Arithmetic

Negative Number Rules

The rules on positive and negative numbers come down to one idea: a minus sign reverses direction. This guide works through addition, subtraction, multiplication and division with a complete sign table, then covers exponents, the number line, and the everyday cases where negatives actually turn up.

The rules on positive and negative numbers, in one table

Addition and subtraction follow the number line: adding moves you right, subtracting moves you left, and a negative sign on the number being added or subtracted reverses that direction. Multiplication and division follow a shorter rule: same signs give a positive answer, different signs give a negative one. Those two sentences cover every case below.

Read the multiplication and division blocks first, because they are the tidy ones. Count the negative signs: an even number of them gives a positive answer and an odd number gives a negative one. (−2) × (−3) × (−4) = −24 has three negatives, so the result is negative.

Addition and subtraction have no such shortcut, and this is where the familiar phrase “two negatives make a positive” does real damage. It is true for multiplication and division. It is true for subtracting a negative. It is false for adding two negatives: −5 + (−3) = −8, which is more negative, not less. Two debts do not make a credit.

The number line with negative numbers

Draw zero in the middle, the counting numbers to the right, and their mirror images to the left. −5 sits five units left of zero, exactly as far from zero as 5 is, on the other side.

Order is decided by position, not by size. Further left means smaller, so −9 < −5 < −2 < 0 < 3. That means −9 is less than −5 even though 9 is bigger than 5, which is the single most common stumble with negatives. A colder temperature and a deeper overdraft are both smaller numbers.

The distance from zero, ignoring which side you are on, is the absolute value. Both 5 and −5 have an absolute value of 5, written |−5| = 5. It answers “how far”, never “which way”, and the absolute value calculator works through the cases where a sign is buried inside an expression.

Two movement rules, and that is all

The gap between any two numbers is the distance between their positions. From −12 to 7 is 19 units, which is 7 − (−12) = 19. If you find yourself adding 12 and 7 to get that answer, you are already using the rule for subtracting a negative without naming it.

Adding a negative, and adding a negative to a negative

Adding a negative is the same as subtracting the positive version of it. The plus and the minus sit next to each other and collapse into a single minus.

Negative plus negative

When both numbers are negative, both moves go left, so the sizes add and the answer stays negative.

Add 5 and 3 as if the signs were not there, then put a minus in front. Two overdraft fees on an already overdrawn account make the balance worse by the total of both.

Signs that disagree

When one number is positive and the other is negative, they pull in opposite directions and partly cancel. Subtract the smaller size from the larger, then take the sign of whichever was larger.

The inverse property of addition

The opposite of a number is also called its additive inverse: 7 and −7 are opposites, and so are −0.25 and 0.25. On the number line they sit the same distance from zero on either side, so moving by one and then the other lands you back where you started.

This is not a piece of trivia. It is the reason subtraction can always be rewritten as adding an opposite, and it is what lets you clear a term when solving an equation: adding 7 to both sides of x − 7 = 12 works precisely because −7 + 7 = 0. It is also why a deposit exactly matching an overdraft returns the balance to zero rather than to some other number.

Subtracting negative numbers

Subtracting a negative is the same as adding the positive. The minus of the operation and the minus of the number cancel each other out.

Two arguments make this feel inevitable rather than arbitrary. The first is the debt reading: if $3 of what you owe is written off, subtracting a debt of 3 leaves you 3 better off. The second is the pattern. Watch the answers as the number being subtracted shrinks one step at a time:

Each line drops the subtracted number by 1 and raises the answer by 1, all the way down. Nothing changes at the moment the numbers cross zero, so 5 − (−3) has to be 8. Any other answer would break a pattern that held perfectly for every line above it.

Subtraction is also how you measure a gap, and this is where negatives earn their keep. The temperature difference between a low of −12°F and a high of 7°F is 7 − (−12) = 19 degrees. Doing it the other way round gives −19, which tells you the same distance and the opposite direction.

Why two negatives multiply to a positive

Multiplying by a negative flips a number to the other side of zero. Multiplying by two negatives flips it twice, which puts it back where it started, and that is the whole reason the answer comes out positive.

The pattern makes it concrete. Take a column of products by −2 and walk the left-hand number down through zero:

Every step down the left column adds 2 to the answer, without exception. Continuing that pattern past zero forces (−1) × (−2) to be 2. There is no choice about it; any other value would put a kink in a sequence that is otherwise perfectly even.

A tighter argument uses the fact that anything times zero is zero:

Division follows for free

Division is multiplication in reverse, so it inherits the same rule. −12 ÷ (−4) = 3 because 3 × (−4) = −12, and −12 ÷ 4 = −3 because −3 × 4 = −12. Checking a division by multiplying back is the fastest way to catch a dropped sign.

For a longer chain of factors, count the negatives instead of tracking them one at a time. An even count gives a positive result and an odd count gives a negative one, whatever the sizes involved. The multiplication calculator shows the sign decision alongside the arithmetic, and the plain four function calculator handles mixed chains of signed addition and multiplication in one expression.

Negative numbers with exponents

These two expressions look almost identical and give different answers:

Order of operations settles it. Exponents are evaluated before the minus sign is applied, so −3² squares the 3 first and negates the result. Parentheses override that, and (−3)² makes −3 the thing being squared. If you want the second meaning, the parentheses are not optional. The order of operations calculator and the PEMDAS calculator both show the step where this decision gets made.

Once the base really is negative, the exponent decides the sign, and the rule is just the even-or-odd count of negatives again:

A negative base raised to an even power is positive, because the factors pair off and each pair of negatives cancels. Raised to an odd power it stays negative, because one factor is left without a partner. Odd powers are the rows where the two notations happen to agree, which is exactly why the disagreement at even powers catches people out.

Where negative numbers actually turn up

Every one of these is the same arithmetic wearing different clothes, and each gives you a way to sanity-check a sign.

Temperature

Below zero is negative. A morning low of −5°F that rises 8 degrees gives −5 + 8 = 3°F. A drop of 8 from −5 gives −5 − 8 = −13°F. The span between a low of −12°F and a high of 7°F is 19 degrees, which is the subtracting-a-negative rule doing something you can feel.

Bank balances

An overdrawn account holds a negative balance. Overdrawn by $45, then hit with two $30 fees:

Deposit $200 into that account and you get −105 + 200 = 95. The deposit first has to cover the $105 hole before any of it counts as money you have, which is the additive inverse showing up in an unwelcome place.

Elevation

Sea level is zero, and below it is negative. Badwater Basin in Death Valley sits about 282 feet below sea level, so its elevation is −282 feet. A point 1,200 feet above sea level is 1200 − (−282) = 1482 feet higher than the basin floor, not 918 feet.

The same shape turns up in golf scores relative to par, floors below ground level in an elevator, and years before a fixed date. In every case the negative is not a smaller quantity but a direction, and the arithmetic is identical.

Common mistakes

• Thinking −9 is bigger than −5. It is further left on the number line, so it is smaller. Only the absolute values run the other way.
• Applying “two negatives make a positive” to addition. −5 + (−3) = −8. The phrase belongs to multiplication, division and subtracting a negative, nowhere else.
• Cancelling signs that are not both there. −5 − 3 is −8. There is nothing to cancel, because only one of those minus signs belongs to a number.
• Losing the sign in a product. −4 × 6 is −24. Multiply the sizes, then decide the sign separately, so it cannot get dropped along the way.
• Writing −3² when you mean (−3)². The first is −9 and the second is 9. The parentheses carry the whole meaning.
• Cancelling only one negative out of three. (−2)(−3)(−4) is −24, not 24. Count the negatives and check whether the count is odd.
• Confusing the opposite with the absolute value. The opposite of 7 is −7, while |7| = 7. They only coincide when the number you start with is negative.
• Adding instead of subtracting to find a gap. The distance from −12 to 7 is 7 − (−12) = 19. It looks like addition because subtracting the negative turns it into one.
• Trusting a calculator with an unbracketed minus. Type (−3)^2 rather than −3^2 whenever the negative is meant to be part of the base.

Frequently asked questions

Calculators for this

Every one of these shows the working, so you can check the method as well as the answer.

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