1. Negative numbers are first introduced for solving equations such as
Chapter 8
Negative Numbers
Negative numbers are first introduced for solving equations such as
4x + 20 = 0
However, most early philosophers considered negative numbers to be absurd and unreal, because a negative number cannot be represented by a collection of objects.
This situation changed slowly when we found that negative numbers can be used to represent quantities in opposite senses. For example,

2. Indication of Opposite Direction
We use positive numbers to indicate the heights of buildings and mountains. For example, Mt. Whitney in CA is 14,495 feet high, because it is 14,495 feet above sea level.
A negative number on the other hand, can be used to indicate the depth of oceans or some places below sea level.

3. Due to the low elevation, this place can be as hot as 134ºF in summer.
The lowest point in North America is in Badwater, Death Valley National Park, CA.
It is 282 feet below mean sea level, and hence we use the negative number -282 (feet) to indicate that its elevation is below (as opposite to above) sea level.
Due to the low elevation, this place can be as hot as 134ºF in summer.

5. SR86 near Brawley, 110 ft below sea level.

6. Indication of Opposite Sensation
Hot and cold are obviously two opposite sensations. If it is very hot, we use a large number to represent the temperature.
When Gabriel Fahrenheit defined the temperature scale in 1714, he wanted to avoid using negative numbers and let 0 ºF to be the coldest temperature he could create with a mixture of ice and salt.
However, he (probably) did not know that there were places on earth much colder than 0 ºF

7. In Antarctica, it is always colder than 0ºF in winter
In Antarctica, it is always colder than 0ºF in winter. Negative numbers will then be useful to indicate how cold it is.
A record low temperature of -121ºF is recorded in 1966.
300 degree club

8. Opposite kind of Wealth
In banking, a negative number can be used to indicate a debt or deficit (as opposite to credit)
In the stock market, a negative number is used to indicate the drop (as opposite to the rise) in the price of a stock
In family law, a debt is considered as a negative asset.

9. More Opposites in Science
There are two kinds of opposite electronic charges, one is called positive, and the other is called negative. Lightening occurs when the opposite charges collide.

10. More Opposites in Science
Scientists also use + and – symbols to label chemical ions with opposite charges.

11. More Opposites in Science
Concave and convex lenses, who have negative and positive focal lengths respectively. Concave lenses are thus also called negative lenses.

12. Section 8.1 Addition and Subtraction of Integers

13. Negative numbers on the Number line

14. Comparing negative numbers
The coldest temp ever recorded in Africa was ˉ9°F at Morocco. The coldest temp ever recorded in Australia was ˉ11°F at New South Wales. Which temp was colder?

15. Conclusion
The ordering of negative numbers is opposite to that for the positive numbers.
For example, 3 is less than 5 , but ˉ3 is greater than ˉ5.

16. Classification of Numbers
Whole numbers: 0, 1, 2, 3, 4 , …
Positive integers: 1, 2, 3, 4, …
Negative integers: ˉ1, ˉ2, ˉ3, ˉ4, …
Integers: … ˉ3, ˉ2, ˉ1, 0, 1, 2, 3, …

17. Color chip Model for Addition of Integers
ˉ1 is always represented by a red chip, but +1 can be represented by a black chip, or blue chip or orange chip.
When we combine a red chip with a orange chip, they cancel each other out.

20. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
When a black chip is combined with a red chip, they will cancel each other (because 1 + (ˉ1) = 0 ).
(In science, we learn that a positive charge will cancel out a negative charge)
(click to see animation)

21. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
When a black chip is combined with a red chip, they will cancel each other (because 1 + (ˉ1) = 0 ).

22. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
When a black chip is combined with a red chip, they will cancel each other (because 1 + (ˉ1) = 0 ).

23. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
On the other hand, a pair of red & black chips can also appear from thin air, because 0 = (ˉ1) + 1.
(click to see animation)

24. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents –1
On the other hand, a pair of red & black chips can also appear from thin air, because 0 = (-1) + 1.

25. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
On the other hand, a pair of red & black chips can also appear from thin air, because 0 = (ˉ1) + 1.

26. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
On the other hand, a pair of red & black chips can also appear from thin air, because 0 = (ˉ1) + 1.

27. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
On the other hand, a pair of red & black chips can also appear from thin air, because 0 = (ˉ1) + 1.

28. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
On the other hand, a pair of red & black chips can also appear from thin air, because 0 = (ˉ1) + 1.

29. Color Chip Model for Operations of Integers
a black chip represents +1
a red chip represents ˉ1
On the other hand, a pair of red & black chips can also appear from thin air, because 0 = (ˉ1) + 1.

30. Color Chip Model for Addition of Integers
Addition – means combining sets of chips
Example: (ˉ3) + 5 − +
(click to see animation)

31. Color Chip Model for Addition of Integers
Addition – means combining sets of chips
Example: (ˉ3) + 5 − +

32. Color Chip Model for Addition of Integers
Addition – means combining sets of chips
Example: (ˉ3) + 5 − + + − + + − +

33. Color Chip Model for Addition of Integers
Addition – means combining sets of chips
Example: (ˉ3) + 5 + +
There are only 2 black chips left, hence the answer is +2.

34. Color Chip Model for Subtraction of Integers
Subtraction – means to take away chips
Example: 4 − 7 means to take away 7 black chips from black chips.
However, there are not enough chips to take away, so we need to bring in pairs of red/black chips. (click to see animation)

35. Color Chip Model for Subtraction of Integers
Subtraction – means to take away chips
Example: 4 − 7 means to take away 7 black chips from black chips.
There is still not enough black chips to take away, so we have to bring in another pair. (click)

36. Color Chip Model for Subtraction of Integers
Subtraction – means to take away chips
Example: 4 − 7 means to take away 7 black chips from black chips.
There is still not enough black chips to take away, so we have to bring in another pair. (click)

37. Color Chip Model for Subtraction of Integers
Subtraction – means to take away chips
Example: 4 − 7 means to take away 7 black chips from black chips.
Now we have enough black chips to take away. (click)

38. Color Chip Model for Subtraction of Integers
Subtraction – means to take away chips
Example: 4 − 7 means to take away 7 black chips from black chips.

39. Color Chip Model for Subtraction of Integers
Subtraction – means to take away chips
Example: 4 − 7 means to take away 7 black chips from black chips.
In the end, there are 3 red chips left, so the answer to 4 – 7 is -3

40. Color Chip Model for Subtraction of Integers
An alternative approach.
Example: 2 − 3
We can start with 2 black chips and lots of zeros (i.e. red/black pairs).
This is okay because the net value of the set is still +2.

41. Color Chip Model for Subtraction of Integers
An alternative approach.
Example: 2 − 3
Next we take away 3 black chips.
Now the net value of the set is -1, hence 2 – 3 = -1.
(Click to see animation.)

42. Color Chip Model for Subtraction of Integers
An alternative approach.
Example: − 1
We can start with 2 red chips and lots of zeros (i.e. red/black pairs).
This is okay because the net value of the set is still -2.

43. Color Chip Model for Subtraction of Integers
An alternative approach.
Example: − 1
Next we take away 1 black chip.
The name of this model is
“Ocean of Zeros”
Now the net value of the set is -3, hence -2 – 1 = -3.
(Click to see animation.)

44. Color chip Model for Subtraction of Integers

46. 1. Use Color chips to perform ˉ3 + 8
Example for Lab
1. Use Color chips to perform ˉ3 + 8
start with 3 red chips
bring in 8 black chips
cancel out blk-red pairs, and there will be 5 black chips left, hence the answer is 5.

47. 2. Use Color chips to perform 4 – (ˉ2)
Example for Lab
2. Use Color chips to perform 4 – (ˉ2)
start with 4 black chips
Want to take away 2 red chips, but there is no red chips, so bring in 2 red-black pairs.
Now take way 2 red chips, and the answer is seen to be 6.

48. Number Line Model (for addition and subtraction of integers)
In this tutorial, we will learn how to add and subtract signed numbers with the help of a rabbit.
The line the rabbit sitting on is called the number line, where the positive numbers are on the right and the negative numbers are on the left.

49. Number Line Model (for addition and subtraction of integers)
The rabbit will stick to the following rules:
1. It always starts at 0 (its home) facing right.
2. If it sees a positive number, it hops forward.
3. If it sees a negative number, it hops backward.
4. If it sees an addition sign, it continues to read the next number.
5. If it sees a subtraction sign, it turns around and then continues to read the next number.
Click to see the first example.

50. Example 1: 2 + 4 Our rabbit starts from 0 facing right.
It then moves 2 units to the right (click to see animation)

51. Example 1: 2 + 4 Our rabbit starts from 0 facing right.
It then moves 2 units to the right

52. Example 1: 2 + 4 Our rabbit starts from 0 facing right.
It then moves 2 units to the right

53. Example 1: 2 + 4 Our rabbit starts from 0 facing right.
It then moves 2 units to the right.
3. Next the rabbit will move 4 more units forward because it sees the number 4. (click to see next animation.)

54. Example 1: 2 + 4 Our rabbit starts from 0 facing right.
It then moves 2 units to the right.
3. Next the rabbit will move 4 more units forward because it sees the number 4.

55. Example 1: 2 + 4 Our rabbit starts from 0 facing right.
It then moves 2 units to the right.
3. Next the rabbit will move 4 more units forward because it sees the number 4.

56. Example 1: 2 + 4 Our car starts from 0 facing right.
It then moves 2 units to the right.
3. Next the car will move 4 more units forward because it sees the number 4.
Since the rabbit now stops at 6, the answer to is 6.
(click to see the next example)

57. Example 2: (ˉ2) + 5 Our rabbit starts from 0 facing right.
It then hops backward 2 units (to the left) because it sees the - sign. (Click to see animation)

58. Example 2: (ˉ2) + 5 Our rabbit starts from 0 facing right.
It then hops backward 2 units (to the left) because it sees the - sign.

59. Example 2: (ˉ2) + 5 Our rabbit starts from 0 facing right.
It then hops backward 2 units (to the left) because it sees the - sign.

60. Example 2: (ˉ2) + 5 Our rabbit starts from 0 facing right.
It then hops backward 2 units (to the left) because it sees the - sign.
Next it will hop forward by 5 units. (click to see animation)

61. Example 2: (ˉ2) + 5 Our rabbit starts from 0 facing right.
It then hops backward 2 units (to the left) because it sees the - sign.
Next it will hop forward by 5 units.

62. Example 2: (ˉ2) + 5 Our rabbit starts from 0 facing right.
It then hops backward 2 units (to the left) because it sees the - sign.
Next it will hop forward by 5 units.
The rabbit stops at 3, hence (-2) + 5 = 3

63. Subtraction
There is a big difference between addition and subtraction.
In addition, our rabbit is always facing right, because that is the positive direction,
but in subtraction, the rabbit has to turn around (180 deg) first. -3 -2 -1 1 2 3 4 5
Click whenever you are ready.

64. Example 3: 5 – 3 Our rabbit still starts at 0 facing right.
It then hops forward 5 units because it sees no negative symbol in front of 5. (Click to see animation.) -3 -2 -1 1 2 3 4 5 6

65. Example 3: 5 – 3 Our rabbit still starts at 0 facing right.
It then hops forward 5 units because it sees no negative symbol in front of 5. -3 -2 -1 1 2 3 4 5 6

66. Example 3: 5 – 3 Our rabbit still starts at 0 facing right.
It then hops forward 5 units because it sees no negative symbol in front of 5. -3 -2 -1 1 2 3 4 5 6

67. Example 3: 5 – 3 Our rabbit still starts at 0 facing right.
It then hops forward 5 units because it sees no negative symbol in front of 5.
3. Now the rabbit sees the subtraction sign, and will turn around. -3 -2 -1 1 2 3 4 5 6

68. Example 3: 5 – 3 Our rabbit still starts at 0 facing right.
It then hops forward 5 units because it sees no negative symbol in front of 5.
3. Now the rabbit sees the subtraction sign, and will turn around. -3 -2 -1 1 2 3 4 5 6

69. Example 3: 5 – 3 Our rabbit still starts at 0 facing right.
It then hops forward 5 units because it sees no negative symbol in front of 5.
3. Now the rabbit sees the subtraction sign, and will turn around.
4. Finally the rabbit sees the number 3, and so
it jumps forward by 3 units. -3 -2 -1 1 2 3 4 5 6
The rabbit now stops at 2, hence 5 – 3 = 2.

70. Example 4: 2 – ˉ3 Our rabbit still starts at 0 facing right.
It then hops forward 2 units because it sees no negative symbol in front of 2. (Click to see animation.) -3 -2 -1 1 2 3 4 5 6

71. Example 4: 2 – ˉ3 Our rabbit still starts at 0 facing right.
It then hops forward 2 units because it sees no negative symbol in front of 2. -3 -2 -1 1 2 3 4 5 6

72. Example 4: 2 – ˉ3 Our rabbit still starts at 0 facing right.
It then hops forward 2 units because it sees no negative symbol in front of 2.
3. Next the rabbit has to turn around because it sees the subtraction sign. (click to see) -3 -2 -1 1 2 3 4 5 6

73. Example 4: 2 – ˉ3 Our rabbit still starts at 0 facing right.
It then hops forward 2 units because it sees no negative symbol in front of 2.
3. Next the rabbit has to turn around because it sees the subtraction sign. (click to see)
Finally our rabbit will hop backward 3 units because it sees the negative number -3 . (click to see animation) -3 -2 -1 1 2 3 4 5 6

74. Example 4: 2 – ˉ3
Finally our rabbit will hop backward 3 units because it sees the negative number -3 . (click to see animation) -3 -2 -1 1 2 3 4 5 6

75. Example 4: 2 – ˉ3
Finally our rabbit will hop backward 3 units because it sees the negative number -3 . -3 -2 -1 1 2 3 4 5 6

76. Example 4: 2 – ˉ3
Finally our rabbit will hop backward 3 units because it sees the negative number -3 . -3 -2 -1 1 2 3 4 5 6

77. Example 4: 2 – ˉ3
Finally our rabbit will hop backward 3 units because it sees the negative number -3 . -3 -2 -1 1 2 3 4 5 6

78. Example 4: 2 – ˉ3
Finally our rabbit will hop backward 3 units because it sees the negative number -3 .
When the rabbit finishes its jumps, it stops at 5, hence the answer to 2 – (-3) is 5. -3 -2 -1 1 2 3 4 5 6