Operations on Positive and Negative Numbers

This lesson covers the basic rules for adding and subtracting positive and negative numbers, then puts them to work through a long set of examples, including ones with brackets.

The rules

There are three things to remember:

• Same signs: if both numbers are positive, or both are negative, add the numbers and keep that common sign.
• Different signs: if one is positive and one is negative, find the difference of the numbers and keep the sign of the larger one.
• Subtraction: to subtract a number, change the subtraction to addition and flip the sign of the number being subtracted, then use the rules above.

Adding numbers with the same sign

Both positive. Positive plus positive stays positive:

$(+8) + (+4) = +12$

All negative. Add the sizes and keep the negative sign:

$(-3) + (-5) + (-8) = -16$

Adding a positive and a negative

When the signs differ, subtract the smaller size from the larger and keep the sign of the larger number.

$(-12) + (+10) = -2$

Here $12 > 10$, the difference is $2$, and the larger number is negative, so the answer is $-2$.

Working with brackets

Deal with each bracket first, then combine.

$\big[(+28) + (+30)\big] + \big[(-2) + (-2)\big] = (+58) + (-4) = +54$

The first bracket gives $+58$, the second gives $-4$. The signs differ, the difference is $54$, and the larger number is positive.

$0 + (-3) + (-8) + (+6) = -5$

The negatives add to $-11$, then $(-11) + (+6) = -5$.

$\big[(-2) + (+8)\big] + \big[(-6) + (+6)\big] = (+6) + 0 = +6$

Subtraction: change it into addition

To subtract, flip the sign of the number being taken away and add.

$(+8) - (-2) = (+8) + (+2) = +10$

$(+20) - (+3) = (+20) + (-3) = +17$

$(-30) - (-4) = (-30) + (+4) = -26$

$(+25) - (-20) = (+25) + (+20) = +45$

Subtraction with brackets

Simplify each bracket, change the subtraction to addition, then combine.

$\big[(+8) + (-2)\big] - \big[(-3) + (-7)\big] = (+6) - (-10) = (+6) + (+10) = +16$

$\big[(+4) + (+5)\big] - \big[(-8) + (+2)\big] = (+9) - (-6) = (+9) + (+6) = +15$

The additive inverse rule

The note for subtraction is: to subtract one number from another, add the additive inverse of the number to be subtracted. In other words, change the subtraction to addition and flip the sign of the second number, then add as usual.

Key takeaways

• Same signs: add the numbers and keep the common sign.
• Different signs: take the difference and keep the sign of the larger number.
• To subtract a number, add its additive inverse (the same number with the opposite sign).