2Multiply 24 by 2 48Add columns12 r102311 112112 ) 2 31 221 1412 124 1 20 r71 where 71 112-41
Vedic MathGeneral DivisionlessonsGeneral DivisionDecimalizing the Remaindersutras
General Division308 515130086 2234 545423341. Create flag, 1, from divisor 2nd digit2. 5 goes into 30 6 times with r03. 08 – 1x6 2 6 r21. Create flag, 4, from divisor 2nd digit2. 5 goes into 23 4 times with r33. 34 – 4x4 184 18 4 r18503 727250137 11. Create flag, 2, from divisor 2nd digit2. 7 goes into 50 7 times with r13. 13 – 2x7 1 72 – 1 71 6 r71
General Division19902 626219 91020236219 913 262191619310109 0213 2211009 0 212 1001. Create flag, 2, from divisor 2nd digit2. 6 goes into 19 3 times with r11. Compute 19 – 2x3 132. 6 goes into 13 2 times with r11. Compute 10 – 2x2 62. 6 goes into 6 1 times with r01. Compute 02 – 2x1 0 (remainder) 321 r0
General Division92054 63639 20541. Create flag, 3, from divisor 2nd digit2. 6 goes into 9 1 times with r30541. Compute 32 – 3x1 292. 6 goes into 29 4 times with r51492 0541. Compute 50 – 3x4 382. 6 goes into 38 6 times with r21 4690 541. Compute 25 – 3x6 72. 6 goes into 7 1 time with r13. Compute 14 – 3x1 11 remainder316663339 2313355254 602211 11 1461 r11
General Division543 686685232523486252 348 271. Create flag, 8, from divisor 2nd digit1. Change flag to a bar number 22. 6 goes into 52 8 times with r41. Compute 43 - 2x8 27 remainder 8 r27
Decimalizing the Reminder40342 737340553534252006 .3 01401303607
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Vedic Math Vedic Maths over 2000 years old Rediscovered in 20th century by Bharati Krishna Comprised of Sutras and sub-Sutras which are aphoristic formulas A system of Mental Mathematics Recommended Reference Book Vedic Mathematics – Teacher’s Manual - Elementary Level Kenneth R. Williams ISBN: 81-208-2774-0. Completing the Whole Vedic Math .
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Mathematics in their curriculum 97 4.2 Teachers views on Vedic Mathematics and its overall influence on the Students Community 101 4.3 Views of Parents about Vedic Mathematics 109 4.4 Views of Educationalists about Vedic Mathematics 114 4.5 Views of the Public about Vedic Mathematics 122 Chapter Five OBSERVATIONS 165
the Vedic literature, but Swamiji got the essence of the subject while he was studying Vedic literature. Being a humble saint, he did not want to claim the ownership of the new system of mathematics. Hence, he named the subject - Vedic Mathematics. Initially, Swamiji had written sixteen volumes on Vedic Mathematics (one volume on each sütra).
next full adder. One way to improve the speed as well as to reduce the power is to employ the ancient methods of mathematics known as Vedic mathematics. The Vedic in Sanskrit means store-house of knowledge. The Vedic mathematics offers shortcuts to quickly perform the arithmetic operations. There is a total of 16 Vedic
4.2 Teachers views on Vedic Mathematics and its overall influence on the Students Community 101 4.3 Views of Parents about Vedic Mathematics 109 4.4 Views of Educationalists about Vedic Mathematics 114 4.5 Views of the Public about Vedic Mathematics 122 Chapter Five OBSERVATIONS 165 5.1 Students' Views 165
INTRODUCTION TO FIELD MAPPING OF GEOLOGIC STRUCTURES GEOL 429 – Field Geology Department of Earth Sciences Montana State University Dr. David R. Lageson Professor of Structural Geology Source: Schmidt, R.G., 1977, Geologic map of the Craig quadrangle, Lewis and Clark and Cascade Counties, Montana: U.S. Geological Survey GQ-1411, 1:24,000. 2 CONTENTS Topic Page Introduction 3 Deliverables 4 .
