Foundations-Halving

To Halve or not to Halve... That is the Question.

As with the last post, let's begin with a problem. Try and mentally calculate the following division problem: 724898/2. As before, we will return to this problem at the end. 

If you have an even number of items—let's say six jelly beans—and you halve them, then you divide them evenly into two groups—three jelly beans for you and three for a friend. The way we are using it here, you can't halve an odd number. Halving is dividing by two. Only even numbers are divisible by two.

As martian mathematics is largely a single-digit technique, we really only need to know how to halve single-digit numbers. We will start with the easy digits: even numbers. Instead of dividing by two, I will denote halving by multiplying by a half.

Halving a number with all even digits is as easy as doubling a number with all digits less than five. You can just read off the answer as quickly as you can read off the digits. Let's try a few problems to illustrate.

Now onto the odd numbers. As I said before, you can't evenly halve an odd number because only even numbers are divisible by two. So we'll need a way around this. Let's consider half of a single-digit odd number. If we were to do long division on 1, 3, 5, 7, and 9 with a divisor of 2 we would get quotients 0, 1, 2, 3, and 4 and a remainder of 1 in each.

Ignoring the remainders for the time being, the blue colored dividend can be shifted red and maintain the same quotient. We redshift the odd digits (my fellow physicists will appreciate that). We will treat an odd digit as if it were the closest even number less than the digit.

That seems simple enough. Now we have to deal with the remainders that we neglected. This time let's consider all the odd digits, but in the form of even numbers. I will multiply them by ten: 10, 30, 50, 70, and 90. I am free to add zero to one side of an equation with impunity, so let's consider half of fifty minus ten plus ten.

Do you see what's going on here? We see the redshift from before, but we also see that we must add five to half of the neighbor. If we add five to any single digit, then it is the same as the five's conjugate of the digit. Be careful to halve first and then take the five's conjugate. When we doubled, we were concerned with neighbors greater than or equal to five. Those neighbors were to the right. That is, the next digit after the working digit. When halving using martian mathematics, we are concerned with odd digits who are our left neighbor. That is the previous digit from the working digit. To summarize, when we encounter an odd digit, treat it like the next lowest even digit and remember to take the five's conjugate after halving the next digit. We can now lay out the halves of the odd numbers. The variable a is the ones' place value and can be any value including another odd number. So don't read 3a as three times a, but rather as thirty-ate [sic].

As usual, let's try a few problems with odd digits. I have placed red dots to the right of every odd digit to remind us to wear our "conjugation goggles" going forward. That way when we half the next digit we will say the five's conjugate of the neighboring value after halving it. Remember, to conjugate only after halving!

Let's have a look back at that division problem at the top of this post. We are now prepared to easily solve it mentally.

Here are some problems for you to practice on.

In my next post, we will put doubling and halving to practice by multiplying by five.
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