# What is the intuition behind Brahmagupta’s rule for multiplying negative numbers?

The rule says:

The product (or quotient) of two debts is a fortune

What I’m struggling with is what exactly is the product of two debts? What accounting need forces one to multiply debts? How do you interpret something like this?

Here’s what Wikipedia has to say but it doesn’t sound right:

Thus (−2) × 3  =  −6 and (−2) × (−3)  =  6. The reason behind the first example is simple: adding three −2's together yields −6: (−2) × 3  =  (−2) + (−2) + (−2)  =  −6. The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six: (−2 debts ) × (−3 each)  =  +6 credit.

The trippy thing is $$(-3$$ each$$)$$ - that makes no sense IMHO.

I’m okay even coming up with a contrived scenario but I can’t swallow the interpretation above.

So, how should one interpret “debt times debt is fortune from an accounting POV?

This question seems to have similar intent but the answers there are inconclusive or not "intuitively correct" (multiplication by $$-3$$ people for example). Hence the focus of this question is to purely understand it from an accounting POV vs. multiplying negative numbers in general.

• Possible duplicate of Historically, how did people define multiplication for negative numbers? – Conifold Dec 17 '18 at 21:33
• According to Mumford, "The predominately oral transmission... has not left us with any record of these discoveries. They just appear full blown in Brahmagupta’s summary." To the extent that the reasons can be guessed, it seems the rule was adopted because it made algebra work, not because of any intuitions. Explanations, such as Wikipedia's, are late inventions to help memorize the already adopted rule. – Conifold Dec 17 '18 at 21:38
• @Conifold- Yes. I'm just trying to rediscover it or even contrive a situation that'd warrant a meaningful output from "debt times debt"... – PhD Dec 17 '18 at 22:04
• For motivation unrelated to history Math SE would be a better place to ask. – Conifold Dec 17 '18 at 22:08
• – Mauro ALLEGRANZA Dec 18 '18 at 7:42

$$3 \times 2 = 6$$ because $$3 \times 2 = 3+3$$.
The same for $$(−3) \times 2 = (−3) + (−3) = −6$$.
Starting from this, we may interpret $$a \times (−2)$$ as "repeated subtraction" : we have to "subtract" twice the quantity $$a$$.
If $$a$$ is a negative quantitiy, i.e. a debt, to subtract a debt is to earn money : if I have a debt of $$-6$$ with you and I give it to you, my account changes from $$−6$$ to $$0$$ while your account changes from $$0$$ to $$-6$$.
The intuition is that since debts cancel credits, $$k0=0$$ determines $$k(-l)$$ from $$kl$$, even for $$k<0$$. Let $$a=(-2)(-3)$$ so $$a-6=(-2)(-3)+(-2)3=(-2)0=0$$. Therefore, $$a=6$$.