What is the reason for putting the restriction on the divisor and remainder in division algorithm?

Question

Division Algorithm :

Given $2$ numbers, $a$ and $b$, with $b>0$, $\exists ! \, q,r \in \mathbb{Z}$ such that $a=bq+r$ with $0≤r<b$

While I understand this statement fully well and what it is trying to say. I fail to understand the reason for having,

1. $b>0$ why do you need $b$ to be greater than $0$. I'm pretty much confident I have seen negative divisor countless times
2. $0≤r<b$ what is the need for having $r≥0$? Once again I'm pretty confident, I have seen negative remainders countless times

Then why is it that Division Algorithm puts these restrictions?

What would happen if these conditions were not put on them?

Answer 1

1. Indeed, the condition is often written $0 \le r < |b|$ to cater for negative divisors.

2. Sure, you can replace $0 \le r < |b|$ by $-|b|/2 \le r < |b|/2$ for instance.

Answer 2

These restrictions are used because

• they make the proofs easier,

• they can easily be circumvented.