Hello,
I'm hoping someone can explain to me why this relationship exists. I'm studying accounting. In accounting, account balances are determined by subtracting credits from debits. This can lead to balances that are either positive (debit balance) or negative (credit balance). Today, I've been learning about elimination entries. These entries eliminate the account's balance. I was told the way to do this is subtract the debits from the credits. I quickly realized this is the same as taking the balance from debit minus credit and multiplying it by negative 1. I have never been good at math. I know this seems simple, but I truly don't understand why this is and hoping someone can explain it.
Actually, I suspect that this is an example of why people do find math difficult. Its rules are not explained as being both logically consistent and practically justified. The whole application of math to reality is downplayed, and the word problems used to teach people how to apply math are usually extremely artificial.
Most mathematicians are Platonists (at least about math). Platonism is the philosophy that human beings for example are not real; they are imperfect copies of the one real human being that exists in the perfect universe. In Plato's metaphor, the people that we meet and react with are illusionary shadows on the wall of a cave of what is real outside the cave. What makes sensible the restricted Platonism of those mathematicians who subscribe to it is that the world of numbers is an ideal world that exists in our minds.
In the physical world, if there are five sheep in a meadow, it makes no sense to ask how many sheep will be left in the meadow if we drive six out of it. The calculation 5-6 makes no physical sense. Double entry bookkeeping, however, is not about physical things. If John owes me 100 dollars and delivers goods to me at agreed price of 150 dollars, he is now trusting me to pay him 50 dollars. Or we can say that he owes me a negative 50 dollars. The whole vocabulary of bookkeeping comes from merchants keeping track of debts, either debit (meaning in Latin he owes [me]) or credit (meaning in Latin he trusts [me]).
Technically, bookkeeping is designed to make arithmetic easy. We add debit balances together; we add credit balances together, and we try to avoid subtraction until we must strike a "balance." But we can decide to call debits positive numbers and credits negative numbers (or since it is perfectly arbitrary we could decide to call debits negative numbers and credits positive numbers except it is a bit counter-intuitive to think of cash as being a negative). So it is exactly the practical needs of keeping track of abstract things like debts that leads to a practical application of negative numbers. The algebraic "rules" for negative numbers are designed to be as consistent as possible with the rules for positive numbers and be of practical use in applications.
[MATH]14 + 21 = 35.[/MATH]
But [MATH]14 = 7 * 2 \text { and } 21 = 7 * 3 \text { and } 35 = 7 * 5 \implies 7 * (2 + 3) = (7 * 2) + (7 * 3).[/MATH]
You can try hundreds of examples and find this true every time. Mathematicians make it a general rule.
[MATH]a,\ b, \text { and } c \text { are whole numbers} \implies (a * b) + (a * c) = a * (b + c).[/MATH]
They do not tell you that this rule
can be viewed as nothing more than a generalization from practical experience.
Second rule.
[MATH]3 + 7 = 10 = 7 + 3.[/MATH]
Again, you could confirm this proposition on millions of examples and never find an exception.
So why would you disagree when the mathematician says it is a general rule that
[MATH]a \text { and } b \text { are whole numbers} \implies (a + b) = (b + a).[/MATH]
When we get to fractions and negative numbers, we want our rules to change as little as possible
[MATH]a,\ b, \text { and } c \text { are rational numbers } \implies[/MATH]
[MATH](a * b) + (a * c) = a * (b + c) \text { and } (a + b) = (b + a).[/MATH]
Basically these rules say that it does not matter in what order we add numbers or whether we multiply first and then add or add first and then multiply.
Rules three and four. But negative numbers are not exactly like positive numbers. Without getting into the reasons we get
[MATH]a - (+\ b) = a + (-\ b),\ a < 0 \implies (-\ 1) * a = -\ a > 0 \text { and } a > 0 \implies (-\ 1) * a = -\ a < 0.[/MATH]
Technically, the difference between a and b equals the sum of a and b's additive inverse and minus one times a equals a's additive inverse.
OK. Let's think about how these rules apply to striking a balance among m debit entries and n credit entries (all positive numbers
[MATH]b = d_1 + d_2 - c_1 - c_2 - c_3 \ ... \ + d_m - c_n.[/MATH]
But I can turn this into a sum because [MATH]- (+\ c) = + (-\ c).[/MATH] So
[MATH]b = d_1 + d_2 + (-\ c_1) + (-\ c_2) + ( - c_3) \ ... \ d_m + (-\ c_n).[/MATH]
And I can break this sum into the sum of two sums because order is irrelevant in addition. So
[MATH]b = (d_1 + d_2 + ... \ d_m) + \{(-\ c_1) + (-\ c_2) + ( - \ c_3) \ ... \ + (-\ c_n)\}.[/MATH]
And I can turn each of those summands in the second sum into products of their additive inverses and minus 1. So
[MATH]b = (d_1 + d_2 + ... \ d_m) + \{(-\ 1)((c_1) + (-\ 1)(c_2) + ( -\ 1)(c_3) \ ... \ + (-\ 1)(c_n)\}.[/MATH]
But remember I can add before I multiply. So
[MATH]b = (d_1 + d_2 + ... \ d_m) + (-\ 1)(c_1 + c_2 + c_3 \ ... \ + c_n).[/MATH]
In other words, you can say that math tells you to do double entry bookkeeping that way. But you could equally well say that math is useful because it is consistent with double entry bookkeeping (and a whole bunch of other practical disciplines). We do not teach double entry bookkeeping to everyone because it is hardly intellectually fascinating and is of use to a very small percentage of the population. We necessarily teach math to people who cannot yet apply it. Thus the rules of math seem arbitrary and forgettable. And then you are left wondering why something is true when the general rule was taught to you in eighth grade. It is the conundrum of teaching math.
I find it fascinating that these series of steps leads one from y - x to -1 * (x - y). I imagine it would take a lot of practice (as a newbie) to come up with this correct series of steps on my own.