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Class 8Algebra6:58Published 14 Jul 2025

Addition and Subtraction of Positive and Negative Numbers

Learn the sign rules for adding and subtracting positive and negative numbers, then apply them to longer expressions with brackets.

This lesson sets out the simple rules for combining signed numbers: same signs add, while a positive and a negative are handled by taking the difference and keeping the sign of the larger number. It then shows how subtraction becomes addition once you flip the sign of the second number, and works through several multi-term examples that mix brackets and both operations.

What you'll learn

How to add two numbers with the same sign and two numbers with opposite signs
Why subtracting a number is the same as adding its opposite, by flipping the sign of the second number
How to simplify longer expressions that combine brackets, addition, and subtraction step by step

Lesson chapters

0:00The sign rules for adding numbers

0:25First addition examples

1:38Longer addition expressions with brackets

3:00Turning subtraction into addition

4:51Mixed examples with two brackets

Lesson notes

Adding and subtracting positive and negative numbers

This lesson covers the basic sign rules for combining positive and negative numbers, and then applies them to longer expressions that mix brackets, addition, and subtraction.

The rules for adding

When the two numbers have the same sign, add them and keep that sign:

$(+a) + (+b) = +(a+b), \qquad (-a) + (-b) = -(a+b)$

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

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

When the numbers have opposite signs, find the difference of their sizes and keep the sign of the larger number:

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

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

In the last two, the numbers have opposite signs, so we take $12 - 8 = 4$ and $12 - 10 = 2$ , and in each case keep the sign of the larger number, which is negative.

Longer addition expressions

Work through the brackets and combine the like signs first.

$(+28) + (+30) + (-2) + (-2) = (+58) + (-4) = +54$

$\big(0 + (-3)\big) + (-8) + (+6) = (-3) + (-8) + (+6) = (-11) + (+6) = -5$

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

Turning subtraction into addition

To subtract one number from another, change the subtraction to addition and change the sign of the second number (add its opposite, the additive inverse):

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

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

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

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

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

Mixed examples with two brackets

Simplify each bracket, then apply the subtraction rule between them.

Example 1

$\big((+8) + (-2)\big) - \big((-3) + (-7)\big)$

Left bracket: $(+8) + (-2) = +6$ . Right bracket: $(-3) + (-7) = -10$ . So

$(+6) - (-10) = (+6) + (+10) = +16$

Example 2

$\big((+4) + (+5)\big) - (-8) - (-2) = (+9) + (+8) + (+2)$

$= +15$

Example 3

$\big(0 + (-2)\big) - \big(9 - (-1)\big)$

The first bracket is $-2$ , and the second is $9 - (-1) = 9 + 1 = 10$ . So

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

Key takeaways

Same signs: add the numbers and keep the common sign.
Opposite signs: subtract the smaller size from the larger and keep the sign of the larger number.
To subtract, change it to addition and flip the sign of the second number, then apply the addition rules.