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Vedic Multiplication

On Friday, timehop told me it was a year since I wrote this post on multiplication methods. I’d forgotten about the post, and tweeted a link to it. A number of people commented and tweeted about it and a nice discussion ensued. Part of the discussion moved to Vedic Multiplication.

I know a few Vedic Multiplication methods, and believe there to be many more. Most of the ones I know link to the most common algorithms but there is this curious one used for numbers close to one hundred.

Start with a multiplication problem:

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First, take bother numbers away from 100:

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Multiply those numbers and make them your last two digits:

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Take one of the differences from 100 away from the other number (it should be the same):

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That becomes your first two digits:

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It’s an interesting little trick. I don’t see it as something that there is any reason to teach, and I don’t think it promotes understanding at all, but it’s interesting nonetheless. I think it may have a use in lessons, as an interesting introduction to algebraic proof.

How would you prove it?

First, consider the product ab, and apply the same steps:

image

The last two digits are (100-a)(100-b). The first two are a-(100-b) which equals a+b-100. Or b-(100-a) which also equals a+b-100. To make these the first two digits of a four digit number we need to multiply the expression by 100.

This gives:

(100-a)(100-b)+100(a+b-100)

Which expands to:

10000-100a-100b+ab+100a+100b-10000

Which cancels to:

ab As required.

image

A nice, accessible, algebraic proof that proves this works. It works for all numbers, not just those close to 100, but if your product (100-a)(100-b) > 99 (ie more than 2 digits) you need to carry the digits.

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  1. November 30, 2014 at 1:36 am

    I taught some Vedic methods as a starter earlier on in the year.The multiplying by 11 and the base 10 multiplication method you list above. It’s more impressive if you use this method to multiply from around the 1000, as in 994 x 991 etc. I told the students that the use was mainly as a party trick but also to explore why it works or to generate ideas.

    The Vedic is amazing for engaging students at the start of class. Weaker students love that they can do massive calculations easily and advanced students want to know the how.

    Based on this, an amazing lesson can be put together on proofs for many maths tricks. A lot of Matt Parker’s maths tricks can be put in algebraic terms. I’m sure you’ve seen this already but the TED Talk on Vedic includes the proofs in it.

    Incidentally, based on some of your other posts my starter on Monday is going to be the story of Fermat’s Last Theorem and then I’ll insert the Simpsons gif with the near solution. Cheers for that 🙂

    • November 30, 2014 at 9:03 am

      Brilliant, I’d love to hear how it goes. And thanks for the link, i had not seen that TED talk.

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