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# Lesson 1.2 multiplication

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### Lesson 1.2 multiplication

1. 1. Urdhva-Tirayk Sutra Multiplication Urdhva-Tirayak Sutra Principle : (ax2+bx+c)*(dx2+ex+f)= x4*ad+x3(ac+bd)+x2(af+bc+cd)+x(bf+ce)+cf Explanation: - Any number can be written as the sum of digits multiplied by a power of 10 e.g. 8 = 100*8 18= 101*1+100*8 218 = 102*2+101*1+100*8 etc. - Replace x by 10 in the above equation, and a, b,c etc by digits. - The left hand side of the equation represents the multiplication of two numbers (First one is the multiplicand and the second one, multiplier) - The right side of the equation is the result (Product) - The coefficient of x4 is the first digit of the product, the coefficient x3 is the second digit of the product and so on. (Contd.) 1
2. 2. Urdhva-Tirayk SutraThumb Rule: A close observation of the equation of the previous slide would reveal the following thumb rules for multiplication of two numbers with three digits each;- The first digit of the product is got by the vertical multiplication of the first digits fromthe left side.- The second digit is got by the cross-wise multiplication of the first two digits and by the addition of the two products- The third digit is obtained by summing the results of multiplication of the first digit ofthe multiplicand by the last digit of the multiplier, of the middle one by the middle oneand the last one by the first one.- The fourth digit is obtained by summing the results of multiplication of the seconddigit of the multiplicand by the third digit of the multiplier and the third digit of themultiplicand by the second digit of the mutiplier-The last digit of the product is obtained by multiplying the last digit of the multiplicand by the first digit of the multiplier. This rule can be extended to numbers containing any number ofdigits. 2
3. 3. Urdhva-Tirayk SutraPictorial Explanation of therule: 142 * 327 First digit: Second digit Third digit 1 4 2 1 4 2 1 4 2 3 2 7 3 2 7 3 2 7 3 14 21Fourth digit Fifth digit1 4 2 1 4 2 = 3 | 14 | 21 | 32 | 143 2 7 3 2 7 32 14 = 3+1 | 4+2 | 1+3 | 2+1 | 4 = 46434 3
4. 4. Urdhva-Tirayk SutraExamples: Note:In Example 3, the sum of the1. 12*13 = 1*1|1*3+2*1|2*3 products gives 30 and the digit in = 1 | 5 | 6 = 156 ten’s place i.e. 3, is Carried forward to the left of the ‘ | ’ and added to 9. Similarly, 7*3 gives 21 and the 2,2. 23*21 = 2*2 | 2*1+3*2 | 3*1 being a carry, is added to the 0 on = 4 | 8 | 3 = 483 the left of the ‘ | “.In these3 37*33 = 3*3 | 3*3+7*3 | 7*3 exercises, the ‘Carry’s’ are shown as = 9 |30 | 21 subscripts and ‘negative digits’ with = 9+3 | 0 +2| 1 a bar on the top of the digit, for = 1221 better understanding.4. 111*111 = 1*1 | 1*1+1*1 | In Example 4, the digits of the 1*1+1*1+1*1 | multiplicand and multiplier are 1*1+1*1 | 1*1 having the same color code, but the Digits in the multiplier are in Bold’. =1 | 2 | 3 | 2 | 1 This would enable a better =12321 understanding of the pattern of cross multiplications of digits from multiplicand and multiplier in a systematic manner. 4
5. 5. Urdhva-Tirayk Sutra 5. 108*108 108 108 1*1 | 1*0+0*1 | 1*8+0*0+8*1 | 0*8+8*0 | 8*8 = 1 | 0 | 16 | 0 | 64 = 1 | 0+1 | 6 | 0+6 | 4 = 1| 1 | 6 | 6 | 4 = 11664 Note the subscripts, which are carried and Added to the previous digits. 6. 321*52 = 321*052 321 052 3*0 | 3*5+2*0 | 3*2+2*5+1*0 | 2*2+1*5 | 1*2 = 0 | 15 | 16 | 9 | 2 = 0+1 | 5+1 | 6 | 9 | 2 = 16692 5
6. 6. Urdhva-Tirayk SutraOwing to their relevance to this context, a few algebraic examples of theUrdhva-Tiryak type are given below:1. (a+b)*(a+9b) a+b a+9b a2+10ab+b22. (a+3b)*(5a+7b) a+3b 5a+7b 5a2+22ab+21b23. (3x2+5x+7)*(4x2+7x+6) 3x2+5x+7 4x2+7x+6 12x4+41x3+81x2+79x+42 6
7. 7. Urdhva-Tirayk SutraThe Use of the VinculumThe multiplications by digits higher than 5 may some times be facilitated by theuse of the vinculum.Note: A negative digit in a number (not negative number) is represented by a bar on the top of thedigit. This is called vinivulumEx: 1. In the number 576, the digits 7 and 6 can be written with viniculum as follows; 500+76 =500+(100-24) =600-20-4 =624Ex 2. The number 73 can be written as follows; 73 = 70+3 = (100-30)+3 = 1+(-3)+3 =1 3 3But the vinculum process is one which one must very carefully practice,before one resorts to it and relies on it. 7
8. 8. Urdhva-Tirayk SutraMiscellaneous Examples:1. 73*37 By urdhva-tirayak method: By vinculum method 73 133 37 043 2181 04519 52 12 2701 2701Obviously, this method isbetter.2. 94*81urdhva-tirayak : Vinculum: Nikhilam: 94 114 94 -6 81 121 81 – 19 72414 13794 (175-100) |114 = 7614 = 7614 = 7614 8
9. 9. Urdhva-Tirayk SutraPractical Applications in Compound Multiplication:A. Area of a rectangle Examples: 1. 5’11” * 7’8” Suppose 12” = x, then the above multiplication can be written as ( 5x+11) * (7x+8) 5x+11 = 35x2+117x+88 7x+8 = 35x2+(108x+9x)+88 = 35x2+(9x*x+9x)+(7x+4) = 35x2+9x2+16x+4 = 44x2+ (12+4)x+4 = 44x2 + (x+4)x + 4 = 45x2 + 4*12 + 4 = 45x2 + 52 = 45sq.ft+ 52 sq.in 2. Similarly, work out 3’7” * 5’10” The answer should be 44 sq.ft + 124 sq.in 9
10. 10. Urdhva-Tirayk SutraVolume of ParallelpipedsExamples:1. 3’7” * 5’10” * 7’2” Suppose 12” = x, then the above multiplication can be written as (3x+7) * (5x+10) * (7x+2) = (15x2+65x+70) * (7x+2) =((15x2+(5x+5)*x+ (5x+10)) * (7x+2) = (20x2+10x+10) * (7x+2) =140x3+110x2+90x+20 = 140x3+(9x+2)x2+(7x+6)x+(x+8) = 149x3+9x2+7x+8 = 149 cuft +9*144 cuin+7*12 cuin+8 cuin = 149cuft and 1388 cuinQuestions relating to paving, carpeting, ornamenting etc can beReadily answered by this method.Example: At the rate of 7 annas, 9 pices per foot, how much 8 yards, 1 foot, 3 inches costs?( Note: 1 Re=16 Annas and 1 Anna = 12pices8 yards and 1 foot = 25 ft)Answer; 25 ft – 3 in 7 annas – 9 pies = 175 annas| 246 pies | 9*3/12 pies = 175 annas + ( 20 annas + 6 pies) + (2 1¼ pies)or 195 annas, 8 1/4 pies or Rs 12/3/8¼ End of chapter 1B – Urdhva-tiravk sutra 10