Is a negative divided by a negative positive? And in ordinary arithmetic, yes: dividing one negative number by another nonzero negative number always produces a positive result. This guide explains the sign rule, why it works, how to solve these divisions, and which exceptions involving zero must be remembered.
Introduction to Dividing Negative Numbers
Division determines how many times one number is contained in another. When signed numbers are involved, the answer has two parts:
- Its magnitude, found by dividing the absolute values.
- Its sign, determined by the signs of the dividend and divisor.
The dividend is the number being divided, while the divisor is the number doing the dividing. As an example, in
[ (-20)\div(-4), ]
(-20) is the dividend and (-4) is the divisor. Both numbers are negative, so their quotient is positive Less friction, more output..
The Negative-Divided-by-Negative Rule
The sign rules for division are:
- Negative ÷ negative = positive
- Positive ÷ positive = positive
- Negative ÷ positive = negative
- **Positive ÷
…positive ÷ negative = negative.
Because of that, these four rules are a direct consequence of the way multiplication interacts with signs: since division is the inverse operation of multiplication, the sign of a quotient must be the sign that, when multiplied by the divisor’s sign, reproduces the dividend’s sign. Think about it: in symbols, if we seek (q) such that (a = b \times q), then (\operatorname{sgn}(q) = \operatorname{sgn}(a) \times \operatorname{sgn}(b)) (where (\operatorname{sgn}(x)=+1) for (x>0), (-1) for (x<0), and (0) for (x=0)). Applying this to the case where both (a) and (b) are negative yields (\operatorname{sgn}(q)=(-1)\times(-1)=+1), confirming that a negative divided by a negative is positive.
Solving a Negative‑Divided‑by‑Negative Problem
- Strip the signs – work with the absolute values.
Example: ((-20)\div(-4)) → (|-20| \div |-4| = 20 \div 4). - Perform the ordinary division – (20 \div 4 = 5).
- Re‑apply the sign rule – because both original numbers were negative, the quotient is positive: (+5).
The same steps work for any pair of non‑zero negatives, whether they are integers, fractions, or decimals: [ \frac{-7.On top of that, 5}{-0. 5}= \frac{7.5}=15,\qquad \frac{-\frac{3}{4}}{-\frac{2}{5}}= \frac{\frac{3}{4}}{\frac{2}{5}}= \frac{3}{4}\times\frac{5}{2}= \frac{15}{8}=1.5}{0.875.
The Zero Exception
Zero is the only number that breaks the usual sign pattern:
-
Zero divided by any non‑zero number (negative or positive) is zero:
(\displaystyle \frac{0}{-7}=0,\quad \frac{0}{3}=0).
The magnitude is zero, and the sign is irrelevant because (0) has no sign Practical, not theoretical.. -
Any non‑zero number divided by zero is undefined in ordinary arithmetic.
Expressions such as (\displaystyle \frac{-5}{0}) or (\frac{0}{0}) have no real‑number value; they either lead to contradictions or to indeterminate forms that require limits in calculus to interpret.
Thus, while the rule “negative ÷ negative = positive” holds universally for non‑zero divisors, one must always check that the divisor is not zero before applying it.
Quick Practice
| Problem | Work (absolute values) | Sign of quotient | Result |
|---|---|---|---|
| ((-12)\div(-3)) | (12\div3=4) | (+) (both negative) | (4) |
| ((-0.9)\div(-0.3)) | (0.9\div0. |
Conclusion
Dividing two negative numbers yields a positive result because the sign rule for division mirrors that for multiplication: the product (or quotient) of two like signs is positive, while unlike signs give a negative. The procedure is straightforward—divide the absolute values and then assign a positive sign when both original numbers share the same sign. Think about it: the only caveat is the presence of zero as a divisor, which renders the expression undefined, and zero as a dividend, which always produces zero regardless of the divisor’s sign. Mastering these principles equips you to handle signed division confidently in arithmetic, algebra, and beyond.
