When negative numbers were first introduced, many professional mathematicians found the numbers troublesome. Usually, people think of positive and negative numbers as the same things going in two different directions on the number line or the Cartesian coordinates, but there are actually some real differences that only show up when people use the numbers in calculations.
Addition and Subtraction
In terms of addition and subtraction, the behaviors of positive and negative numbers are exact opposites. Adding positive numbers increases the value of anything. On the number line, adding a positive number moves anything to the right. Adding negative numbers decreases the value of anything. On the number line, adding a negative number moves anything to the left. For subtraction, the roles of positive and negative numbers are reversed. A standard technique for keeping the signs straight during subtraction is to reverse the sign of a number and add. For example, to subtract -5 from -3, change the problem to that of adding +5 to -3.
Multiplication and Division
The multiplication and division rules for signed numbers are: when the signs are the same the result is positive, and when the signs are different the result is negative. This is different from what happens to signed numbers with addition and subtraction. The magnitudes of the factors are what determines the sign of the results in addition and subtraction. The sign of the largest factor will determine the sign of the results in addition and subtraction, but magnitude has nothing to do with the sign of the results in multiplication and division.
Roots and Powers
Roots and powers are the places where positive and negative numbers behave in different ways -- not just opposite, but different in more complex ways. The powers of positive numbers are positive and raising a positive number to a power increases its magnitude. Raising a negative number to a power depends on the parity of the power. Raising a negative number to an even power results in a positive number, but the results are negative if the power is odd. The absolute value increases in either case. The root of a positive number is positive, but the root of a negative number is a complex number. The number system had to be extended to accommodate the roots of negative numbers.
Inequalities are like equations in many respects. You can add or subtract a positive or negative number to both sides of an inequality as you can for an equality. You can multiply or divide through an inequality by a positive number as you can for an equation without changing the validity of the inequality or equation. The difference comes when you multiply or divide an equality through by a negative number. Multiplying through an equation does not effect the equal sign, but multiplying or dividing through an inequality by a negative number reverses the inequality sign. "Less than" becomes "greater than" and "greater than of equal to" becomes "less than or equal to."