The set of real numbers can be thought of as all possible distances from a fixed point, represented by [latex]0[/latex] on the number line below.  The distance of points to the right of [latex]0[/latex] are represented by positive [latex](+)[/latex] numbers.  The distance of points to the left of [latex]0[/latex] are represented by negative [latex](-)[/latex] numbers.  The sign of a number represents its direction relative to [latex]0[/latex]. Numbers are assumed to be positive if no sign is specified: [latex]2[/latex] means [latex]+2[/latex].

The integers are counting numbers and their negatives, as well as zero:

[latex]...,-3, -2, -1, 0, 1, 2, 3, ...[/latex]

The set of real numbers includes fractions and decimals, as well as the integers.

The absolute value of a real number [latex]x[/latex], represented by [latex]x[/latex], is its distance from [latex]0[/latex] without regard to direction. Since it represents distance, the absolute value of a number is never negative. For example, since [latex]-2[/latex] is located [latex]2[/latex] units to the left of [latex]0, |-2|=2[/latex]. Since [latex]2[/latex] is located [latex]2[/latex] units to the right of [latex]0, |2|=2[/latex].

If we add two positive numbers, such as [latex]1[/latex] and [latex]2[/latex], we can think of beginning at [latex]0[/latex] and moving [latex]1[/latex] unit to the right, and then [latex]2[/latex] more units to the right. So,

[latex]1+2=3[/latex]

If we add two negative numbers, such as [latex]-1[/latex] and [latex]-2[/latex], we proceed in the same way but move to the left each time. So,

[latex](-1)+(-2)=-3[/latex]

Suppose we wish to add two numbers with different signs. If we add [latex]-2[/latex] and [latex]3[/latex] we move from [latex]0[/latex] to the left [latex]2[/latex] units to [latex]-2[/latex], and then to the right [latex]3[/latex] units, ending at [latex]1[/latex].

[latex]-2 + 3 = 1[/latex]

If we add [latex]2[/latex] and [latex]-3[/latex] we move from [latex]0[/latex] to the right [latex]2[/latex] units to [latex]2[/latex], and then to the left [latex]3[/latex] units, ending at [latex]-1[/latex].

[latex]2 + (-3) = -1[/latex]

One way to think of subtraction is to consider the distance between two numbers. [latex]5-3=2[/latex] since we would need to move [latex]2[/latex] units to the right of [latex]3[/latex] to get to [latex]5[/latex]. But we can also think of subtracting a number as the addition of its opposite.

[latex]5-3=5+(-3)=2[/latex]

We can rewrite subtraction as the addition of a number’s opposite.

[latex]a-b=a+(-b)[/latex]

The following video explains how to subtract two signed integers.

In the following video are examples of adding and subtracting signed decimals.