Transcript Arithmetic Operations Revisited

VEDIC MATHEMATICS :
Arithmetic Operations
T. K. Prasad
http://www.cs.wright.edu/~tkprasad
Prasad
Arithmetic Operations Revisited
1
Positional Number System
TENTHOUSANDS
THOUSANDS
HUNDREDS
TENS
UNITS
43210
=
4 * 10,000
+ 3 * 1,000
+ 2 * 100
+ 1 * 10
+0
Prasad
Arithmetic Operations Revisited
2
Two Digit Multiplication (above the
base) using Vedic Approach
1.
Prasad
Method : Vertically and Crosswise Sutra
2. Correctness and Applicability
Arithmetic Operations Revisited
3
Method: Multiply 13 * 12
• Write the first number to be multiplied and
excess over 10 in the first row, and the
second number to be multiplied and excess
over 10 in the second row.
13
12
Prasad
3
2
Arithmetic Operations Revisited
4
13
3
12
2
• To determine the 3-digit product:
– add crosswise to obtain the left digits
• (13 + 2) = (12 + 3) = 15
– and
– multiply the excess vertically to obtain the
right digit.
• (3 * 2) = 6
• 13 * 12 = 156
Prasad
Arithmetic Operations Revisited
5
Another Example
• 12 * 14 =
•
12
•
14
•
16
2
4
8
• 12 * 14 = 168
Prasad
Arithmetic Operations Revisited
6
Questions
• Why do both crosswise additions yield the
same result?
• Why does this method yield the correct
answer for this example?
• Does this method always work for any pair
of numbers?
Prasad
Arithmetic Operations Revisited
7
Proof Sketch
• (12 + 4) = (14 + 2) = 16
• Why are they same?
• That is, the sum of first number and excess over 10 of the
second number, and ….
• (12 + (14 – 10)) = (12+14 – 10) = (26 – 10) = 16
• (14 + (12 – 10)) = (14+12 – 10) = (26 – 10) = 16
Prasad
Arithmetic Operations Revisited
8
Correctness Argument:
Two possibilities
• 12 = (10 + 2)
• 14 = (10 + 4)
• 12 * 14
= (10 + 2) * 14
= 10 * 14 + 2 * 14
= 10 * 14 + 2 * (10 + 4)
= 10 * 14 + 2 * 10 + (2 * 4)
= 10 * (14+2) + 8
Right digit
= 10 * 16 + 8
[Vertical
Left digits
Product]
= 168 [Crosswise
Addition]
Prasad
• 12 = (10 + 2)
• 14 = (10 + 4)
• 12 * 14
= 12 * (10 + 4)
= 12 * 10 + 12 * 4
= 12 * 10 + (10 + 2) * 4
= 12 * 10 + 10 * 4 + (2 * 4)
= 10 * (12 + 4) + 8
Right digit
= 10 * 16 + 8
[Vertical
Left digits
Product]
= 168 [Crosswise
Addition]
Arithmetic Operations Revisited
9
Another Example
• 15 * 12
15
12
17
18
Prasad
5
2
10
0
Arithmetic Operations Revisited
10
Yet Another Example
• 17 * 15
17
15
22
22+3
25
7
5
35
5
5
Need proof to feel comfortable!
Prasad
Arithmetic Operations Revisited
11
Method: Multiply 113 * 106
• Write the first number to be multiplied and
excess over 100 in the first row, and the
second number to be multiplied and excess
over 100 in the second row.
113
106
Prasad
13
6
Arithmetic Operations Revisited
12
113
13
106
6
• To determine the 5-digit product:
– add crosswise to obtain the left digits
• (113 + 6) = (106 + 13) = 119
– and
– multiply the excess vertically to obtain the
right digits.
• (13 * 6) = 78
• 113 * 106 = 11978
Prasad
Arithmetic Operations Revisited
13
Questions
• Why do both crosswise additions yield the
same result?
• Why does this method yield the correct
answer for this example?
• Does this method always work for any pair
of 3 digit numbers?
Prasad
Arithmetic Operations Revisited
14
Proof Sketch
• (113 + 6) = (106 + 13) = 119
• Why are they same?
• (113 + (106 – 100)) = (113 + 106 – 100) = 119
• (106 + (113 – 100)) = (106 + 113 – 100) = 119
Prasad
Arithmetic Operations Revisited
15
Correctness of Product :
Two possibilities
• 113 = (100 + 13)
• 106 = (100 + 6)
• 113 * 106
= (100 + 13) * 106
= 100 * 106 + 13 * (100 + 6)
= 100 * 106 + 13 * 100 + (13 * 6)
= 100 * (106 + 13) + 78
= 100 * 119 + 78
= 11978 Left digits Right digits
[Crosswise
Addition]
Prasad
[Vertical
Product]
• 113 = (100 + 13)
• 106 = (100 + 6)
• 113 * 106
= 113 * (100 + 6)
= 113 * 100 + (100 + 13) * 6
= 113 * 100 + 100 * 6 + (13 * 6)
= 100 * (113 + 6) + 78
= 100 * 119 + 78
= 11978 Left digits Right digits
Arithmetic Operations Revisited
[Crosswise
Addition]
[Vertical
Product]
16
Another Example
• 160 * 180
160
60
Breakdown?!
180
80
240
4800
288
00
• Note that, the product of the excess over
100 has more than two digits. However, the
weight associated with 240 and 48 are both
100, and hence they can be combined.
Prasad
Arithmetic Operations Revisited
17
Yet Another Example
• 190 * 199
190
199
289
289+89
378
90
99
8910
10
10
Breakdown?!
This approach is valid with suggested modifications!
Prasad
Arithmetic Operations Revisited
18
More Shortcuts
Prasad
Arithmetic Operations Revisited
19
Quick squaring of numbers
that end in 5
• 15 * 15
= 225
= (1*2) (5*5)
• 75 * 75
= 5625
= (7*8) (5*5)
• 95 * 95
= 9025
= (9*10) (5*5)
Prasad
• Proof: Let the two digit number
be written as D5.
• D5 * D5
= (D*10 + 5) * (D*10 + 5)
= (D*D*100) + (D*2*50) + 5*5
= (D*(D+1))*100 + 25
Arithmetic Operations Revisited
20
Quick Multiplication : Special Case
• Proof: Let two digit numbers be AB and AC.
• AB * AC
= (A*10 + B) * (A*10 + C)
= (A*A*100) + (A*10*(B+C)) + B*C
= (A*A)*100 + (A)*(B+C)*10 + (B*C)
• For B+C=10, this reduces to
A*(A+1)*100 + B*C
• For A=12, B=8 and C=2, this reduces to
(12)*(13)*100 + 16 = 15616
Prasad
Arithmetic Operations Revisited
21
Quicking squaring of numbers that
begin with 5
• 51 * 51
= (5*5+1)*100 + (1*1)
= 2601
• 57 * 57
= (5*5+7) *100 + (7*7)
= 3249
• 59 * 59
= (5*5+9) *100 + (9*9)
=3481
Prasad
• Proof: Let the two digit number
be written as 5D.
• 5D * 5D
= (50 + D) * (50 + D)
= (25 + D)*100 + (D*D)
Arithmetic Operations Revisited
22
Quick squaring of two digit numbers
• Proof: Let two digit numbers be AB.
• AB * AB
= (A*10 + B) * (A*10 + B)
= (A*A)*100 + 2*(A*10)*B + B*B
= (A*A)*100 + 20*(A*B) + (B*B)
• For AB=79, this reduces to 4900+20*63+81
= 4981+1260 =6241
• For AB=116, this reduces to 12100+20*66+36
= 12136+1320 =13456
Prasad
Arithmetic Operations Revisited
23
Generalized Multplication
Using Working Base
Prasad
Arithmetic Operations Revisited
24
23
24
+3
+4
• To determine the product, choose working base as 20:
– add crosswise to obtain the left digits with weight 20
• (23 + 4) = (24 + 3) = 27
– multiply the excess vertically to obtain the right digits.
• (3 * 4) = 12
• 23 * 24 = 27 * 20 + 12
•
= 540 + 12
23 * 24 = 552
Prasad
Arithmetic Operations Revisited
25
723
724
+23
+24
• To determine the product, choose working base as 700:
– add crosswise to obtain the left digits with weight 700
• (723 + 24) = (724 + 23) = 747
– multiply the excess vertically to obtain the right digits.
• (23 * 24) = 552
• 723 * 724 = 747 * 700 + 552
•
= 522900 + 552
723 * 724 = 523452
Prasad
Arithmetic Operations Revisited
26
783
775
-17
-25
• To determine the product, choose working base as 800:
– add crosswise to obtain the left digits with weight 800
• (783 - 25) = (775 - 17) = 758
– multiply the excess vertically to obtain the right digits.
• (17 * 25) = 425
• 783 * 775 = 758 * 800 + 425
•
= 606400 + 425
783 * 775 = 606825
Prasad
Arithmetic Operations Revisited
27
532
472
+32
-28
• To determine the product, choose working base as 1000/2:
– add crosswise to obtain the left digits with wt. 1000/2
• (532 - 28) = (472 + 32) = 504
– multiply the excess vertically to obtain the right digits.
• (+32) * (-28) = 896
• 532 * 472= (504 / 2)*1000 + (104 -1000)
•
= 252000 + 104 - 1000
532 * 472= 251104
Prasad
Arithmetic Operations Revisited
28