Summary

1. There are numbers that are less than zero. They are written with a '–' sign in front of them (e.g., – 2), and are called . They lie to the of zero on the number line.

2. The numbers ..., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, ... are called . The numbers 1, 2, 3, 4, ... are called and the numbers …, – 4, – 3, – 2, – 1 are called . Zero (0) is neither nor .

3. Every given number has another number associated to it which when added to the given number gives zero. This is called the of the number. For example, the additive inverse of 7 is and the additive inverse of – 543 is .

4. Addition can be interpreted as Starting Position + Movement = .

5. Addition can also be interpreted as the combination of movements or increases/decreases: Movement 1 + Movement 2 = .

6. Subtraction can be interpreted as Target Position – Starting Position = .

7. In general, we can add two numbers by following for Addition:

a. If both numbers are positive, add the numbers and the result is a number (e.g., 2 + 3 = 5).

b. If both numbers are negative, add the numbers (without the signs), and then place a sign to obtain the result (– 2 + (– 3) = – 5).

c. If one number is positive and the other is negative, subtract the number (without the sign) from the number (without the sign), and place the sign of the number to obtain the result (e.g., – 5 + 3 = – 2).

d. A number plus its additive inverse is (e.g., 2 + (– 2) = 0).

e. A number plus zero gives back the (e.g., – 2 + 0 = – 2).

8. We can subtract two integers by converting the problem into an addition problem and then following the rules of addition. Subtraction of an integer is the same as the of its .

9. Integers can be compared: … – 3 < – 2 < – 1 < 0 < +1 < +2 < +3 < ... numbers are to the of numbers on the number line.

10. We can give meaning to positive and negative numbers by interpreting them as and . We can also interpret positive numbers as distances a reference point like the ground level. Similarly, negative numbers can be interpreted as distances the ground level. When measuring temperatures in degrees Celsius, positive temperatures are those the freezing point of water, and negative temperatures are those the freezing point of water.

11. The concept of negative numbers was first developed in , particularly in and , thousands of years ago. (628 CE) was the first to give systematic rules for operations with positive, negative, and zero numbers that we still use today!