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The sign rules, and where they come from.
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In this lesson you multiply and divide positive and negative numbers, including fractions and decimals, and write fractions as decimals. The sign rules are easy to state and worth understanding rather than memorizing: a negative times a positive is repeated addition of something negative, and a negative times a negative comes out positive because it has to, for the distributive property to keep working. A rule you can rebuild is a rule you cannot misremember.
You can add and subtract positive and negative numbers, and you know that $-5$ is 5 units to the left of 0 on a number line. You know that multiplying whole numbers is repeated addition, so $4 \times 3 = 3 + 3 + 3 + 3$, and that division undoes multiplication. You can multiply and divide fractions and decimals. In this lesson you multiply and divide when some of the numbers are negative, and you learn why the sign rules must be what they are.
| Term | What it means |
|---|---|
| Rational number | A number that can be written as a fraction of two integers, such as $\tfrac{3}{4}$, $-2$ or $-0.5$. |
| Factor | One of the numbers being multiplied: in $(-3)(4)$ the factors are $-3$ and 4. |
| Product | The answer to a multiplication. |
| Quotient | The answer to a division. |
| Absolute value | The distance of a number from 0, its size without its sign: $\vert -7\vert = 7$. |
| Distributive property | The rule $a(b + c) = ab + ac$, which holds for every kind of number. |
| Terminating decimal | A decimal that ends, such as 0.375. |
| Repeating decimal | A decimal in which a digit or block of digits repeats forever, such as $0.\overline{3} = 0.333\ldots$ |
Multiplying and dividing signed numbers takes two separate decisions.
| Signs | Product or quotient |
|---|---|
| same signs: $(+)(+)$ or $(-)(-)$ | positive |
| different signs: $(+)(-)$ or $(-)(+)$ | negative |
So $(-6) \times 4 = -24$, $(-6) \times (-4) = 24$, $(-24) \div 4 = -6$ and $(-24) \div (-4) = 6$. The same rules hold for fractions and decimals: $(-0.5) \times (-8) = 4$ and $-\tfrac{1}{2} \div \tfrac{1}{4} = -2$.
The sign rules are not made up. Each one is forced, because it is the only choice that keeps the ordinary rules of arithmetic working. The sections below show why.
Another way: picture
On a number line, $3 \times (-4)$ is three jumps of 4 to the left, starting at 0, landing on $-12$. Multiplying by a negative number reverses direction: $(-3) \times (-4)$ is the mirror image of $3 \times (-4)$ across 0, landing on $+12$.
Another way: story
Think of a video game where you lose 4 points every time you hit a wall. Hitting the wall 3 times changes your score by $3 \times (-4) = -12$. Now rewind the video 3 hits: you undo three losses of 4, so your score changes by $(-3) \times (-4) = +12$.
Multiplication by a whole number is repeated addition, and that works for negative numbers too:
$$4 \times (-3) = (-3) + (-3) + (-3) + (-3) = -12$$
If you owe 3 dollars to each of 4 friends, you owe 12 dollars in all, a balance of $-12$. Because multiplication can be done in either order, $(-3) \times 4 = -12$ as well.
A pattern shows the same thing. Watch the products as the first factor goes down by 1 each time: $2 \times 4 = 8$, $1 \times 4 = 4$, $0 \times 4 = 0$. Each product is 4 less than the one before. Keep going and the next products must be $-4$ and $-8$, so $(-1) \times 4 = -4$ and $(-2) \times 4 = -8$.
Use the same pattern with a negative second factor: $2 \times (-4) = -8$, $1 \times (-4) = -4$, $0 \times (-4) = 0$. Now each product is 4 more than the one before. Continue: $(-1) \times (-4) = 4$ and $(-2) \times (-4) = 8$.
The distributive property gives a proof. We know that $4 + (-4) = 0$, so
$$(-3) \times \big(4 + (-4)\big) = (-3) \times 0 = 0$$
Distributing gives $(-3)(4) + (-3)(-4) = 0$, that is, $-12 + (-3)(-4) = 0$. The only number that adds to $-12$ to make 0 is 12, so $(-3)(-4) = 12$. If a negative times a negative were anything else, the distributive property would break, and so would most of algebra.
Division undoes multiplication. Asking $-20 \div 4$ means asking "what times 4 gives $-20$?" The answer is $-5$. Asking $-20 \div (-4)$ means "what times $-4$ gives $-20$?" The answer is 5. So division has exactly the same sign rules as multiplication, and you can always check a quotient by multiplying back.
One division is never allowed: dividing by zero. No number times 0 gives $-20$, so $-20 \div 0$ has no answer. It is undefined, whatever the sign of the number on top.
A fraction is a division, so a negative sign can sit in three places without changing the value:
$$-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$$
Each has exactly one negative sign, so each equals $-0.75$. But $\frac{-3}{-4}$ has two negative signs, and a negative divided by a negative is positive, so it equals $+0.75$. Most people write the sign in front, as $-\frac{3}{4}$, because it is easiest to read.
The rules do not change when the numbers are fractions or decimals. Find the size with the fraction or decimal methods you already know, then decide the sign.
When you divide by a fraction, the reciprocal keeps the same sign as the fraction: the reciprocal of $-\tfrac{9}{10}$ is $-\tfrac{10}{9}$, because $\left(-\tfrac{9}{10}\right)\left(-\tfrac{10}{9}\right) = 1$. Deciding the sign at the very end, after the arithmetic, keeps these steps simple.
To multiply several signed numbers, you can multiply two at a time. There is also a quicker way to find the sign: count the negative factors. Negatives cancel in pairs, so an even number of negatives gives a positive product and an odd number gives a negative product.
$(-1)(-2)(-3) = -6$ has three negatives, so it is negative. $(-1)(-2)(-3)(-4) = 24$ has four, so it is positive. Positive factors never change the sign, and if any factor is 0 the whole product is 0.
Every rational number can be written as a decimal by long division of the top by the bottom. Decide the sign first, then divide the sizes.
Some decimals terminate: $-\frac{3}{8} = -(3 \div 8) = -0.375$. Others repeat: $-\frac{2}{3} = -0.666\ldots$, written $-0.\overline{6}$. There are only as many possible remainders as the bottom number, so the remainders must eventually repeat, and then the digits repeat too. A fraction in lowest terms terminates exactly when its bottom number has no prime factors except 2 and 5. That is why eighths, fifths and twentieths end, while thirds, sixths and sevenths repeat.
Use the same steps for every multiplication or division.
Keeping the size and the sign as separate steps means you only ever do arithmetic you already know, and the sign is decided by one simple rule.
How to check. Check a quotient by multiplying it by the divisor: $(-24) \div 6 = -4$ because $(-4) \times 6 = -24$. Estimate the size: $(-0.5) \times 18$ should be about half of 18 in size. And in a story, ask whether the sign makes sense: a total loss should be negative, and a number of minutes should be positive.
On January 23 and 24, 1916, the town of Browning, Montana, went from 44 °F to $-56$ °F in about 24 hours, the largest one-day temperature drop recorded in the United States. The change was $-56 - 44 = -100$ degrees. Spread evenly over 24 hours, that is a change of $-100 \div 24 \approx -4.17$ degrees every hour. The quotient is negative because a negative change was divided by a positive number of hours. Weather forecasters use the same kind of division to describe a cold front: a drop of 18 degrees over 6 hours is a rate of $-18 \div 6 = -3$ degrees per hour, and they can multiply that rate by the hours still to come to predict the temperature later that night.
Bank statements show money leaving an account as negative numbers. Suppose a family's streaming service costs 15.49 dollars a month and is paid automatically. Each month the account changes by $-15.49$ dollars, so over a year it changes by $12 \times (-15.49) = -185.88$ dollars. If they cancel the service for 4 months of the year, they remove four of those negative changes, and the balance ends up $(-4) \times (-15.49) = 61.96$ dollars higher than it would have been. Taking away a negative change is a positive change, the same idea as a negative times a negative.
In golf, each score is reported compared with par, the number of strokes a good player is expected to need. A score of $-3$ means 3 strokes under par, which is good. A golfer who plays four rounds at $-3$, $+1$, $-4$ and $-2$ has a total of $-8$. Her average per round is $-8 \div 4 = -2$, two strokes under par each round. If she keeps up that average for 18 more rounds in a season, she expects to be about $18 \times (-2) = -36$ strokes against par for those rounds.
The most common mistake is mixing up the rules for adding and multiplying. "Two negatives make a positive" is true for multiplying, but $(-3) + (-4) = -7$, not 7. When you add, two negatives make a bigger negative.
The second is thinking a product with any negative factor is negative. $(-5)(-2) = 10$: two negatives give a positive.
The third is deciding the sign from the bigger number. That idea belongs to adding numbers with different signs. For multiplying and dividing, only the signs matter: $(-2) \times 50 = -100$ and $2 \times (-50) = -100$.
The fourth is putting two negative signs in a fraction and thinking it is still negative. $\frac{-3}{-4}$ is positive.
The last is forgetting that dividing by zero is undefined. $0 \div (-5) = 0$ is fine, but $-5 \div 0$ has no answer.
Write $4 \times (-3)$ as repeated addition.
$4 \times (-3) = (-3) + (-3) + (-3) + (-3)$
Multiplying by 4 means adding four copies of the number.
Add the first two copies.
$(-3) + (-3) = -6$
Adding negatives moves further left on the number line.
Add the other two copies.
$-6 + (-3) + (-3) = -12$
Each copy moves 3 more to the left.
Switch the order of the factors.
$(-3) \times 4 = 4 \times (-3)$
Multiplication gives the same answer in either order.
Write both products.
$4 \times (-3) = (-3) \times 4 = -12$
Different signs give a negative product of size 12.
Start from a sum that is zero.
$4 + (-4) = 0$
A number plus its opposite is 0.
Multiply both sides by $-3$.
$(-3) \times \big(4 + (-4)\big) = (-3) \times 0 = 0$
Anything times 0 is 0.
Use the distributive property on the left.
$(-3)(4) + (-3)(-4) = 0$
Multiply $-3$ by each part of the sum.
Replace $(-3)(4)$ with its value.
$-12 + (-3)(-4) = 0$
A negative times a positive is negative.
Ask what adds to $-12$ to make 0.
$-12 + 12 = 0$
Only the opposite of $-12$ does that.
State what the product of the two negatives must be.
$(-3)(-4) = 12$
Any other answer would make the distributive property fail.
Decide the sign first.
$-\tfrac{7}{12} = -(7 \div 12)$
One negative sign makes the decimal negative; now divide the sizes.
Start the long division: 12 does not go into 7.
$7 \div 12 = 0.\ldots$
The top is smaller than the bottom, so the decimal starts with 0.
Bring down a zero and divide.
$70 \div 12 = 5 \text{ remainder } 10$
The first digit after the point is 5.
Bring down a zero and divide again.
$100 \div 12 = 8 \text{ remainder } 4$
The second digit is 8.
Bring down another zero.
$40 \div 12 = 3 \text{ remainder } 4$
The third digit is 3.
Notice the remainder has repeated.
$\text{remainder } 4 \to 40 \div 12 = 3 \text{ remainder } 4$
The same remainder gives the same digit forever, so the 3 repeats.
Write the decimal with the sign.
$-\tfrac{7}{12} = -0.58\overline{3}$
The bar goes only over the digit that repeats.
Check with a rough multiplication.
$12 \times 0.5833 \approx 7.0$
Multiplying back gives about 7, so the size is right.
Divide the sizes.
$2.4 \div 0.6 = 24 \div 6 = 4$
Multiplying both numbers by 10 clears the decimals without changing the quotient.
Decide the sign.
Write the quotient and check it.
A product has $5$ negative factors, and all of its other factors are positive. None of the factors is zero. What is the sign of the product?
Complete the worked solution to find $(-8) \times 4 \times (-2)$.
Multiply the first two factors.
$(-8) \times 4 =$ p
A negative times a positive is negative.
Multiply that result by the last factor.
$\text{that product} \times (-2) =$ r
A negative times a negative is positive.
Check the sign by counting the negative factors.
$\text{two negatives} \to (+)$
An even number of negative factors gives a positive product.
Which fraction is equal to $-\dfrac{2}{11}$?
Find $(-3) \times 7$.
answer
Find $(-0.6) \times (-4)$.
answer
Find $15 \div (-3)$.
answer
Overnight, the temperature changed by a total of $-38^\circ\text{F}$ in $5$ hours, changing by the same amount each hour. What was the change per hour, as a decimal?
answer °F per hour
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A scuba diver starts at the surface and descends at a steady rate, so her depth changes by $-1.5$ feet every minute. How many minutes does it take her to reach $-19.5$ feet?
Answer:
You can multiply and divide with negative numbers. Without looking: what is the sign of a negative times a negative, why does it have to be that, and is $-\frac{5}{6}$ a terminating or a repeating decimal?
20. Your turn: $(-2.4) \div 0.6$, step 2
$(-) \div (+) = (-)$
Different signs give a negative quotient.
20. Your turn: $(-2.4) \div 0.6$, step 3
$(-2.4) \div 0.6 = -4, \quad (-4) \times 0.6 = -2.4$
Multiplying back gives the number you started with.