## Operations With Integers: Overview

Students know that plus and minus signs are used to indicate addition and subtraction, respectively. These signs are also used with integers to indicate specific points on a number line. A positive number means a move to the right on the number line, while a negative number means a move to the left. These concepts are key to mastering computation with negative numbers. The rules for operations with negative numbers may seem mysterious — but they are not arbitrary. To understand the rules, you first need to understand what it means to do operations with negative numbers.

A number line is one of the most useful models for understanding negative numbers. To add two integers on a number line, go to the first addend and then go in the direction represented by the second addend. Thus, 5 + (^{-}3) is done by starting at 5 and going ^{-}3 units from there — that is, three units to the left — to wind up at ^{+}2.

Using this method, students will readily see that when adding two positive numbers the sum is positive, and when adding two negative numbers the sum is negative. When adding a positive and a negative number, you can tell whether the sum is positive or negative by using the number line to visualize which number is farther from zero. For example, ^{+}5 + (^{-}8) has a negative sum, since ^{-}8 is farther from zero than ^{+}5. Since ^{-}8 is 8 units from zero and ^{+}5 is 5 units from zero, the sum is ^{-}(8 − 5) = ^{-}3.

When subtracting with negative numbers on a number line, the subtraction sign means to go in the opposite direction of the sign that follows. Thus, to find ^{+}2 − (^{-}4), start at ^{+}2, then do the opposite of ^{-}4— that is, go 4 units to the right ending at ^{+}6. This can confuse students. Up to now, subtraction has always meant “go left on the number line.” This is true when you're subtracting a positive number.
It should be noted that you would have received the same result if you had added ^{+}4 to ^{+}2. In fact, by examining the examples below, you can see that subtracting an integer gives the same result as adding the opposite integer.

^{+}2 − (^{-}4) = ^{+}6 |
^{+}2 + (^{+}4) = ^{+}6 |

^{+}5 − (^{+}7) = ^{-}2 |
^{+}2 + (^{-}7) = ^{-}2 |

^{-}3 − (^{-}6) = ^{+}3 |
^{-}3 + (^{+}6) = ^{+}3 |

^{-}8 − (^{+}6) = ^{-}14 |
^{-}8 + (^{-}6) = ^{-}14 |

That is, to subtract two signed numbers we could change the sign of the number being subtracted and add.

The rules for adding and subtracting are fairly easy to see on a number line. The rules for multiplying and dividing with negative integers can seem mystifying, but they needn't be. The key is to remember that multiplication can be seen as repeated addition and division as repeated subtraction. Remember that 3 x 4 means to add 4 three times, namely 4 + 4 + 4. The product of 3 and ^{-}4 can be thought of as adding ^{-}4 three times: ^{-}4 + (^{-}4) + (^{-}4) = ^{-}12. The product of ^{-}4 and 3 can be understood by remembering that multiplication with integers is commutative. That is, ^{-}4 x 3 = 3 x (^{-}4). Notice that the product of a negative number and a positive number is negative.

Look at these patterns:

3 x 2 = 6 | 3 x (^{-}2) = ^{-}6 |

2 x 2 = 4 | 2 x (^{-}2) = ^{-}4 |

1 x 2 = 2 | 1 x (^{-}2) = ^{-}2 |

0 x 2 = 0 | 0 x (^{-}2) = 0 |

^{-}1 x 2 = ^{-}2because the pattern decreases by 2 each time Thus, ^{-}2 x 2 = ^{-}4 |
^{-}1 x (^{-}2) = 2because the pattern increases by 2 each time ^{-}2 x (^{-}2) = 4 |

Note that the product of two negative numbers is positive, and of course the product of two positive numbers is positive.

To understand the division rules, just remember that division is the inverse of multiplication. For example, the product of two negative numbers is a positive number in multiplication (^{-}3 x (^{-}2) = 6). Therefore, the quotient of a positive number divided by a negative number must be negative (6 ÷ (^{-}3) = ^{-}2). Write the family of products and quotients for several examples (see below) to see the relationships and make sense of the rules.

^{-}3 x 5 = ^{-}15 |
^{-}4 x (^{-}6) = 24 |

5 x (^{-}3) = ^{-}15 |
^{-}6 x (^{-}4) = 24 |

^{-}15 ÷ 5 = ^{-}3 |
24 ÷ (^{-}4) = ^{-}6 |

^{-}15 ÷ (^{-}3) = 5 |
24 ÷ (^{-}6) = ^{-}4 |

Thus, when the dividend and divisor are both positive or both negative, the quotient is positive, and when one is positive and the other negative, the quotient is negative.