1、Introduction : Vedic mathematics that includes the famed Gayatri mantra, which adds to 108 when decoded. How important is Speed? How fast your can solve a problem is very important. There is a race against time in all the competitions. Only those people having fast calculation ability will be able t
2、o win the race. Time saved can be used to solve more problems or used for difficult problems. Is it useful today? Given the initial training in modern maths in todays schools, students will be able to comprehend the logic of Vedic mathematics after they have reached the 8th standard. It will be of i
3、nterest to every one but more so to younger students keen to make their mark in competitive entrance exams. Indias past could well help them make it in todays world. It is amazing how with the help of 16 Sutras and 16 sub-sutras, the Vedic seers were able to mentally calculate complex mathematical p
4、roblems. Introduction : Learn to calculate 10-15 times faster. 满分网（http:/）更多SAT资料：http:/ is a system of reasoning and mathematical working based on ancient Indian teachings called Veda. It is fast , efficient and easy to learn and use. Example 1 : Finding Square of a number ending with 5 To find the
5、 square of 75 Do the following Multiply 5 by 5 and put 25 as your right part of answer. Multiply 7 with the next higher digit ie (7+1)=8 gives 56 as the left part of the answer, Answer is 5625 Example 2 : Calculate 43 X 47 The answer is 2021 Same theory worked here too. The above rule works when you
6、 multiply 2 numbers with units digits add up to 10 and tenth place same Example 3 : Find 52 X 58 ? Answer = 3016 How long this take ? Example 4: Multiply 52 X 11 Answer is 572 Write down the number being multiplied and put the total of the digits between 2 digits 52 X 11 is 5 and 5+2=7 and 2 , answe
7、r is 572 Example 5: Can you find the following within less than a minute? a) 1001/13 ? b) 1/19 ? 满分网（http:/）更多SAT资料：http:/ you can learn Fastmaths techniques with ease at your home in your spare time Chapter 1 : Numbers 1.1 Numbers Numbers begins at 1. All other numbers come from one. There are only
8、 nine numbers and a zero. NUMBERS 0 ? ZERO 5 ? FIVE 1 ? ONE 6 ? SIX 2 ? TWO 7 ? SEVEN 3 ? THREE 8 ? EIGHT 4 ? FOUR 9 ? NINE ? Starting from number 1 all whole numbers are generated using “ By one more than one before“. 2 is more than 1; 4 is more than 3; 6 is more than 5 and so on. ? Whole numbers a
9、re also called Natural Numbers Assignments 1. Which Number is 1 more than a) 19 b) 40 c) 189 d) 23 e) 4589 2.Which number is 1 less than a) 29 满分网（http:/）更多SAT资料：http:/ 48 c) 2339 d) 5 e) 65320 Assignments Answers 1. Which Number is 1 more than a) 20 b) 41 c) 190 d) 24 e) 4590 2.Which number is 1 le
10、ss than a) 28 b) 47 c) 2338 d) 4 e) 65319 满分网（http:/）更多SAT资料：http:/ 1 : Numbers 1.2 Place Value Since there are only 9 numbers and a zero we count in groups of 10. Ten Units make a TEN, Ten Tens make a HUNDRED . Ten Hundreds make a THOUSAND. PLACE VALUE X X X ?X? Thousand Hundred Ten Units The firs
11、t seven place values are UNITS, TENS, HUNDREDS, THOUSANDS,TEN-THOUSANDS,HUNDRED-THOUSANDS, and MILLIONS. In any number the value of a digit depends upon its position The 4 in 41 stands for four Tens The two in 42 stands for two Units The value of the digit 5 in 452 is five Tens, because it is in the
12、 tens column. The following Number can be written as 54321 = 54 X 1000 + 3 X 100 + 2 X 10 + 1 X 1 since The 54 in 54321 stands for 54 Thousands The 3 in 54321 stands for 3 Hundreds The 2 in 54321 stands for 2 Tens The 1 in 54321 stands for 1 Units The number 54,321 says fifty four thousand, three hu
13、ndred 满分网（http:/）更多SAT资料：http:/ twenty one ? Assignments 1.Find the value of 4 in the following a) 430 b) 947 c) 14 d) 125004 2. Write the following numbers in Words a) 57 b) 7002 c) 405 d) 9 3. Fill in the blanks a) 243 = _ X 100 + 4 X _+ _X 3 b) 45 = 1000 X _ + 100 X _ + 10 X _+ 1 X _ c) 9 = 100 X
14、 _ + 10 X _+ 1 X _ 4. Write the following numbers in Figures a) Two hundred and thirty five b) Nine thousand and twenty nine c) Four million d) Sixty-eight e) Twenty four thousand Assignments Answers 1.Find the value of 4 in the following a) HUNDRED b) TEN c) UNITY d) UNITY 满分网（http:/）更多SAT资料：http:/
15、 Write the following numbers in Words a) Fifty Seven b) Seven thousand two c) Four hundred Five d) Nine 3. Fill in the blanks a) 243 = 2 X 100 + 4 X 10+ 1 X 3 b) 45 = 1000 X 0 + 100 X0 + 10 X 4+ 1 X 5 c) 9 = 100 X 0 + 10 X 0+ 1 X 9 4. Write the following numbers in Figures a) 235 b) 9029 c) 4000000
16、d) 68 e) 24000 Chapter 1 : Numbers 1.3 9-Point Circle The basic numbers always remain one to nine. 满分网（http:/）更多SAT资料：http:/ Point Circle We can represent 9 numbers as shown above. This circle is called a nine-point circle. The number 1 is the absolute and is inside everything. The number 1 is a fa
17、ctor of every number and every number is a factor to itself. ? Where do we add 10 on a nine-point Circle?. Now where do we add 0 ? Nine Point Circle 满分网（http:/）更多SAT资料：http:/ Chapter 1 : Numbers 1.3.2 Product: When two numbers multiplied together the answer is called product. Example The product of
18、3 and 6 is 18? The product of 5 and 9 is 45? ? Multiplying by 1 brings about no change Any number when multiplied by 0 gives 0 Assignments Find the Product of a) 5 X 4 b) 7 X 9 c) 6 X 2 d) 1 X 0 e) 12 X 1 Assignments Answers Find the Product of 满分网（http:/）更多SAT资料：http:/ 20 b) 63 c) 12 d) 0 e) 12 Ch
19、apter 1 : Numbers 1.3.3 Factors: Numbers, which multiplied together to give a product, are called factors. 3 and 8 are factors of 24, because 24= 3 X 8? A number may also be seen as a factor of itself.?Some numbers have more than one pair as factors All numbers have one and themselves as a factor. E
20、xample 1: Find Factors of 36 ? Factors of 36 36 can be expressed as 1 X 36 = 36, 2 X 18 = 36, 3 X 12 = 36, 4 X 9 = 36, 6 X 6 = 36 Factors of 36 are 1,2,3,4,6,9,12,18,36. The number 1 is a factor of every number 1.3.3.1 Factor pairs Number 18 has 6 factors; 1,2,3,6, 9,18 18 can be expressed as 1 X 18
21、 = 18, 2 X 9 = 18, 3 X 6 = 满分网（http:/）更多SAT资料：http:/ Arrange Pair factors like (1X18),( 2X9), (3X6).?These pair of numbers is called factor pairs. Factor pairs of 18 are (1X18),( 2X9), ( 3X6) If you know one factor of a number, you can get another using factor pairs. If you know 44 can be divided by
22、 4, than another factor of 44 must be 11 since 11X4 = 44 Assignments List all factors and list factor pairs if any. a) 64 b) 48 c) 128 d) 27 e) 37 Assignments Answers List all factors and list factor pairs if any. a) 64 Factors 1,2,4,8,16,32,64 Factor Pairs (1,64) (2,32) ( 4,16) (8,8) b) 48 Factors
23、1,2,3,4,6,8,12,16,24,48 Factor Pairs (1,48) (2,24) (3,16) (4,12) (6,8) c) 128 Factors 1,2,4,8,16,32,64,128 Factor Pairs (1,128) (2,64) (4,32) (8,16) 满分网（http:/）更多SAT资料：http:/ 27 Factors 1,3,9,27 Factor Pairs (1,27) (3,9) e) 37 Factors 1,37 Factor Pairs (1,37) Chapter 1 : Numbers 1.3.3.2 Highest com
24、mon factor (HCF) Suppose we have 2 numbers 70 and 99 70 = 2 X 5 X 7 99 = 3 X 3 X 11 Looking at the factors, there is no common factor except number 1. There is no factor of one number, which is also a factor of the other number, except for 1. Such pair of numbers is called relatively prime; they are
25、 prime in relation to each other. Example 1: Check 18 and 30 18 = 2 X 3 X 3 30 = 2 X 3 X 5 So 18 and 30 are not relatively prime, they have factors in common Both numbers can be divided by 2, 3 and 2 X 3 = 6 Of these three factor numbers the number 6 is the highest Common Factor (HCF) Example 2: Che
26、ck 48 and 72 满分网（http:/）更多SAT资料：http:/ = 2 X 2 X 2 X 2 X 3 72 = 2 X 2 X 2 X 3 X 3 ? So 48 and 72 are not relatively prime, they have factors in common. Of these factor numbers the number 2 X 2 X 2 X 3 = 24 is the highest Common Factor (HCF) Example 3: Check 140 and 27 140 = 2 X 2 X 5 X 7 27 = 3 X 3
27、X 3 So 140 and 27 are relatively prime. The highest Common Factor (HCF) = 1 When numbers are close together the HCF will also be a factor of the sum and of the difference of the numbers ? Example 4: Find HCF of 411 and 417? The above note means the HCF will divide into 411 and 417 also 411 + 417 = 8
28、28 417 ? 411= 6 This means that HCF is either 6 or a factor of 6 ( 6 or 3 or 2 or 1). Since 6 is not a factor of 411 and 417 , test for 3 or 2 HCF(411,417)= 3 Example 5: Find HCF of 90 and 102 This means the HCF will divide into 102 and 90 also 102 + 90 = 192 102 ? 90 = 12 This means that HCF is eit
29、her 12 or a factor of 12 (12, 6,4,3,2,1) 3 is a common factor of 90 and 102 And 2 also, but not 4 ,Therefore 2X3 = 6, HCF = 6 满分网（http:/）更多SAT资料：http:/ 6 Assignments 1. Find the following a) HCF(80,20)= b) HCF(68,24)= c) HCF(88,38)= d) HCF(88,82)= e) HCF(63,18)= Assignments Answers 1. Find the follo
30、wing a) HCF(80,20)=20 80 = 2 X 2 X 2 X 2 X 5 20 = 2 X 2 X 5 ? So 80 and 20 are not relatively prime, they have factors in common. Of these factor numbers the number 2 X 2 X 5 = 20 is the highest Common Factor (HCF) b) HCF(68,24)=4 68 = 2 X 2 X 17 24 = 2 X 2 X 6 ? So 68 and 24 are not relatively prim
31、e, they have factors in common. Of these factor numbers the number 2 X 2 = 4 is the highest Common Factor (HCF) c) HCF(88,38)= 2 88 = 2 X 2 X 2 X 11 38 = 2 X 19 So 88 and 38 are not relatively prime, they have factors in common. Of these factor numbers the number 2 is the highest Common Factor (HCF)
32、 满分网（http:/）更多SAT资料：http:/ HCF(88,82)=2 88 = 2 X 2 X 2 X 11 82 = 2 X 41 So 88 and 82 are not relatively prime, they have factors in common. Of these factor numbers the number 2 is the highest Common Factor (HCF) e) HCF(63,18)=9 63 = 3 X 3 X 7 18 = 2 X 3 X 3 So 63 and 18 are not relatively prime, the
33、y have factors in common. Of these factor numbers the number 3 X 3 = 9 is the highest Common Factor (HCF) Chapter 1 : Numbers 1.3.4 Divisibility The number 1 is not a product and cannot be divided. A number, which is a product, is divisible by any one of its factors. 10 is a product of 2 and 5 and
34、so 2 and 5 are factors of 10. 10 can be divided by 2 or 5 without any reminders 10 /5 =2 or 10/2 = 5 1.3.5 Prime Numbers Some numbers will have only one pair of factors 11 = 11 X 1 and there are no other numbers which multiply together to give 11. Such numbers are called prime numbers. ? 满分网（http:/）
35、更多SAT资料：http:/ first few prime numbers are 1, 3, 5, 7, 11, 13, 17, 19 ? Assignments Find Prime Number from the following 31, 49, 147, 97, 81 Assignments Answer Find Prime Number from the following 31, 49,147,97, 81 Answer : 31, 97. All other numbers have more than 1 factor. 49 can be written as 7X7
36、and 81 can be written as 3X 27 or 9X9 Chapter 1 : Numbers 1.3.6 The number 2. The number two stands for 2 types of beings in the creation, good and evil. So the number two divides the creation into two types of beings. It also divides the number into two sorts, odd and even. 1.3.6.1 Odd and Even Nu
37、mbers ? Numbers which have 2 as a factor are called Even Numbers, which do not have 2 as a factor, are called Odd. The even numbers are 2,4,6,8,10,12 14, and so on. 满分网（http:/）更多SAT资料：http:/ number which ends in a 2,4,6,8,or 0 is even. The odd numbers are 1,3,5,7,9,11,13, and so on. Any number, whic
38、h ends in a 1,3,5,7 or 9, is an odd number. An odd number cannot be divided into two equal parts. 1.3.6.2. Multiples Multiple means many. If we take number 1 many times, we arrive at 1,2,3,4,5 Similarly if we take number two many times, we arrive at 2,4,6,8 These are all multiples of two. A multiple
39、 of a number is that number multiplied by any number ? ? Assignments 1. Find the Odd numbers from the following 3, 6, 7, 12, 15, 19, 21, 10,100 2. Find the Even numbers from the following 13, 26, 70, 12, 19, 39, 61, 102,150 Assignments Answers 1. Find the Odd numbers from the following 3, 7,15, 19,
40、21 2. Find the Even numbers from the following 26, 70, 12, 102,150 满分网（http:/）更多SAT资料：http:/ Chapter 1 : Numbers 1.3.7. The Number 9 In our number system number nine is the largest digit The digital root of a number can be obtained by summing the digits of the number, for example, for 23, digital ro
41、ot is 2 + 3 = 5. ? We will learn more about digital roots in chapter 3. The digit sum or Digital root of a number is unchanged if 9 is added to it or subtracted from it. Table of 9 9 X 1 ?= 9 9 X 2 ?= 1 8 9 X 3 = 2 7 9 X 4 ?= 3 6 9 X 5 ?= 4 5 9 X 6 = 5 4 9 X 7 ?= 6 3 9 X 8 ?= 7 2 9 X 9 = 8 1 If you
42、read the answers as two columns the left column goes up by one more than one before (1,2,3,4,5) and the right column goes down by one less than the one before ( 9,8,7,6,5.) 1.3.7.1 By Addition and By Subtraction? When adding or subtracting numbers which end in 9 or 9s use the following method. Examp
43、le : Find 75 + 39 满分网（http:/）更多SAT资料：http:/ 40 to 75 and take 1 off. 75 + 39 = 75 + 30 - 1 = 114 Example : Find 122 -59 Subtract 60 from 122 and put 1 back. 122 - 60 + 1 = 63 Assignments Find the following a) 132 + 49 = b) 34 + 29 = c) 63 - 19 = d) 56 - 9 = e) 79 +19 = Assignments Answers Find the f
44、ollowing a) 132 + 49 = 181 Add 50 to 132 and take 1 off. 132 + 49 = 132 + 50 - 1 = 182 - 1 = 181 b) 34 + 29 = 63 Add 30 to 34 and take 1 off. 34 + 29 = 34 + 30 - 1 = 64 - 1 = 63 c) 63 - 19 = 44 Subtract 20 to 63 and add 1. 63 - 19 = 63 - 20 + 1 = 43 + 1 = 44 d) 56 - 9 =47 满分网（http:/）更多SAT资料：http:/ S
45、ubtract 10 to 56 and add 1. 56 - 9 = 56 - 10 + 1 = 46 + 1 = 47 e) 79 +19 = 98 Add 20 to 79 and take 1 off. 79 + 19 = 79 + 20 - 1 = 99 - 1 = 98 Chapter 1 : Numbers 1.3.8. The Number 10 The number ten is 1 with a zero next to it. So multiplying a number by ten the answer is the same but with a zero on the end. Example 1 Find 84 X 10 84
