Before we get to anything complicated like subtraction, we
should cover every facet of addition we can. We’ve got a blueprint for any
expression that looks like “_____ + _____ + _____...” if the blanks are natural
numbers, but what about cases where they’re other kinds of numbers, or other
ways to write symbols?
PLUS vs. +
We’ve been writing “+” in our Leta expressions, but there’s
no reason why the + sign (which isn’t a letter) can’t be written as PLUS. The
word PLUS contributes -4 to the Leta Score of an expression – we can think of
it as a ZERO (no value, four letters) that needs to be put in between any pair
of words. This doesn’t complicate the expressions too much, but now we’re
looking for sums of two words that have a Leta Score of 4, or sums of three
words that have a Leta Score of 8, and so on.
Some examples of these are “SEVEN PLUS SEVEN”, “FIVE PLUS
SIX”, “TWO PLUS THIRTEEN”, “FOUR PLUS EIGHT PLUS NINE”, and “SIX PLUS SIX PLUS
SIX PLUS SIX”. As you can see, it’s the same as the earlier case, but you have
to take into account how many terms you’re adding together.
An interesting little sidenote here, before we move into
anything more complicated: when we were working only with “+” and not “PLUS”, all
Leta expressions had to have a ZERO, ONE, TWO, or THREE in them, because those
numbers were the only sources of negative Leta score. If you use “PLUS”, there’s
a guaranteed source of negative score between each number, so you can have
expressions like SEVEN PLUS SEVEN” and “FIVE PLUS SIX”. Now, is there a new set
of numbers that Leta expressions must have at least one of? Well, let’s extend our
table of Leta scores a bit.
It's pretty clear that the scores are just going to keep
increasing as we look at higher and higher numbers, so there must be a point at
which any expression made of only larger numbers can’t have be Leta. It looks
like NINE is that point – all numbers greater than NINE have Leta scores that are
5 or higher. A PLUS can only contribute -4 per number (and there will always be
a number left over), so there can’t be a Leta expression that looks like “_____
PLUS _____ PLUS _____ PLUS ….” where all the blanks are NINE or more. So our
theoretical maximum is EIGHT – all Leta expressions of this type must have at
least one ZERO, ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, or EIGHT. Can we get
this maximum lower, to SEVEN? No, because I can think of a counterexample that
works with only numbers EIGHT or higher. “EIGHT PLUS EIGHT PLUS EIGHT PLUS
EIGHT” works. So, we know that every Leta expression in this form must have a
number that’s less than NINE!
Negative Numbers
The word NEGATIVE contributes -8 to the Leta Score of an
expression. So, as you get more and more negative, the scores leap negative
quickly, with NEGATIVE ONE having a Leta Score of -1 – 11 = -12. Let’s look at
the first few negative numbers:
Because every Leta score for a negative number is of the
form (negative number) – (positive number), we know that they’ll always be
negative. So we can use the same rule as before to generalize a statement for
all Leta expressions that look like “_____ + _____ + _____ + …”. Any of these
are possible as long as the Leta Scores add up to zero.
Here’s a few more interesting expressions that are Leta, now
that we can use negative numbers. “NEGATIVE ONE + TEN + THIRTEEN”, “NEGATIVE
ONE + TWENTY-ONE”, “NEGATIVE TWO PLUS TWENTY-SIX”, and “FOUR HUNDRED AND TWENTY-FOUR
+ NEGATIVE THREE HUNDRED AND SEVENTY”.
And now that we’re dealing with negative numbers, there’s no
analogue to our nice “all Leta Expressions must have these numbers” limitation.
Any number, no matter how arbitrarily high, can be countered with an arbitrarily
low number, and vice versa.
That’s enough for this time, but as we’ve only scratched the
surface so far!
