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In the previous chapter we explained with pictures how to get multiplication tables of 6 & above. Now we will explain the logic behind why this method works.
Modulo 5 Representation
In this method numbers are represented on the palm by extending fingers such that the number of fingers extended equals the number less 5. For convenience fingers are extended starting sequentially from the little finger.
Hence 8 is represented by 3 fingers – the Little, Ring & the Middle fingers. 6 is represented by 1 finger – the Little finger.
Multiplication Using Fingers - Explanation
Let us take example of 6 X 8. It can be modified into an expression as shown below. For simplicity we will use the “.” To represent the multiplication sign.
8 X 6
= (10 – 2).(10 -4)
= 100 – 10.2 – 10.4 + 2.4
= 10 (10 – 2 - 4) + 2.4
= 10 {(5-2) + (5 – 4)} + 2.4
Let us see the significance of each of the terms.
(5-2) represents 2 fingers folded from one of the palms and 3 extended, which is mod5 representation of 8.
(5 – 4) represents 4 fingers folded from the other palm and 1 extended, which is the mod5 representation of 6.
If both the palms are brought together in front of our face, as if in prayer, then a total of 4 fingers would be extended in both the palms – 3 from one palm and 1 from the other.
Think of this 4 as if it is in the ten’s place – its value is 40.
This is the value of the expression 10 {(5-2) + (5 – 4)}
In 2.4, 2 represents the number of folded fingers in the palm representing 8 in mod5 mode – since 3 are extended 2 must be folded – the thumb & the index finger.
Similarly, in 2.4, 4 represents the number of folded fingers in the palm representing 6 in mod5 mode – since 1 is extended 4 must be folded – all fingers except the little finger.
2.4 is the product of the number of fingers folded in each of the palms. The product is 8.
The value of 8 X 6 is 40 + 8 – 40 from Step 4 & 8 from Step 8
It becomes easy of the folded fingers are folded in such a way that all of them touch at the tip of the thumb.
Multiplication Using Fingers – Procedure
Now that we have understood the logic behind the process, let us just understand the procedure with another example. Let us take 7 X 8.
Represent 7 on the left palm – Extend 2 fingers (little & ring)
Fold the other 3 fingers on the left palm to touch the tip of the left thumb.
Represent 8 on the right palm – Extend 3 fingers (little, ring & middle)
Fold the other 2 fingers on the right palm to touch the tip of the right thumb.
Bring the 2 palms together in front of the face.
Touch the tips of the left & right thumb (together with the other folded fingers). This may require a little practice especially for children.
The number of extended fingers facing the eyes is 2 + 3 = 5 which is actually 50
The product of the folded fingers touching one another is 3 X 2 which is 6
Hence 7 X 8 is 50 + 6 = 56
A Slight Variation
If we perform 6 X 6, we will get 2 fingers extended in both palms, which is 20.
The folded touching fingers in each palm would be 4 each. Hence the product is 16
Hence 6 X 6 = 20 + 16 = 36.
Here principle is the same but the figures are slightly different.
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