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## Leapfrog Division

Published on Sunday, February 03, 2013 in , , ,

Today, I'm going to get you playing with numbers in a strange yet fun way. I'll save the destination for a surprise, so trust me and play along, and I think you'll like where we wind up.

Even in blogs and other resources that cover math shortcuts thoroughly, division doesn't get a lot of attention, simply because few shortcuts exist. Let's see what we can discover about division, shall we?

STARTING POINTS: Remember when you were first learning about division? First, they taught you about problems that came out evenly, such as 8 ÷ 2 = 4.

Next, they probably introduced you to problems that didn't come out evenly, and taught you about “remainders” They would give you a problem such as 13 ÷ 2, and you were expected to give the answer as “6 remainder 1.” That's the kind of division we're going to work with today. Always think about your answers as “x remainder y” for the purposes of this post.

Even if the reminder is 0, for this post, you'll need to think of the answer as including “x remainder 0” for the practices you'll be taught in this post.

“LEAPFROG” DIVISION: Let's return to the earlier example of 13 ÷ 2 = 6 remainder 1. Write down the quotient (the first number in the answer), the 6 in that example, and keep the remainder (1 in this example) in mind.

Here's the bizarre type of math to which I referred earlier. Imagine the remainder, 1, leapfrogs from its place on the left of the quotient (the 6) and lands to the quotient's immediate left. This would now look like the number 16, wouldn't it?

Let's try dividing that number by 2. What is 16 ÷ 2? It's 8. Don't forget, though, I want you to include the remainder in all answers, even if the remainder is 0. Better stated, 16 ÷ 2 = 8 reminder 0.

Once again, write down the quotient (the 8 in this last problem), and then imagine the remainder (0) leapfrogging over to the left of the quotient.

If we keep repeating this process, we wind up with a list of numbers like this:

• 13 ÷ 2 = 6 (remainder 1)
• 16 ÷ 2 = 8 (remainder 0)
• 08 ÷ 2 = 4 (remainder 0)
• 04 ÷ 2 = 2 (remainder 0)
• 02 ÷ 2 = 1 (remainder 0)
• 01 ÷ 2 = 0 (remainder 1)
• 10 ÷ 2 = 5 (remainder 0)
• 05 ÷ 2 = 2 (remainder 1)

...and so on. It's not hard to imagine that you could take this leapfrogging process as far as you like. For many numbers, but not all, the sequence will eventually repeat at some point.

Remember, you've only been writing down the quotients. Let's take a look at the sequence of quotients from the above series of problems: 68421052. Actually, we could have stopped earlier or later in that process, so it's probably more effective to think of that sequence as a decimal number: 0.68421052.

Let's put that decimal number into Wolfram|Alpha and see what comes up. Probably the most interesting result returned by Wolfram|Alpha is the first one in the Possible closed forms pod, wher it reads 1319 = 0.6842105263157. We started with a 13, so it's very interesting that a fraction with 13 would show up. Is this a coincidence?

Let's pick up from the last division we did above, and see where it takes us:
• 05 ÷ 2 = 2 (remainder 1)
• 12 ÷ 2 = 6 (remainder 0)
• 06 ÷ 2 = 3 (remainder 0)
• 03 ÷ 2 = 1 (remainder 1)
• 11 ÷ 2 = 5 (remainder 1)
• 15 ÷ 2 = 7 (remainder 1)
So, the whole sequence of quotients we've gotten from dividing 13 by 2 gives us 6824105263157 and 1319 = 0.6842105263157. After 13 decimal places, it's looking a lot less like coincidence.

DECIMAL PRECISION: This is actually a reliable division shortcut most identified with performing lightning calculator Alexander Craig Aitken (1895-1967), who used it when dividing by numbers ending in 9.

When given a problem such as 1319 to convert to decimal, when the divisor/denominator (the bottom number) is 9, simply round up the divisor/denominator to the nearest multiple of 10, and ignore the 0. For 1319, you'd mentally change this to 1320, then to 132, and proceed via the leapfrog division approach above.

As it turns out, this doesn't just work for 19, but for ANY number ending in a 9! If you're dividing by 29, you'd round up to 30, and focus on division by 3 repeatedly. When dividing by 39, you round up to 40, and do your leapfrog division by 4, and so on with higher numbers.

Naturally, if the dividend/numerator (the top number) is larger than the divisor/denominator, you'll want to reduce the number to a mixed fraction, first. For example, 6729 should be reduced to 2929 first, and then worked out as taugh above. Let's try that:
• 67 ÷ 29 = 2 + (9 ÷ 29) = 2.????? (do leapfrog division with 9 ÷ 3)
• 9 ÷ 3 = 3 (remainder 0)
• 03 ÷ 3 = 1 (remainder 0)
• 01 ÷ 3 = 0 (remainder 1)
• 10 ÷ 3 = 3 (remainder 1)
• 13 ÷ 3 = 4 (remainder 1)
• 14 ÷ 3 = 4 (remainder 2)
• 24 ÷ 3 = 8 (remainder 0)
• 08 ÷ 3 = 2 (remainder 2)
• 22 ÷ 3 = 7 (remainder 1)
So, if we did this correctly, 6729 should equal roughly 2.310344827 Wolfram|Alpha tells us that, yes, this is roughly correct! If you're rusty on working with mixed fractions, Math Dude has some wonderful refresher courses in the form of free podcasts: What Are Mixed Fractions?, How to Turn Mixed Fractions Into Improper Fractions, How to Add and Subtract Mixed Fractions, and How to Multiply and Divide Mixed Fractions.

Thanks to this approach, if you're comfortable dividing by, say, 8, then you can now feel just as comfortable dividing by 79. In other words, if you're comfortable dividing by a number n in your head, you now know how to easily divide 10n - 1, as well. If you can divide by 12, think how impressed people will be when you divide by 119!

TIPS: This trick can also help with numbers ending in 1, 3, and 7, as well. for numbers ending in 9, multiply both the dividend/numerator and divisor/denominator by 9. For numbers ending in 3, multiply both numbers by 3, and for numbers ending in 7, multiply both numbers by 7.

For example, if you need to work out 813, simply change the fraction to 2439, and proceed from there. What's 617? It's the same as 42119, and if you can mentally divide by 12, you already know how to do this!

Few numbers tend to be evenly divisible by numbers ending in 9, so the decimal equivalents tend to go on forever (usually repeating at some point). Due to this, you can take the leapfrog division process out to as many or as few places as you deem appropriate. If you're presenting this against a calculator, you'll generally want to go beyond the number of places that can be displayed by the calculator.

Take a look at the Mental Division tutorial in the Mental Gym, especially where the numbers ending in 9 are concerned, and you can learn further shortcuts for specific numbers ending in 9, such as 9 and 99.

Practice this, and you'll have an impressive new ability to demonstrate for your family, friends, and any teachers you may know!