In Robert Recorde\'s Grounde of Artes (1541), an arithmetic text for algorists a

Question

In Robert Recorde's Grounde of Artes (1541), an arithmetic text for algorists and abacists, Recorde gave an rule for multiplying tow one digit numbers greater than 6.

Here is the rule: first subtract each number from 10. The product of the two differences you get is the units digit in the product of the original numbers and if you subtract one of the differences from the other original number, you get the tens digit in the product of the original numbers.

Please prove it works in general (Not to give an example)

Explanation / Answer

proof of above rule:

lets take 2 one digit numbers greater than 6:    x,y > 6

first lets subtract them from 10

x-difference = 10-x

y-difference = 10-y

product of 2 differences = (10-x)(10-y) = 100-10y-10x+xy = 10(10-y-x)+xy

this product of 2 differences is the unit digit of the original product lets say x*y = 10a+b

so 10(10-y-x)+xy = b

lets subtract 10-x from y so that => y-10+x is the tens digit in product of original numbers

so    y-10+x = 10a      (a, b are the digits of the original product)

so   10-y-x = 10a    ----1

substitute above equation 1 in   10(10-y-x)+xy = b

we get

10(10a) + xy = b

100a + xy = b

xy = b-100a = b + 10a

so product is 10a+b which same as original.

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