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"Obvious" is the most dangerous word in mathematics.

Are you a writer who wants to buck the trend and do math correctly? You've come to the right place! Here you will find tips on how to understand mathematics better in your works, and maybe learn a bit too.

Counting is simple, right? Kids can do it! Well, kids also argue about it daily... and then they grow up and argue about it daily, too. Mathematicians have had to put a lot of thought into exactly how to count, and non-mathematicians have put just as much thought into ignoring any math that doesn't work in their favor.

Fact is, it's simple... and it's easily screwed up and confused.

You may be asking: "Why do these people care about the numbers? What's important in my story is Bob's relationship with Alice and how they're going to get by after the loss of their puppy, and not how many years ago Alice was an investment banker! That was a throw-away line! I was thinking about the puppy!"

The answer is in The Law of Conservation of Detail. The reader doesn't know which direction a story is going in the first time they read it. For all they know, it will be important. If the readers read the story again (something all authors would love!), they'll be looking for the subtleties they didn't catch before and any math errors will certainly not stand up to that. Unless you're planning on opening each story you write with Content Warnings about how much you suck at math, you want to be sure you at least get your counting right.

Here's a deeper look at counting issues, which will help you understand why simple addition and subtraction aren't so simple sometimes.

Using a calculator and/or doing calculations

Despite what your teacher may have said, there is no shame whatsoever in counting on your fingers or using a calculator.

Not everyone can do math well in their heads. Psychologists are still in debate on if a "math mind" can be taught in math class or if math classes are just weeding out the non-math people by attrition and then pointing at the calculus students and saying "look what we made".

Everyone CAN do math though, even if not very fast. Math is a series of logical steps, and counting, adding and subtracting are pretty easy steps. You can do it, take your time - there's no due date for this "homework" you assign for yourself.

Please be aware that a calculator is just Dumb Muscle. It just calculates, and won't tell you the best mathematical approach, or what units you're using, or that you forgot something.

Garbage In, Garbage Out

It's helpful to remember is the old programming term GIGO or "Garbage In, Garbage Out". If you're working with the wrong information, you'll get a wrong answer. Make sure you stop and think about the numbers you're using.

  • Are you frustrated and just want to get the math over with?
    • This is the most common cause of GIGO in math. Take your time, write out the problem. Rushed math instead of thought-out math merely increases the probability of a correct answer, but doesn't guarantee it.
  • Are you referring to the correct information when you put it in the calculator?
    • If you're looking at the numbers on the wrong piece of paper, you're going to get the wrong answer.
  • Are you using the calculator correctly?
    • If you're using a scientific calculator that allows multiple steps at once, still do the steps one at a time.
  • Does your answer make sense?
    • If you're calculating the age of a child characternote  and you get 56, your answer doesn't make sense and you should check your math.

Zero is a number

Things sometimes get confusing because of the fact that "zero" is indeed a number and can be used as one, even though it has no "value". More simply, zero represents "nothing", but it isn't "nothing" itself.

If you happen to label a group of cookies with the icing numbers zero to nine you still actually have 10 cookies, not 9. That cookie with a zero on it is just as tasty and desirable - putting a zero on it doesn't make it an un-cookie! You've includednote  the zero in your group of cookies.

The concept of "zero" as the "first number" is the source of a lot of nitpicking in math. When a child counts to ten, they are taught:

"1 2 3 4 5 6 7 8 9 10"

However in mathematicsnote  the first group of ten numbers is:

"0 1 2 3 4 5 6 7 8 9"

Ten numbers, just like our cookie example. After nine, we add the tens column to the numbers, and ten has a zero in the ones place.

But it continues to be complicated. As you know, negative numbers go in reverse from zero and include zero. Zero is neither negative or positive. There's no negative zero for the negative numbers, like "-2, -1, -0, 0 1, 2". This makes counting across zero to negatives confusing - something you do often once you get to algebra.

Decades can be dicey because most people view the first year of a decade to be the one ending in zero and the final one ending in 9, such as 1990-1999 being "the nineties". However there was never a "year zero" between the BC and AD years of 1 BC and 1 AD (the time had already passed over the thousand years previously, no one really cared), so counting from the start of the numbering system the 10-year periods since that start of AD dates start with a 1 and end with 0. This is leads to off by one errors.

Off By One Errors

There's a commonly ignored issue with counting called an OBOE or "Off By One Error", where your answer is one off of what it should be.

It's also referred to as The Fence-Post Problem, which goes like this:

You are building a straight fence (not one that connects to itself like an enclosure) and have five fenceposts. How many sections of fence do you need?

A gut answer is "five", one for each post - but that's incorrect. If you build your fence that way, you'll leave a section of fence without a post on one side - here's an illustration:

POST fencing POST fencing POST fencing POST fencing POST fencing

See how the "Fencing" on the far right doesn't have a post to hold it up on the end? It's gonna flop down. You need only four sections of fence with five fenceposts. That fifth section is gonna collect dust in your garage.

This sort of OBOE error is rampant in anything having to do with time periods when you don't consider when in the year the event happened. Remember, you're only X age in a certain year until your birthday, and very few people are born at New Years on the dot. If all you know is that a certain person is 25 in 2020, you don't know if their birthday has already passed or not unless we are told when in the year. This leads to OBOE errors - in 2020 are they 25 about to turn 26? Or are they 25 having just recently been 24? Trying to calculate this person's birth year will give you either 1994 or 1995, which is it? You can't know from the information given.

You run into this with time periods in fiction when dates are thrown around casually by a writer without much thought. In Real Life people doing genealogy argue endlessly if a person should be granted "an extra bonus birthday" the year they died, or if they died just short of their birthday.

Inclusive and exclusive counting.

Put up the numbers 1 to 5 like so:

1 2 3 4 5

The "normal" way people count things (objects, years, whatever) is to assign one number to each object. Hence the above represents five objects numbered one through five.

Now inclusive and exclusive counting are how the end numbers are handled.

  • Inclusive counts or "includes" the end numbers "1 2 3 4 5" means five objects.
  • Exclusive does not count or "excludes" the end numbers so "1 2 3 4 5" so it means here that there are three objects between objects one and five.

Another way to put it would be to make a row of objects, say numbered 0-9. You mean the same thing when you say both of these quotes:

  • "I want to take 3 through 6" (Inclusive - you want numbers 3, 4, 5 and 6.)
  • "I want all the numbers between 2 and 7" (Exclusive - you want numbers 3, 4, 5 and 6 - but 2 and 7 are only mentioned as bookends - you don't want either of them.)

Why does anybody care about these issues?

  • Well, you'll run into disputes when you include zero in your set of numbers. Zero is usually excluded when counting for this reason.
  • You'll also run into this in board games when someone will roll the die and then count the space they were already on as a space after rolling the dice so they can get to a "good" space on the board which is one less than what they rolled. They're cheating by exploiting OBOE errors by inclusively counting the space they already counted on the last turn.

This brings us to what we call "normal counting" - the kind you do every day. Why didn't "normal counting" get explained first? As with so many things in math, definitions need to be laid down first so everyone is on the same page. You'll need to know those two concepts as both are included in our "normal" counting definition.

Normal counting is when you exclude the number you were already on before starting counting and include the last.