"Historical roots of the justification for the rule for multiplication of negative numbers"

This was an on going evolution process, it's impossible to pinpoint someone and give him all the credit. For hundreds of years (maybe thousands) merchants and accountants rely a lot on such rules for their bookkeeping, even without a formal description from the mathematicians about what negative numbers was.

Mentioned this, probably the first one to state such rules was the Indian mathematician Brahmagupta at 7th century:

The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.

At that time there was no concept of number line (John Wallis at 17th century) or vectors (20th century). His proof should have been based only on simpler concepts. The most simple proof is that all the properties of the elementary arithmetic operations must be the same whether the number is positive or negative. In other words, the distributive property must be valid also for negative numbers:

(-1) . 0 = 0
(-1) . (1 + (-1)) = 0
(-1).1 + (-1).(-1) = 0
(-1) + (-1).(-1) = 0
(-1).(-1) = 1

"Historical roots of the justification for the rule for multiplication of negative numbers"

This was an on going evolution process, it's impossible to pinpoint someone and give him all the credit. For hundreds of years (maybe thousands) merchants and accountants relied a lot on such rules for their bookkeeping, even without a formal description from the mathematicians about what negative numbers was.

Mentioned this, probably the first one to state such rules was the Indian mathematician Brahmagupta in the 7th century:

The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.

At that time there was no concept of number line (John Wallis at 17th century) or vectors (20th century). His proof should have been based only on simpler concepts. The most simple proof is that all the properties of the elementary arithmetic operations must be the same whether the number is positive or negative. In other words, the distributive property must be valid also for negative numbers:

(-1) . 0 = 0
(-1) . (1 + (-1)) = 0
(-1).1 + (-1).(-1) = 0
(-1) + (-1).(-1) = 0
(-1).(-1) = 1

"Historical roots of the justification for the rule for multiplication of negative numbers"

This was an on going evolution process, it's impossible to pinpoint someone and give him all the credit. For hundreds of years (maybe thousands) merchants and accountants rely a lot on such rules for their bookkeeping, even without a formal description from the mathematicians about what negative numbers was.

Mentioned this, probably the first one to state such rules was the Indian mathematician Brahmagupta at 7th century:

The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.

At that time there was no concept of number line (John Wallis at 17th century) or vectors (20th century). His proof should have been based only on simpler concepts. The most simple proof is that all the properties of the elementary arithmetic operations must be the same whether the number is positive or negative. In other words, the distributive property must be valid also for negative numbers:

1 . 0 = 0
1 . (1 + (-1)) = 0
1.1 + 1.(-1) = 0
1 + 1.(-1) = 0
1.(-1) = -1

"Historical roots of the justification for the rule for multiplication of negative numbers"

This was an on going evolution process, it's impossible to pinpoint someone and give him all the credit. For hundreds of years (maybe thousands) merchants and accountants rely a lot on such rules for their bookkeeping, even without a formal description from the mathematicians about what negative numbers was.

Mentioned this, probably the first one to state such rules was the Indian mathematician Brahmagupta at 7th century:

The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.

At that time there was no concept of number line (John Wallis at 17th century) or vectors (20th century). His proof should have been based only on simpler concepts. The most simple proof is that all the properties of the elementary arithmetic operations must be the same whether the number is positive or negative. In other words, the distributive property must be valid also for negative numbers:

(-1) . 0 = 0
(-1) . (1 + (-1)) = 0
(-1).1 + (-1).(-1) = 0
(-1) + (-1).(-1) = 0
(-1).(-1) = 1