Addition and Subtraction of Integers and Rational Numbers

Addition of Integers and Rational Numbers

Using the number line, we can represent moving to the right as adding a positive integer and moving to the left as adding a negative integer.

  1. (Positive Integer) (Positive Integer)
    For , start at the point corresponding to and move units to the right. The result corresponds to , so:
  1. (Positive Integer) (Negative Integer)
    For , start at the point corresponding to and move units to the left. The result corresponds to , so:
  1. (Negative Integer) (Positive Integer)
    For , start at the point corresponding to and move units to the right. The result corresponds to , so:
  1. (Negative Integer) (Negative Integer)
    For , start at the point corresponding to and move units to the left. The result corresponds to , so:

Rules for Addition of Numbers

  1. When adding two numbers with the same sign, the sum is the absolute value of the two numbers added together, with the common sign.
  2. When adding two numbers with different signs, the sum is the absolute value of their difference, with the sign of the number with the larger absolute value.
  3. When adding two numbers with equal absolute values but opposite signs, the sum is .
  4. When adding any number to , the sum is the number itself.

Commutative and Associative Properties of Addition

In the addition of two numbers, and , we observe that:
The result remains the same regardless of the order in which the numbers are added. This is called the commutative property of addition.
Similarly, for the addition of three numbers , , and , we observe that:
The result remains the same regardless of which two numbers are added first. This is called the associative property of addition.

Rules for Addition

For three numbers , , and :
  1. Commutative Property of Addition:
  2. Associative Property of Addition:
When adding three or more numbers, it is often convenient to use the commutative and associative properties to rearrange the terms for easier calculation. For example:

Subtraction of Integers and Rational Numbers

The addition of two natural numbers, , can be expressed as subtraction: . This relationship also holds for the addition and subtraction of integers.

  1. (Positive Integer) (Positive Integer)
    , so .
    Additionally, since , we can write:
    .
    This shows that subtracting from is the same as adding to .

  2. (Positive Integer) (Negative Integer)
    , so .
    Additionally, since , we can write:
    .
    This shows that subtracting from is the same as adding to .
Thus, the subtraction of two integers can be simplified by changing the sign of the subtracted number and performing addition instead.

Subtraction of Rational Numbers

  1. The subtraction of two numbers can be rewritten as the addition of the first number and the opposite (negated) sign of the second number.
  2. Subtracting from any number leaves the number unchanged.