Negative Numbers — Rules for Adding & Multiplying

A thermometer reading $-8$, a bank balance of $-50$, a step to the left of zero on a number line — negative numbers are simply numbers less than zero, placed on the opposite side of zero from the positives. The trick to working with them is that addition and multiplication follow different sign rules.

The core rules and their symbols

A negative number carries two pieces of information: the sign says which side of zero you are on, and the size says how far from zero. With that in mind:

Adding, same sign — add the distances from zero and keep that sign:

Adding, different signs — subtract the smaller distance from the larger, then keep the sign of the number farther from zero:

Multiplying — same signs give a positive product, different signs give a negative product:

Why the addition rule works — picture the number line

Adding becomes intuitive as movement: adding a positive means move right, adding a negative means move left.

starts at $-2$ and moves $3$ units left, landing at $-5$. With different signs the moves compete:

starts at $-7$ and moves $4$ right. You do not reach zero, so the answer is $-3$. This is why, with opposite signs, the larger distance from zero decides the sign — it is the move that wins the tug-of-war.

Why a negative times a negative is positive

Multiplication does not use the tug-of-war logic at all. You multiply the sizes first, then set the sign purely by whether the signs match.

Different signs, negative product:

Same signs, positive product:

You can feel why the same-sign case stays positive: multiplying by $-1$ flips a number to the opposite side of zero, so multiplying by two negatives flips it twice — back to where it started, on the positive side. For most school problems the rule is enough: same signs give positive, different signs give negative.

Worked example: $-6 + 9$

The signs differ, so do not add $6$ and $9$ directly. First compare distances from zero: $9$ is farther than $6$, so the answer will be positive.

Now subtract the distances:

So

The key idea for signed addition: with opposite signs, the larger distance from zero decides the sign.

Practice these, then check the sign first

Before calculating each, say the rule out loud, then compute: $-4 + 11$, $-9 + (-2)$, and $(-5)(3)$. For the additions, name which sign rule applies; for the product, name whether the signs match. The first answer is positive (different signs, $11$ farther), the second is $-11$ (same signs, add distances), the third is $-15$ (different signs).

Calculation traps with signs

• Using the multiplication rule for addition. In addition with different signs, the sign depends on which number is farther from zero. In multiplication it does not.
• Dropping parentheses. Note that $-3^2 = -(3^2) = -9$, but $(-3)^2 = 9$. The parentheses change what is being squared.
• Keeping the wrong sign. In $-10 + 6$, since $10$ is farther from zero than $6$, the result is $-4$, not $4$.

When negative numbers are used

They appear in temperature, elevation, finance, coordinate geometry, algebra, and physics. Once the sign rules feel natural, reading equations correctly gets much easier, and small sign errors that flip an entire answer become rare.