Why negative integer divided by positive is same as positive integer divided by negative

Question

My query is - why -3/4 is same as 3/-4? For -3/4, -3 is less than 4 so when -3 is divided into 4 parts, the result will be less than -3 which is correct. In this case, it will be -0.75 And, for the case of 3/-4, 3 is more than -4. So, a larger number is divided by smaller number. So, in this case, the result should be a more larger number than 3. But, as per mathematics rulebooks, the answer in this case is also -0.75 Kindly someone help to elaborate on this.

Answer 1

The actually rules we are given are: If $a < b$ then $a+c < b+c$ and if $a < b$ and if $c >0$ (that is very important that $c > 0$) then $ac < bc$.

From those we can derive other rules. If $a < b$ and $c < 0$ then $ac > bc$ (the old flipperoo). And if $0< a$ then $\frac 1a > 0$ and if $a < 0$ then $\frac 1a < 0$. And as $a\div b = a \times \frac 1b$ then if $a < b$ and $c > 0$ then $\frac ac < \frac ab$ and if $a< b$ and $c < 0$ then $\frac ac > \frac ab$ (fliperoo redux).

And we have if $0 < a < b$ then $\frac ba < 1$ and $\frac ba > 1$.

[Not to mention some really basic ones that are so basic that it might never occur to us that the *couldn't be true, such as $1 > 0$ and $-1 < 0$ and $-a = (-1)\times a$ etc.]

.......

Okay.... so how does this work with $(-3)\div 4$ and $3\div -4$.

So have $-3 < 0$ and $4 > 0$ so $(-3)\div 4 < 0\div 4 = 0$. So $(-3)\div 4 < 0$. That works if $\frac {-3}4 = -0.75$.

And we have $3 > 0$ and $-4 < 0$ so $3\div(-4) < 0\div (-4)= 0$. So $3\div(-4) < 0$. That's fine.

Further more we hav $-3 > -4$ and $4 > 0$so we should have $-3\div 4 > -4 \div 4$ and we should have $\frac {-3}4 >-1$ and that is what we have if $\frac{-3}4= -0.75 > -1$.

[Don't get confused and think $-0.75 < -1$ We have $.75 < 1$ and $-1 < 0$ so the old fliperoo gives us $-.75> -1$. If we want to see that directly we can do $0.75 < 1$ so $0 = 0.75 - 0.75 < 1 -0.75 = 0.25$ so $-1 = 0-1 < 0.25 - 1 = -0.75$]

And we have $3 < 4$ and $-4 < 0$ so we should have $3\div(-4) > 4\div (-4)$ (DOn't ever forget the fliperoo) and so $\frac 3{-4} > -1$. Which is just fine if $\frac 3{-4} = -0.75$

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So all the above tells you why we don't have a contradiction. It doesn't tell us why we must have it.

But it's simple enough.

If $\frac {-3}4 = ?????$ then $\frac{-3}4\times 4 = -3 = 4\times ?????$

So $-3+3 = 0 = 4\times ????? + 3$ so

$0 - 4\times ????? = 4\times ????? + 3- 4\times ?????= 3$ so

$-4\times ???? =3$ so

$?????? \times (-4) = 3$ so

$?????? \times (-4)\div (-4) = 3\div (-4)$ so

$????? \times 1 = ????? = 3\div (-4)$.

So $\frac {3}{-4}$ and $\frac {-3}4$ are the same thing.

But how do we know that $\frac 3{-4}$ or $\frac {-3}4 = -\frac 34$?

Let $\frac 3{-4} = A$ and $\frac 34 = B$ then

$3 = -4A$ and $3 =4B$ so $-4A = 4B$ so $A(-4) = B(4)$ and so $A(-4)\div (-4) = B(4)\div (-4)$ so $A\times(1) =A= B\times(-1)=B$ so $A = -B$ and $\frac 3{-4} = -\frac 34$.