Successfully reported this slideshow.
Upcoming SlideShare
×

0 views

Published on

Vedic Mathematics is a system of mathematics that allows problems to be solved quickly and efficiently. It is based on the work of Sri Bharathi Krishna Thirthaji Maharaja (1884 – 1964), who devised the system from a close study of the Vedas. The Vedas are ancient scriptures of India that deal with many subjects. It is based on 16 sutras (aphorisms) from the Vedas that provide a principle or a rule of working to solve a problem. These sutras may be ancient in origin, but are still relevant to modern day mathematics.

Published in: Education
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

1. 1. Vikram Devatha First Edition Vedic Addition
4. 4. iii To the curious minded...
5. 5. 4 Preface Vedic Mathematics is a system of mathematics that allows problems to be solved quickly and efﬁciently. It is based on the work of Sri Bharathi Krishna Thirthaji Maharaja (1884 – 1964), who devised the system from a close study of the Vedas. The Vedas are ancient scriptures of India that deal with many subjects. It is based on 16 sutras (aphorisms) from the Vedas that provide a principle or a rule of working to solve a problem. These sutras may be ancient in origin, but are still relevant to modern day mathematics. Vedic Math provides many different methods to solve any given problem. The choice of method depends on the conditions that the given problem satisﬁes. This is very much like planting a tree – the choice of which tree to plant has to depend on the nature of the soil and the environment. It is impossible to plant the same tree everywhere without considering the surroundings. Vedic math works in a similar manner. Conventional mathematics generally provides a single method to solve a mathematical problem. This method is applied “blindly” whenever the student comes across the problem. However, in vedic math, the student chooses which method to employ. In multiplication, for instance, there are almost ﬁve different methods that can be used, and the choice of method depends entirely on what the student is comfortable with. Learning such a system of mathematics at an early age can greatly help in dispelling fears of mathematics in children and can even make it more fun. Vedic math also allows us to develop the ability of lateral thinking, enabling us be faster at calculations and even to rely less on the calculator. ! ! This series of books is an attempt to present the material in a modular fashion. Each book focusses on one speciﬁc arithmetic operation - addition, subtraction, multiplication and division. These books can be read in any order, but it is recommended that addition and subtraction be read before multiplication and division. This particular book is related to addition only, and subsequent books will cover the other arithmetical operations. Some of the vedic methods apply to speciﬁc sets of numbers, while others are general methods and can be used in all cases. How to use this book Each chapter introduces one or two ideas, and takes you from the simple to the more advanced methods. At times, you will be posed a question, and I suggest that you pause, think and arrive at an answer before continuing.
7. 7. Some notes
8. 8. 7 Some notes Patterns in numbers Ever seen a pattern in nature, such as the recurring phases of the moon, and wondered at the beauty of nature? These patterns exist in numbers too. Patterns such as the Fibonacci Numbers and the Golden Mean are well known examples. Learning how to recognize these patterns and using them to solve problems is what Vedic Math is all about. Number line and negative numbers The number line is a straight line with zero at the center and extending to infinity on either side. Numbers to the left of zero are negative while those to the right are positive. Zero, a number discovered in ancient India, is neither positive nor negative. Negative numbers are used extensively in Vedic Mathematics. You will notice that in the above diagram, the negative sign is placed above the digit rather than to its left as in conventional math e.g. -3 has been written as 3̅̅. Similarly, –9 will be written 9̅̅, -32 as 3̅̅2̅, –10 as either 1̅0̅ or 1̅0 (since 0̅ = 0). Number tables There are two kinds of number tables that are essential for mathematics – addition & subtraction tables and multiplication & division tables. Today, schools generally advocate addition tables till 10 + 10, and multiplication tables until 12 x 12. For Vedic Math, you only need to know the tables up to 5 x 5. Tables of higher numbers are not required. However, knowledge of tables till 20 x 20 and 16 x 16 will be useful. Answers in parts Answers are normally obtained in parts, namely, the left hand side (LHS), middle (mid) and the right hand side (RHS). Each of these are obtained using different methods. For example, 998 x 992 = 990 / 016 Here the answer to the problem 998 x 992 has been obtained mentally in two steps – one giving the left hand side of the answer (990) and the second giving the right hand side (016). The method used will be discussed in a later chapter. Bases There are two kinds of Bases – Standard Base and Special Base. Examples of standard bases are 10, 100, 1000, 10000 and so on i.e. numbers start with a 1 and followed by zeroes. Multiplication and division with these numbers are very simple – the decimal point is shifted, either to the right or to the left 0 1 2 33̅ 2̅ 1̅
9. 9. 8 respectively. Vedic math also uses Special bases. These can be any number, such as 50, 500, 5000, 25, 250, 2500 and so on. More on this later. Place value Place-value notation, or positional-notation is a way of representing numbers. The value of a digit, depends on its place or position in the number. Beginning with the ones place at the right, each place value is multiplied by increasing powers of 10. Place value for the number 24.759 is shown below To the left of the decimal point, digits to the right have smaller place value than those to the left by a factor of 10. However, to the right of the decimal point, digits to the left have higher place value than those to the right. Columns We will use the terms “Place value” and “Columns” interchangeably. For instance, while adding the numbers, we will refer to columns. Direction In conventional math, most arithmetic operations are performed Right to Left, i.e. starting with the Units column, and moving leftward to the Tens column, Hundreds column and so on. For instance, while adding 2 numbers, the Units column are added first, then the tens and so on. While subtracting numbers, again the units column is subtracted first, then the tens. Direction of operation is Right to Left. In Vedic Math, arithmetical operations are performed Left to Right. In so doing, digits with a higher Place Value are processed first, and rightfully so, as they have a larger value. 2 4 . 7 5 9 4 5 7 9 8 6 + 3 4 5 Units column Tens columnHundreds column Tens Units Decimal Tenths Hundredths Thousandths
10. 10. 9 Carryover Unlike conventional math, carryover in Vedic Math can be made either to the left or to the right. When a digit is to be carried over, it is written in small case. For instance, in the following number, the ‘2’ is a carry-over which is added to the 5. 5 2 4 6 = 7 4 6
11. 11. 1 Conventional method
13. 13. 2 Column-less method
14. 14. 13 Column-less addition Add the following numbers in the method that you are familiar with. It is likely that you started adding from the Units column (7+6+5), and then moved to the Tens column (5+8+9). Sometimes, having to start from the units column may not be a such a great idea. There may be cases where you need to start from the Leftmost column (since that column has the highest place value). In the following chapter, you will learn a method of adding numbers from the Leftmost column. For now, let us see if it is possible to add numbers starting from any column. Example 1 Watch the screencast below to see the column-less method of adding these numbers. This can also be viewed at http:// youtu.be/JmZQdFCLqvQ Screencast 2.1 Column-less addition 4 5 7 9 8 6 + 3 9 5
15. 15. 14 Step 1: Numbers are added column by column, starting with the column of your choice. Lets add numbers in Column 2, then Column 1 and lastly Column 3. Adding digits in Column 2, 5+8+9 gives 22, written as a ‘small 2’ and a ‘big 2’. Adding digits in Column 1, 4+9+3 gives 16, written as a ‘small 1’ and a ‘big 6’. Adding digits in Column 3, 7+6+5 gives 18, written as a ‘small 1’ and a ‘big 8’. Step 2: All the small digits are carried over to the previous column, 1 is carried over to the 2 giving 3, 2 and 6 give 8, and the 1 is carried over 0 to give 1. The answer is 1838. Example 2 In some cases, there will be multiple carryovers. Try adding the following numbers using the column-less method. You will see that you will need to carryover twice to arrive at the ﬁnal answer. Column1 Column2 Column3 Step 1 Step 2 4 5 7 9 8 6 + 3 9 5 16 2 2 18 1 8 3 8 2 6 7 7 7 8 + 5 5 9
16. 16. 15 Watch the screencast below to see the solution. This can also be viewed at http://youtu.be/BvyKC3SEfog Step 1: Numbers are added column by column, starting with the column of your choice. Adding digits in Column 2, 6+7+5 gives 18, written as a ‘small 1’ and a ‘big 8’. Adding digits in Column 1, 2+7+5 gives 14, written as a ‘small 1’ and a ‘big 4’. Adding digits in Column 3, 7+8+9 gives 24, written as a ‘small 2’ and a ‘big 4’. Step 2: All the small digits are carried over to the previous column, 2 is carried over to the 8 giving 10. This is written as a ‘small 1’ and a ‘big 0’. The existing ‘1’ to the left of the 8 is carried over to the Hundreds column giving 5, and the ‘1’ with the ‘4’ is carried over to the Thousands column, 0+1 giving 1. Step 3: The ‘small 1’ is carried over the 5, giving 6. The ﬁnal answer is 1606. Screencast 2.2 Double carry-over Step 1 Column1 Column3Column2 Step 2 2 6 7 7 7 8 + 5 5 9 1 4 18 2 4 1 5 10 4 1 6 0 4Step 3
17. 17. 3 Two-digit method
18. 18. 17 Two-Digit Method In this chapter, you will learn a method of adding numbers quickly and efﬁciently. You will see that numbers don’t have to be added from the right to the left only, which is most likely what you are accustomed to. Lets review two points from before. Place value Place-value notation, or positional-notation is a way of representing numbers. The value of a digit, depends on its place or position in the number. Beginning with the ones place at the right, each place value is multiplied by increasing powers of 10. To the left of the decimal point, digits to the right have smaller place value than those to the left by a factor of 10. However, to the right of the decimal point, digits to the left have higher place value than those to the right. Direction In conventional math, most arithmetic operations are performed Right to Left, i.e. starting with the Units column, and moving leftward to the Tens column, Hundreds column and so on. For instance, while adding numbers, the Units column are added first, then the Tens column, then the Hundreds. While subtracting numbers, again the Units column is subtracted first, then the Tens. Direction of operation is Right to Left. In Vedic Math, arithmetical operations are performed Left to Right. Digits with a higher Place Value are processed first, and rightfully so, as they have a larger value. Let us examine this more closely with Addition. Add these two sets of numbers. It is likely that you added the above numbers from the Right to the Left, i.e. starting with the Units column in both cases. Now see if you can ﬁnd a way to add these numbers from the Left to the Right. i.e. start adding from the Hundreds column in both cases! 2 5 6 + 8 9 4 6 3 8 + 1 9 4
19. 19. 18 Example 1 In the following example, we add 769 and 583, from the Left to Right i.e. we start with the Hundreds column, or the leftmost column, and move column by column, to the right. Watch the screencast below for an explanation. This can also be viewed at http://youtu.be/qSd_glxGfDo The process is broken down into 6 steps, shown in roman numerals. Step i: Start by adding the Hundreds column, 7+5 to get 12. This is written as a 1 in the preceding column (thousands column), and the 2 is carried over to the subsequent column (tens column). Step ii: The 2 and the 6 in the tens column are combined as “26”. Add 26 and 8 to get 34. Step iii: 34 written as a 3 in the preceding column (the Hundreds column), and the 4 is carried over to the subsequent column (Units column). Step iv: The 4 and 9 are combined as 49. Add 49 and 3 to get 52 7 2 6 4 9 + 5 8 3 1 3 5 2 Step i Step iii Step v Step vi Step ii Step iv Screencast 3.1 Adding 769 and 583
20. 20. 19 Step v: 52 is written as 5 in the previous column (tens column), and the 2 is carried over to the subsequent column. Step vi: Since there is no column to the right of the units column, the 2 from the 52 is written in the units column. Read the above steps again carefully, and apply this method to add the following 325 and 948. The steps are explained in the next page, but spend a few minutes trying to ﬁgure this out yourself, before continuing. Hint: start writing your answer one column ahead. Example 2 Were you able to add 325 and 948 from Left to Right? These numbers are again added Left to Right i.e. we start with the leftmost column, and move to the right. Watch the screencast below for an explanation. This is also available at http:// youtu.be/3mNpth3NwQ0 3 2 5 + 9 4 8 Screencast 3.2 Adding 325 and 948
21. 21. 20 Here is the solution for this exercise: Step i: Start by adding the hundreds column, 3+9 to get 12. This is written as a 1 in the preceding column (Thousands column), and the 2 is carried over to the subsequent column (Tens column). Step ii: The 2 that was carried over, and the 2 in the tens column are combined as “22”. Add 22 and 4 to get 26. Step iii: 26 written as a 2 in the preceding column (the Hundreds column), and the 6 is carried over to the subsequent column (Units column). Step iv: The 6 and 5 are combined as 65. Add 65 and 8 to get 73 Step v: 73 is written as 7 in the previous column (Tens column), and the 3 is carried over to the subsequent column. Step vi: Since there is no column to the right of the units column, the 3 is written in the units column. Example 3 Try adding 4658 and 7589 using this method on your own now. Make sure you add Left to Right. The solution is explained in the next page, however, try to ﬁgure this out before seeing the solution. Step i Step iii Step v Step vi Step ii Step iv 3 2 2 6 5 + 9 4 8 1 2 7 3 4 6 5 8 + 7 5 8 9
22. 22. 21 Watch the screenshot below for the solution. This is also available at http://youtu.be/GyRFgxb-gy8 Step i and ii: Start by adding the leftmost column, 4+7 to get 11. This is written as a 1 in the previous column, and 1 is carried over to the subsequent column (Tens column). Step iii: The 1 that was carried over, and the 6 in the tens column are combined to get “16”. 16 and 5 are added to get 21. Step iv: 21 written as a 2 in the preceding column, and the 1 is carried over to the subsequent column. Step v: The 1 and 5 are combined as 15. 15 and 8 are added to get 23 Step vi: 23 is written as 2 in the previous column, and the 3 is carried over to the subsequent column. Step vii, viii, ix: 38 and 9 are added to get 47 the last two digits in the answer. Step ii Step iv Step vi Step viii Step iii Step vStep i Step vii Step ix 4 1 6 1 5 3 8 + 7 5 8 9 1 2 2 4 7Screencast 3.3 Adding 4658 and 7589
23. 23. 22 Exercises in Two-Digit Addition Try the following exercises, starting from the leftmost column, and move to the right. Answers are provided in Chapter 6. Attempt the following mentally, i.e. keep the carryovers in your mind, and write down only the digits of the ﬁnal answer. 5 1 4 3 + 2 6 2 9 4 3 5 8 + 7 2 0 9 7 6 5 8 + 1 2 7 9 3 0 2 5 7 3 4 2 6 + 2 5 8 Click for solution 7 5 6 1 2 2 9 2 3 + 4 0 Click for solution Click for solution Click for solution 3 0 9 2 + 7 1 7 4 Click for solution Click for solution 6 8 0 2 + 5 4 1 6 Click for solution 9 3 2 + 4 8 7 6 1245 + 3529 = 4427 + 1903 = 12.54 + 23.56 = 45.95 + 45.95 = 8.695 + 3.795 = Click for solution Click for solution Click for solution Click for solution Click for solution 34.50 + 88.50 = Click for solution Click for solution
24. 24. 23 Why is this called the Two-digit Method? You may have wondered why this technique is called the “Two- Digit method” of addition. It is so called because any column that is added, must yield 2, and only 2 digits. For example, in the following, the hundreds column sums to 3, but should be written as ‘03’, with the ‘0’ in the thousands place, and the ‘3’ carried over to the tens column. Again while adding the Units column, 02+7 gives 9, but must be written as ‘09’. The answer will be incorrect otherwise. What would happen if the sum of any column results in 3 digits? This will result in a Reverse carryover. Example 1 Try the following for instance, using the Two-Digit method of addition. You will have three digits when you sum Column 2, but in the 2- digit method, you must have two, and only two digits as you sum each Column. The ﬁnal answer in this case is 1035. How would you handle the three digits of Column 2 to arrive at this answer? 2 3 6 0 2 + 1 4 7 0 4 0 9 2 5 7 3 4 2 + 4 3 6
25. 25. 24 Watch the screenshot below for an explanation, and solution of this issue. This is also available at http://youtu.be/gREz1ilAMcY The following explains this in detail. Step 1: Adding digits in Column 1, we get 9, a single digit. Hence, a zero is inserted in front of the 9, making it a ‘09’. ‘0’ is written in the preceding column, and the ‘9’ is carried over to the Tens column. Adding digits in Column 2, we get 95+4+3=102, three digits, written as a ‘small 1’ and ‘big 0’ in the Hundreds column, and ‘2’ is carried over to the Units column. Adding digits in Column 3, 27+2+6 gives 35, written as 3 in the tens column and 5 in the Units column. Step 2: The ‘small 1’ in the Hundreds column needs to be carried over to the Thousands column. 0+1=1, giving the ﬁnal answer as 1035. This is the Reverse Carryover since it moves Screencast 3.4 Reverse Carryover 2 9 5 2 7 3 4 2 + 4 3 6 0 10 3 5 1 0 3 5
26. 26. 25 from the right to the left, i.e. in the reverse direction as compared to the other carryovers (the ‘9’ was carried over from the left to the right, and so was the ‘2’). Example 2 Try another example involving Reverse Carryover. Try solving this on your own, before viewing the screenshot given in the next page. Here’s a hint: the Reverse Carryover in the above example arises due to the Units column. The ﬁnal answer in this case is 1035. How would you handle the three digits of Column 2 to arrive at this answer? Watch the screenshot below for an explanation, and solution of this issue. This is also available at http://youtu.be/ 1pIpNeYS8Q0 1 6 6 2 4 3 5 8 + 3 7 8 6 Screencast 3.5 Reverse carryover
27. 27. 26 Step 1: Adding digits in Column 1, we get 8, a single digit. Hence, a zero is inserted in front of the 8, making it a ’08’. ‘0’ is written in the preceding column, and the ‘8’ is carried over to the Hundreds column. Adding digits in Column 2, we get 86+3+7=96, ‘9’ is written in the preceding column, and the ‘6’ is carried over to the Tens column. Adding digits in Column 3, 66+5+8 gives 79, written as 7 in the Hundreds column and 9 is carried over to the Units column. Adding digits in Column 4, 92+8+6 gives 106. This is three digits, and is written as a ‘small 1’ and ‘big 0’ in the Tens column, and 6 in the Units column. Step 2: The ‘small 1’ in the Tens column needs to be carried over to the Hundreds column. 7+1=8, giving the ﬁnal answer as 9806. This is the Reverse Carryover since the 1 is carried over from the right to the left, i.e. in the reverse direction as compared to the other carryovers (the ‘8’ was carried over from the left to the right, and so were the ‘6’ and ‘9’). Although this still works, the Reverse Carryover prevents us from arriving at the ﬁnal answer mentally, and hence, a paper and pen become necessary. The alternative is to use the Three-Digit method of addition. How would that work? What would be the steps? Try the following, and formulate a Three-Digit method of addition. The goal is to sum these numbers mentally, and avoid the Reverse Carryover. 1 8 6 6 6 9 2 4 3 5 8 + 3 7 8 6 0 9 7 10 6 0 9 8 0 6 Step 1 Step 2 2 0 2 3 3 4 + 5 6 7
28. 28. 4 Three-digit method
29. 29. 28 Three-Digit method The Three-Digit method of addition is a variation of the Two- Digit method. In some cases, the Two-Digit method leads to a Reverse Carryover in the ﬁnal step, which can be avoided using this method. Example 1 Let us revisit the question from the previous chapter. Were you able to formulate the Three-Digit method for summing the numbers below? The answer in this case is 1103. How can we add these numbers, from left to right (starting with the Hundreds column), and arrive at the answer mentally? Watch the screencast below for an explanation of the Three- Digit method. This can also be viewed at http://youtu.be/ 9EdIMGNAJqc Screencast 4.1 Three-Digit method 2 0 2 3 3 4 + 5 6 7