# Positive and Negative Numbers

Here are rules you can use to operate on positive and negative numbers.

Operation | Case | Algebraic | English | Example |
---|---|---|---|---|

a + b | Both positive | a + b | Add in the usual way. | 3+4 = 7 |

Both negative | −(|a| + |b|) | Add the absolute values, then make the result negative. | -3 + (-4) = -(|-3| + |-4|) = -(3 + 4) = -7 | |

One positive, one negative | (Assuming |a| ≥ |b|.) If a is positive, then |a| − |b|, otherwise −(|a| − |b|) | Take the difference of the absolute values; if the one with the greater absolute value is positive then the result is positive; otherwise the result is negative. | 3 + (-4): First compute the difference of the absolute values: |-4| - |3| = 4 - 3 = 1. Then, since |-4| >|3|, the sign of the result is negative so the result is -1. | |

a − b | a + (−b) | Subtraction is the same as adding the opposite, sometimes called "keep change change" (keep the sign of the first term; change the subtraction operator to an addition operator, and change the sign of the second term.) | 3 - (-4) = 3 + (+4). Then use addition rules above to compute the result. | |

a × b | Same sign | |a| × |b| | Multiply the absolute values. | -3 × (-4) = |-3| × |-4| = 3 × 4 = 12 |

Opposite signs | −(|a| × |b|) | Multiply the absolute values, then make the result negative. | 3 × (-4) = -(|3| × |-4|) = -(3 × 4) = -12 | |

a ÷ b | Just like multiplication but divide instead of multiply. | 12 ÷ (-4) = -(|12| ÷ |-4|) = -(12 ÷ 4) = -3 |

If you are comfortable with the algebraic column in the table above, you may be ready for the proof challenge!