Ok, I’m probably not giving enough info here, but let me try to explain. First, I was the typical “not good at math” kid all through school. So, please explain this like I’m 5.
This formula is for figuring out damage in the game The Division 2.
Base damage x (1 + CWD) x (1 + (CHC x CHD) + (HSC* x HSD)) x (1 + sum of total weapon damage talents) x (1 + amplify1) x (1 + amplify2) x (1 + amplify3) x (1 + DtOOC x 0.8*) x (1 + DtA x 0.44*) x (1 + DtH x 0.56*)
What I don’t understand is why the number 1 is needed? It seems like an insignificant number that wouldn’t make much of a difference.
I’m going to guess that the other numbers can range from 0 to 1 and represent bonuses to the base amount, which is 1. Some might even be negative, to take away some of the base amount.
Let’s pare the formula down to something much simpler. Let’s just say you’ve got base damage, and you’ve got a skill that gives you 10% bonus damage. The formula for that is
Base damage x (1 + .10)
If you don’t have the “1 +” in the formula, your 10% isn’t a bonus, it’s a massive penalty. Or in other words, what “10% bonus” really means is that you’re going to do 110% of base damage.
The formula you’ve given looks like it has several types of bonuses, some of which multiply each other instead of just being additive.
My guess is to prevent any of the terms from being 0, which would result in the entire product being 0. This assumes that base damage has to be non-zero.
Edit: The bonus explanations posted above, with the variables being percentages, make a lot more sense and are more likely the right answer than what I said in a gaming context.
Or to put it another way, whether 1 is “small and insignificant” depends on what you’re adding it to. If those other variables are typically, say, 10 or more, then adding 1 is fairly small and insignificant. If they’re typically 0.1, though, then adding 1 to that isn’t insignificant at all.
I suspect that the misconception might be because that very first variable, the “base damage”, might be very large. But that’s multiplying all of the 1s (and everything else), not adding.
The formula goes:
FinalDamage = BaseDamage x (1 + BonusType1) x (1 + BonusType2) etc etc
If you have no bonuses at all, that reduces to
ActualDamage = BaseDamage x 1 x 1
So with no bonuses, base damage is your actual damage. So the formula is working, because that’s what we want. Now say you get your 10% bonus. What that means is you’ll do your base damage plus an extra 10%, i.e. 110% total. The role of the “1 +” in the formula is to be the 100% of base value to which the 10% bonus is being added.
If it helps, the 1’s in the OP’s formula seem to be similar to the 1’s in the formulas for calculating (simple or compound) interest.
Say an account starts with P dollars and earns 5% interest. After a year, 5% of P would get added to the amount in the account, so it would become P + 0.05P. Or whatever the interest rate is—call it r—the amount after one year would be P + rP. But another way to write that would be as P(1 + r). You have P1 (the amount you started with) plus Pr (the amount of interest that gets added on).
And now I see that the asterisks that I meant as multiplication symbols got interpreted as “italicize this” symbols. Stoopid Discourse. (Yeah, I know there’s a way to make it display them literally; I just didn’t think of or notice that there was a problem, until it was too late.)
The answer has been covered above, but it underlines a useful point. The number one is a really important number in mathematics. It is the multiplicative identity. Multiply by one and you get what you started with. In the OP’s initial formulae, the 1 is representing the quantity you start with. One is neither large nor small, it has the specific semantics of preserving the original thing, whatever value it may have.
You see this pattern all over the place. A number multiplied by 1 plus another value. You always read that as the initial value plus a change defined by that second number.
You will see the opposite as well. Multiply by 1 minus another value - means the initial value less some fraction defined by that second number. Interest calculations are another common place to see this pattern.
Another place you see the pattern is in probability, where you forever see 1 - p. This is because if something will happen with the probability is p, the chance of it not happening is 1 - p. So formulae are forever cluttered with 1 - p, but where it has a very specific meaning beyond just the arithmetic it represents.
Let’s apply a little algebra to (hopefully) clarify this a bit.
Suppose you have an initial (or “base”) amount of something. Call it B. And you then increase that by, say, x%, that is, by a factor of x/100 – How much do you have now?
The formula commonly given is B * ( 1 + x/100 ), where it might not be immediately obvious what the “1” is doing there.
Work it out this way: You start with the amount B. Then you compute the percent increase of B * (x/100). For example, to increase by 5%, you would compute B * 0.05 – Then you ADD that increase to the original amount to get the new amount:
B + (B * x/100).
With a bit of algebra, this can be “factored” to give the equivalent formula:
B + (B * x/100) = B * ( 1 + x/100 )
For example, if the increase is 5%, or 0.05, this would be:
B + (B * 0.05) = B * (1 + 0.05) = B * (1.05)
(BTW: If you always put a blank space on each side of the * then Discourse doesn’t take it to mean italicize.)
You math folks amaze me. I don’t remember how this kind of stuff was explained to me in school (I’m 50 and don’t remember a whole lot anyway ). I was too busy not paying attention for any number of reasons. Social studies was more my forte anyway. I do remember not being given real world examples of when this weird math stuff would be used. I’m not saying figuring out damage in a video game is “real world,” but it’s something many kids could, at least, relate to.
Math didn’t start making sense to me until I got my first job, and had money to spend. Budgeting, splitting the dinner check with my friends, figuring out sales tax on three comics and a paperback, etc. When I was a cashier at Kroger I would try to run subtotals in my head. When my dad taught me how to figure out tipping, I was amazed. It seems simple now, and I’m sure a math teacher went over moving a decimal point to easily figure out 10%, but it must have gone over my head, or again, I wasn’t paying attention.
Math has a lot going on “under the hood.” I think the ideas are fairly simple, but converting those to notation (equations) adds a whole layer of difficulty. Then, being able to describe what is going on, like in a story problem, can tie it all together. But for many, it’s the last straw. They got lost somewhere in the translation of the idea into notation and vice versa.
For sure, in video games and pretty much any other form of programming, math rules the roost. In programming, we are just converting “real world” ideas into notation, math first, then computer code.
The Division 2’s damage formula is likely the same as most other similar games. As others have pointed out, expressions like this are written to handle a wide variety of possible bonus or penalty factors, but also have to default to a baseline when there are no bonuses or penalties.
The final result of such a formula would give you values ranging from (typically) zero to some large number, and it is all determined by these fractional factors. But when assembled, it looks a little mysterious to a non math person, i.e. D = 1 x (1 + A) x (1 + B) etc.
While a “math” person can recognize such a formula for what it is, I believe even a non-math person can understand the basic ideas underpinning the formula.
Your example doesnt happen to be a good example of why it is needed…
WIth your example, the bonus’s could be factors like 1.1 … and avoid doing all that 1 + stuff… but … what if you have two separate bonuses of 10% … so the new total is 20% bonus… Its not 1.1 + 1.1 = 2.2 … its ( 1 + 0.1 + 0.1 ) = 1.2 . But thats just the authors choice on how to do it. He could also do it as ( 1.1 + ( 1.1 - 1 ) ) = 1.2 … that is, when adding the extra factor, deduct the “1”… and you would be here asking why there is -1 everywhere ???
). I was too busy not paying attention for any number of reasons. Social studies was more my forte anyway. I do remember not being given real world examples of when this weird math stuff would be used. I’m not saying figuring out damage in a video game is “real world,” but it’s something many kids could, at least, relate to.