Foundations-Doubling

Doubling is ×2

Before we begin, I would like you to try your hand at the following multiplication problem (in your head): 2 × 734829. If you would like to be able to multiply this as easily as reading of the digits, then read on. 

We are all familiar with doubling. If you have N items in a bag and you double it, then it is like adding a second identical bag so that now we have twice as many as we did before (2N). In martian mathematics you need to know the following doubles:

What about the numbers 5-9? We will get to that in a moment, but first let's take what we have out for a ride. If we want to double a number of any size then we just read it from left-to-right one digit at a time, doubling as we go.

Easy, huh? Now let's tackle the other five doubles you need to know. For the most part, martian math is a single-digit technique and we can ignore anything not in the ones' place. For this reason, when we double a number greater than or equal to five, we only need to double its five's conjugate. Doing so gives us only the ones' place value of the product. I will write the five's conjugate of 6 as 6*, which you should remember is equal to one. So in 2 × 6 = 12, 2 × 6* = 2 × 1 = 2 gives us the ones' place of 12. Here are the missing doubles:

Memorizing the five's conjugate values should be paying off now. Using this notation, you will note that 0 = 5* and 0* = 5, so we can simplify our table of doubles as follows:

If we try using these doubles just as we did before we run into a problem. Consider 2 × 28. If you just double each digit like we did before then we get 46 as our product. However, if we punch 2 × 28 into our calculator we see that the product should have been 56. Our tens' place value is off by one. 

In martian math, we double each digit as before, but if our right-hand neighbor is 5 or more, then we add one after doubling. Now when we double 28 we see that we should lead with 4 + 1 = 5 instead of just 4. If the leading digit is 5 or more, then imagine a zero to the left, double it to get zero and add 1. Then 2 × 84 is 2 × 084 = 168. Let's try a few more.

Finally, let's return to the multiplication problem at the beginning of this post.

Hopefully, you found it much easier this time. Here are some more problems for practice:

In my next post, we will cover the next most important foundational technique in martian math—halving.
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