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A Negative Number Divided By a Negative Number

When you are dividing a negative number by a negative number, the quotient will be a positive number.

Let's do another example.

Step One: Determine the Sign

Since we are dividing $-34,$ which is negative, by $-17,$ which is negative, our quotient will be positive.

Step Two: Pretend the Numbers are Positive and Divide

$$34 \div 17 = 2$$

Step Three: Solution!

$$\text{positive } 2 = 2$$

When dividing one negative integer by another, there is one central idea: the negative signs cancel out. This is special for this particular case. A positive number divided by a negative number is actually negative. So, you can't divide any two numbers and expect to get a positive result. Positive and negative numbers have different behaviors in general.

One way to visualize this on a number line is to interpret the minus sign as movement to the left and the (implied) plus sign as movement to the right.

Dividing positive integers: a refresher

Let's say we want to divide $4$ by $2$. We can mark both of these numbers on the number line:

Then, we can answer our question by seeing how many jumps of size $2$ we need to get to $4$.

This image has two jumps of size $2$ (represented by the purple arrows facing right) to get to $4$; thus, $\color{#29A603}{4} \div \color{#8925AE}{2} = \boxed{2}$. Positive numbers divided by positive numbers give more positives.

Negative Integer Division

Let's try dividing $-4$ by $-2$. We can start with the same setup by marking $-2$ and $-4$ on the number line. The $-4$ will be to the left of $-2$.

And now we figure out how many jumps of size $-2$ (or, jumps of size $2$ to the left) it takes to get to $-4$.

This image has two jumps of size $-2$ (represented by the purple arrows facing left) to get to $-4$; thus, $\color{#29A603}{-4} \div \color{#8925AE}{-2}=\boxed{2}$.

So we get a positive number again! Visually, we see that the arrows for $4\div 2$ and $-4\div -2$ are simply flipped across $0$; this goes along with the idea that the minus signs of $-4$ and $-2$ cancel out.

Like we saw before, we can cancel out the minus signs: $$\begin{align} -30\div -5&=30\div 5\\ &=\boxed{6} \end{align}$$

(Video) Dividing Integers

by mathman1024

Here is a video by mathman1024 that goes over dividing positive integers, dividing negative integers, and dividing a negative and a positive integer.

$\large{\text{Summary}}$

Division will normally be written one of two ways: $$\begin{align} 15\div3&=5\\ \\ {\text{dividend}\rightarrow\atop \text{divisor}\rightarrow }\frac{15}{3}&=5\leftarrow \text{quotient} \end{align}$$

Division follows the sign rules given by multiplication. $$\begin{align} \frac{+}{+} &= +\\ \\ \frac{-}{-}&=+\\ \\ \frac{+}{-}&=-\\ \\ \frac{-}{+}&=-\end{align}$$ Dividing two numbers with the same sign will give a positive answer, while dividing two numbers with different signs will give a negative answer.

His example for the case of dividing two negative numbers is $(-56)\div (-7)$. We know that the answer will be positive because both the divisor and dividend are negative. We are trying to find how many groups of $7$ go into $56$.

$$(-56) \div (-7) = 8$$

Try one for yourself: