Calculating with Positive and Negative Numbers

Positive, negative and zero

Positive and negative numbers describe values on opposite sides of zero. On a number line, positive numbers lie to the right of zero and negative numbers lie to the left. Moving right increases a value; moving left decreases it.

The negative sign in $-5$ is part of the number: it tells us that the number is five units below zero. This differs from the subtraction sign in $8-5$, which tells us to perform an operation. In $8-(-5)$, the first sign means subtract and the second belongs to the negative number.

A useful way to think about signed numbers is through their magnitude, or distance from zero. The numbers $7$ and $-7$ have the same magnitude but opposite signs. Together they make a zero pair:

$7+(-7)=0$

Adding signed numbers

Addition can be represented by movement along a number line. Start at the first number:

• adding a positive number moves to the right;
• adding a negative number moves to the left.

For example, $-3+8$ starts at $-3$ and moves eight units right, arriving at $5$. By contrast, $4+(-7)$ moves seven units left from $4$, arriving at .

You can also reason using zero pairs. In $-3+8$, three of the positive units pair with the three negative units and cancel to zero, leaving five positive units.

When the numbers have the same sign, add their magnitudes and keep that sign:

$-6+(-4)=-10$

When their signs differ, compare their magnitudes. Subtract the smaller magnitude from the larger and use the sign of the number with the larger magnitude:

$-11+7=-4$

The magnitude $11$ is greater than $7$, so the result is negative.

Subtracting signed numbers

Subtracting a number is equivalent to adding its opposite:

$a-b=a+(-b)$

This gives two important transformations:

$5-8=5+(-8)=-3$

$5-(-8)=5+8=13$

The second calculation increases because subtracting a negative removes a negative quantity. On a number line, subtracting a positive moves left, whereas subtracting a negative moves right.

Do not rely only on the phrase “two negatives make a positive”. It applies when one negative is the subtraction operation and the other is the sign of the number being subtracted. Rewriting the calculation as addition of the opposite makes the reasoning clear. For example,

$-9-(-4)=-9+4=-5$

The answer is still negative because moving four units right from $-9$ does not reach zero.

Multiplying positive and negative numbers

For multiplication, first multiply the magnitudes. Then decide the sign:

• numbers with the same sign have a positive product;
• numbers with different signs have a negative product.

Thus,

$(-6)\times 4=-24$

and

$(-6)\times(-4)=24.$

The negative-by-negative rule is consistent with multiplication patterns. In the sequence

$3\times(-2)=-6,\quad 2\times(-2)=-4,\quad 1\times(-2)=-2,\quad 0\times(-2)=0,$

each product rises by $2$ as the first factor falls by $1$. Continuing the same pattern gives $(-1)\times(-2)=2$.

With several factors, each pair of negative signs produces a positive factor. An even number of negative factors gives a positive product; an odd number gives a negative product. For example,

$(-2)\times(-3)\times(-5)=-30.$

Zero multiplied by any number is zero.

Dividing positive and negative numbers

Division uses the same sign rules as multiplication because it is the inverse operation:

• same signs give a positive quotient;
• different signs give a negative quotient.

For example,

$-42\div 6=-7$

because $6\times(-7)=-42$, while

$-42\div(-6)=7$

because $(-6)\times7=-42$.

Zero divided by any non-zero number is zero. Division by zero is not defined, so an expression such as $8\div0$ has no numerical value.

Applying the rules to decimals and fractions

The sign rules do not change when the numbers are decimals, proper or improper fractions, or mixed numbers. Use the appropriate arithmetic method for their magnitudes, then interpret the signs.

For example,

$-2.7\times0.4=-1.08$

because the factors have different signs. For fractions,

$-\frac34+\frac12=-\frac34+\frac24=-\frac14.$

Here the magnitudes are combined using a common denominator, while the signs determine the direction of the calculation.

A negative mixed number such as $-1\frac12$ means the negative of the whole mixed number:

$-1\frac12=-\left(1+\frac12\right)=-\frac32.$

Therefore,

$-1\frac12+2\frac14=-\frac32+\frac94=-\frac64+\frac94=\frac34.$

Brackets are especially helpful whenever a negative number is an input to another operation. They make it clear which sign belongs to the number: $(-3)\times5$, $7-(-2)$ and .