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Vedic addition

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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.

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Vedic addition

  1. 1. Vikram Devatha First Edition Vedic Addition
  2. 2. Vedic Addition
  3. 3. ii This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/ 4.0/. All Things Vedic Auroville Collaborative Saracon, Auroville, TN 605111, INDIA. Tel: +91-413-2622571 Web: www.allthingsvedic.in/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.
  6. 6. 5 Content has been optimized for the iPad. There are essentially two kinds of interactivity available in the iPad version: screencasts and buttons. A screencast is a digital recording of computer screen output, explaining the Vedic Math techniques, along with an audio narration. Screencasts are natively available on an iPad, and in some versions of the pdf. If the screencasts do not play on your computer or tablet, these are also available on www.youtube.com/VedicAddition for reference. At times, it is easier to explain orally, than in written words. Hence, each method is illustrated using a screencast, as well as a written explanation. Buttons are used to display the solutions to exercises. If the buttons do not work in your version of this eBook, all solutions are provided in Chapter 6 as well. Note for the teacher If you plan on using this material in your classes, I would suggest a minimum of two hours on each chapter. Supplement the exercises presented in this book with your own. Though not essential, I would suggest that the students understand the concept of negative sign, before being introduced to Vedic Math. Prerequisites No prior knowledge of Vedic Math is necessary to read and understand the material I have presented in this book. I start with the basics, and proceed to the more advanced techniques of Vedic Math. It will be helpful to know the addition tables till 20+20 to fully grasp the techniques presented in this book.
  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
  12. 12. 11 Conventional method Before we study the Vedic Math techniques of addition, let us review the method most commonly used today. Add the following numbers: It is likely that you started the addition from the Units column. The main elements in this method of addition are: 1. Addition starts with the rightmost column, usually the Units column, unless there are decimals. 2. The carry over is added on top of the column to the left. 3. All the columns are added, and the answer is given from the left. Although the conventional method can be applied to all cases of addition, it is not an efficient method. 1. Addition starts with the column of least importance i.e. the Units column, and in cases with decimal figures, addition starts with the decimal digits which have even lower place value. In day-to-day situations, it is far more important to sum the columns with the higher place value i.e. the hundreds column, or the thousands column rather than the units column. 2. Addition moves from the rightmost column to the left, however the final answer is given starting with the leftmost digit. This becomes a problem if a paper and pen are not available, since you will need to remember the digits in reverse order while giving the answer. This makes mental addition cumbersome. In the following chapters, you will learn the vedic techniques of addition that will overcome the problems that arise with conventional addition. With practice, you will perfect these new techniques. 3 1 8 1 9 4 2 5 + 6 7 0 1 4 8 4 Carryovers Addition begins with the units column 3 8 9 4 2 5 + 6 7 0
  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
  30. 30. 29 Here are the detailed steps for the Three-Digit method Step i: Adding digits in Column 1, 2+3+5 gives ‘10’, a two digit number. In this method, the sum of each column must be written as 3 digits. Hence, a zero is inserted in front of the ‘10’, making it a ‘010’. Step ii: The ‘0’ from the ‘010’ is written in the column before the preceding column (Hundred-thousands column), and the ’10’ is carried over the existing ‘0’ in the Tens column, now read as ‘100’. Step iii and iv: Adding digits in Column 2, we get 100+3+6=109, ‘1’ is written in the column before the preceding column (the Thousands column), and the ’09’ is carried over to the Units column. Step v: Adding digits in Column 3, 92+4+7 gives 103, written as ‘1’ in the Hundreds column, ‘0’ in the Tens column and ‘3’ in the Units column, giving 1103 as the final answer. Did you notice how the Three-Digit method helped us avoid the Reverse Carryover? We were able to solve the addition moving only from the left to the right. Here is another example. Example 2 Sum the following numbers using the Three-Digit method. The answers are given in the next page, but try to solve this on your own before continuing. Hint: start writing your answer two columns ahead. 2 10 0 09 2 3 3 4 + 5 6 7 0 1 1 0 3 Step i Step iii Step v Step iv Step viStep ii 4 2 5 9 + 8 7 9 3
  31. 31. 30 Watch the screencast below for the solution. This can also be viewed at http://youtu.be/N_qd2Z4kZ6o Here is the detailed solution. Step i: Adding digits in Column 1, 4+8 gives ’12’, a two digit number. In this method, the sum of each column must be written as 3 digits. Hence, a zero is inserted in front of the ’12’, making it a ‘012’. Step ii and iii: The ‘0’ from the ‘012’ is written in the column before the preceding column, and the ’12’ is carried over to the Hundreds column, next to the the existing ‘2, making it ‘122’. Step iii, iv and v: Adding digits in Column 2, we get 122+7 giving 129. ‘1’ is written in the column before the preceding column, and the ‘29’ is carried over to the Tens column next to the ‘5’. Screencast 4.2 Three-Digit method 4 12 2 29 5 04 9 + 8 7 9 3 0 1 3 0 5 2 Step iii Step vStep i Step iv Step vi Step viiStep vStep ii
  32. 32. 31 Step v and vi: Adding digits in Column 3, 295+9 gives ‘304’, written as ‘3’ in the Thousands column, and ’04’ is carried over to the Units column. Step vi and vii: Adding digits in the Units column, 49+3 gives ‘52’. This is a two digit number, but we are using the Three-Digit method. Hence, a ‘0’ is inserted in front of the number, making it ‘052’, written along the Hundreds, Tens and Units column. The answer is 13052. Now try both these examples with the Two-Digit method, and see which method may be more efficient with practice. Exercises in Three-Digit Addition Try the following exercises using the Three-Digit method. Start from the leftmost column, and move to the right. Write the sum of each column as three digits by inserting a ‘0’ in case the sum is two digits, and inserting two ‘0’s if it is a single digit. Answers are provided in Chapter 6. 3 6 7 + 4 3 7 Click for solution 1 4 5 + 8 7 4 Click for solution 3 4 2 5 + 8 2 7 8 Click for solution 1 3 0 4 + 3 9 8 Click for solution 2 3 9 7 8 5 + 2 3 Click for solution 2 5 7 1 2 8 + 3 2 3 Click for solution
  33. 33. 32 Common errors Here is a list of some common errors that we make while using the Vedic Math methods. Check if any of these apply to you. Are there any other kinds of errors that you seem to make often? Share your comments. Concluding remarks on the Three-Digit method One question that is likely to arise is, how does one decide when to use the Three-Digit method over the Two-Digit method? The Two-Digit method is surely easier, as there are fewer digits to work with, but it may result in a Reverse Carryover in some cases. Can we know in advance whether a sum will give a Reverse Carryover? If so, then we can use the Three-Digit method from the start, rather than having to switch from the Two-Digit to the Three-Digit method mid way. The answer is ‘yes’, there is a way to determine if a sum will result in a Reverse Carryover by examining the numbers to be added. But try to figure this out yourself, and mail me your answer. A hint: Read up on “Digital Roots” presented in the Book on Vedic Subtraction. Writing the sum of a column as a single digit in the Two-! Digit method, or as two digits in the Three-Digit method. Make sure that the sum of every column is written as 2 digits in the Two-Digit method, and as 3 digits in the Three-Digit method. Starting the answer in the same column as the column which is summed. This will give errors. The answer must be written in the preceding column in the Two-Digit method, or the column before the preceding column in the Three-Digit method. Making an error while summing a column. At times we sum 2 and 3 as 6. Check your calculations and search where you may have gone wrong. Mistaking the Column-less methods with the Two-Digit or Three-Digit methods. Unlike the conventional system, where there is only one method to remember, Vedic Math provides several methods for each arithmetic operation. We must understand each method clearly, and apply accordingly. X X X X Summing digits across different columns, rather than across a single column. This usually happens when the numbers are not written in an organized manner, most common in real world situations. X
  34. 34. 33 Another question that is likely to arise is, how to choose among the different Vedic Math techniques when faced with a problem. You have learnt three methods of adding numbers - the Column-less method, the Two-Digit method and the Three-Digit method. Should you choose any of these methods for a particular sum, or stick with the Conventional method? Or chuck them all and pull out the calculator? You will find the answer to this question yourself, when you have given the Vedic Math techniques a fair share of practice, and attention. Attempt some real world problems using these methods in the following chapter.
  35. 35. 5 Real world problems
  36. 36. 35 Real world problems One of the aims of this book is to make math easy to apply in day-to-day situations when a calculator, or a paper and pen are not available. This could be in the grocery store, or while paying the bill in the restaurant, or while collecting change from the pizza delivary guy. For the Vedic Math techniques to become second nature, practice is necessary. Answer the following real-world questions, using the techniques you have learnt in the previous chapters. You will soon see, that you don’t need to perform the entire addition, and from the Units column as well, in most cases. Answers are provided in Chapter 6. Irresponsible clerk John visited the medical diagnostic centre, and had a few x- rays taken. While clearing the bill, the receptionist’s computer crashed and she had the clerk make the bill manually. John has a feeling the clerk has made an error while summing the figures. Check to see if John is right! Click for solution
  37. 37. 36 The Woodchucks The Woodchucks may have won the 2014 wood chucking competition. The Iron Ladies chucked 1500 kg, a record in itself. The data entered by the referees is given below. Did the Woodchucks win? Concluding remarks You have now completed learning Addition in Vedic Math. You learnt the Column-less method which can be used to sum any column independently of the other columns, as well as the Two- Digit and Three-Digit methods with which you can sum numbers from Left to Right. All of these methods can be explained using Algebra. In fact Algebra is the foundation on which Vedic Math can be built. If you are familiar with Algebra, try proving the Vedic Math techniques using Algebra. Use these methods in your day to day situations, and you will improve with practice. You will find that the other books in this series, viz. Vedic Subtraction, Vedic Multiplication and Vedic Division will assist you in doing other mental calculations in real world situations. Click for solution
  38. 38. 6 Answers to exercises
  39. 39. 38 Answers to exercises in Chapter 3 Here are solutions to the exercises on the Two-Digit Method of addition. 5 7 1 7 4 7 3 + 2 6 3 9 0 7 7 8 2 4 1 3 5 5 5 8 + 7 2 0 9 1 1 5 6 7 7 8 6 8 5 2 8 + 1 2 7 9 0 8 9 3 7 3 0 0 1 9 6 2 + 7 1 7 4 1 0 2 6 6 6 1 8 2 0 3 2 + 5 4 1 6 1 2 2 3 8 4 9 7 3 0 2 + 4 8 7 6 0 5 8 0 8 3 4 0 4 2 5 7 3 4 2 6 + 2 5 8 1 5 5 9 7 7 5 3 6 1 2 2 9 2 3 + 4 0 1 8 4 1 14 27 46 5 + 3529 = 04774 45 43 23 7 + 1903 = 06340 13 2.5 50 4 + 23.56 = 036.10 48 5.0 98 5 + 45.95 = 091.90 8.1 63 98 5 + 3.795 = 12.490 31 4.2 50 0 + 88.50 = 123.00
  40. 40. 39 Answers to exercises in Chapter 4 Here are solutions to the exercises on the Three-Digit Method of addition. 3 07 6 79 7 + 4 3 7 0 0 8 0 4 1 09 4 01 5 + 8 7 4 0 1 0 1 9 3 11 4 16 2 69 5 + 8 2 7 8 0 1 1 7 0 3 1 01 3 16 0 69 4 + 3 9 8 0 0 1 7 0 2 2 06 5 69 7 1 2 8 + 3 2 3 0 0 7 0 8 2 09 3 03 9 7 8 5 + 2 3 0 1 0 4 7
  41. 41. 40 Answers to exercises in Chapter 4 Here are solutions to the exercises on Real World problems Irresponsible clerk The clerk has indeed made an error. Add the balance column starting from the left. 2+2 gives 04, with the ‘4’ carried over to the tens column. In the Tens column we see there is a 4 and 8, which is greater than 10. Hence, the Hundreds digit now becomes 5. However, the clerk has mentioned the Total Balance as 443, which is a mistake. The Woodchucks Summing the Hundreds column, we get 14. The ‘4’ is carried over tot he Tens column. Summing the Tens column, we get 42+7+4+1 which exceeds 50. Hence, it is clear the Woodchucks have won!
  42. 42. References
  43. 43. xlii References Glover, J., T. (2002) “Vedic Mathematics For Schools, Book 1”, Motilal Banarsidass Publishers Pvt. Ltd., New Delhi. Glover, J., T. (2003) “Vedic Mathematics For Schools, Book 2”, Motilal Banarsidass Publishers Pvt. Ltd., New Delhi. Glover, J., T. (2003) “Vedic Mathematics For Schools, Book 3”, Motilal Banarsidass Publishers Pvt. Ltd., New Delhi. 14 Gupta, A. (2006) “The Power of Vedic Maths – For Admission Test, Professional & Competitive Examinations”, Jaico Publishing House, India. Jagadguru Swami Bharathi Krishna Thirthaji Maharaja (1992) “Vedic Mathematics”, Motilal Banarsidass Publishers Pvt. Ltd., New Delhi.

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