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Positive and Negative Numbers

Positive and Negative Numbers

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Positive and Negative Numbers

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  1. 1.1 Integers and Their Symbols A +5 0 -5 +5 -5 0 Positive and Negative Numbers The numbers 5 and -5 can be drawn on a vertical or a horizontal line as follows. thermometer Horizontal number line • You may write: • the temperature of 5 degrees above zero as 5 ºC, • the temperature of 5 degrees below zero as -5 ºC. Verticalnumber line

  2. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Positive integers Negative integers 0 is neither positive nor negative A set of integers; ..., 3, 2, 1, 0, 1, 2, 3, ... , can be symbolized as I. • The complete number line of integers can be drawn as follows. Natural numbers ( bilangan Asli ) : 1, 2, 3, 4, …Whole numbers ( bilangan Cacah ) : 0, 1, 2, 3, 4, …Integers ( bilangan Bulat ) : …, -2, -1, 0, 1, 2, …

  3. a. Determine all integers between 5 and 4. Example 1 Solution: 4, 3, 2, 1, 0, 1, 2, 3 b. Write all even integers between 6 and 11. Solution: The even integers between 6 and 11 are 4, 2, 0, 2, 4, 6, 8, and 10 c. Write all odd integers between 7 and 8. Solution: The odd integers between 7 and 8 are -5, -3, -1, 1 , 3 , 5, and 7

  4. B -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Comparing and Ordering Integers -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 increase decrease What if we use a vertical number line? Change the sign  to <, >, or = on 4 7 Solution: Example 2 The number 4 is located on the right side of 7. Therefore, 4 > 7.

  5. Remember • = is readequal to • < is readless than • > is readgreater than • 3 < 5 is readthree is less than five • 5 > 3 is readfive is greater than three

  6. Problem 1 Arrange the following numbers from the least to the greatest. a. 5, 3, 6, 6, 2, 4, 1 b. 9, 5, 6, 12, 17, 8, 14 Problem 1 Solution : a. -6, -3, -1, 2, 4, 5, 6 b. -14, -12, -5, 6, 8, 9, 17

  7. In a mathematics test, the score for the right answer is 4, for the wrong answer is -1, and 0 for failing to answer. Complete the table below (use a spreadsheet, if possible) and then sort the students based on their scores from the highest to the lowest. Problem 2 17 20 18 30 31 32 27 25

  8. TASK 1.1 1. Draw a number line. Put each of the following numbers on your number line. a. –1 b. 4 c. –7 d. –9 e. 2 f. 8 2. Write an integer expressing a temperature of 14 degrees Celsius below zero. 3. Write the inverses of the following integers. a. 13 b. –8 c. 150 d. 212 4. Write three opposing pairs of situations. For example, stepping up two steps and stepping down two steps on a staircase.

  9. 5. Change the sign  to <, >, or =. a. 0 8 b. 1 7 c. 12 5 d. –3 7 e. 66 5 f. 76 239 g. 999 99 h. 45 45 6. Write the following integers from the least to the greatest. a. –2, 3, 4, 1 b. 3, 2, 0, 7 c. 4, 5, 2, 3, 1 d. –12, 0, 3, 9, 98, 10, 54 e. –1, 0, 11, 101, 111, 101, 11 7. Write the following integers from the least to the greatest. a. –10, 8, 0, 6, 5 b. 56, 56, 40 c. 0, 12, 3, 5, 64 d. 75, 3, 4, 12, 0, 9, 10

  10. 8. Write an integer which is between the integers below. a. –7 and 3 b. 0 and 6 c. –5 and 13 9. Critical Thinking. Why is any negative integer less than any positive integer? Explain. 10. Critical Thinking. Write down the steps to determine whether an integer is greater or less than another integer.

  11. Operations of Integers 1.2

  12. Operations of Integers 1.2 • Addition ( penjumlahan ) • Subtraction ( pengurangan ) • At the end of the session, you will be able to : 1.finish addition of Integers 2.finish subtraction of Integers

  13. Addition Example • 1. Determine the sum of : -3 + 5 = …. Solution : If : model negative number : model positive number : model 0 - 3 + 5 = …. + = …. 0 0 0 2 - 3 + 5 = 2

  14. Addition Example • Determine the sum of -3 + 5 = …. Solution : • So -3 + 5 = +5 Answer -3 i i i i i i i i i i i 0 -5 -4 -3 -2 -1 1 2 3 4 5 2

  15. Substraction • Example • 1. Calculate : • 3 - 5 = …. • Solution : • If : model negative number • : model positive number • : model 0 • 3 + (- 5 ) = …. • + = …. • 0 0 0 - 2 • 3 - 5 = -2

  16. Addition and Subtraction of Integers In this tutorial, we will learn how to add and subtract signed numbers with the help of a toy car. The line the car sitting on is called the number line, where the positive numbers are on the right and the negative numbers are on the left.

  17. Addition and Subtraction of Integers The car will stick to the following rules: 1. It always starts at 0 (its home) facing right. 2. If it sees a positive number, it moves forward. 3. If it sees a negative number, it backs up. 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.

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

  19. Example 1: 2 + 4 • Our car starts from 0 facing right. • It then moves 2 units to the right

  20. Example 1: 2 + 4 • Our car starts from 0 facing right. • It then moves 2 units to the right

  21. Example 1: 2 + 4 • Our car starts from 0 facing right. • It then moves 2 units to the right.

  22. 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.(click to see next animation.)

  23. 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.(click to see next animation.)

  24. 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 car now stops at 6, the answer to 2 + 4 is 6. (click to see the next example)

  25. Example 2: (-2) + 5 • Our car starts from 0 facing right. • It then backs up 2 units (to the left) because it sees the - sign.(Click to see animation)

  26. Example 2: (-2) + 5 • Our car starts from 0 facing right. • It then backs up 2 units (to the left) because it sees the - sign.

  27. Example 2: (-2) + 5 • Our car starts from 0 facing right. • It then backs up 2 units (to the left) because it sees the - sign.

  28. Example 2: (-2) + 5 • Our car starts from 0 facing right. • It then backs up 2 units (to the left) because it sees the - sign. • Next it will move forward by 5 units.(click to see animation)

  29. Example 2: (-2) + 5 • Our car starts from 0 facing right. • It then backs up 2 units (to the left) because it sees the - sign. • Next it will move forward by 5 units.

  30. Example 2: (-2) + 5 • Our car starts from 0 facing right. • It then backs up 2 units (to the left) because it sees the - sign. • Next it will move forward by 5 units.

  31. Example 2: (-2) + 5 • Our car starts from 0 facing right. • It then backs up 2 units (to the left) because it sees the - sign. • Next it will move forward by 5 units.Now it stops at +3, therefore the answer to (-2) + 5 is 3. Please go to the next tutorial for subtractions.

  32. Subtraction • There is a big difference between addition and subtraction. • In addition, our car is always facing right, because that is the positive direction, • but in subtraction, the car has to turn around (180 deg) first. Click when you are ready.

  33. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units.(Click to start animation)

  34. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units.

  35. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units.

  36. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units.

  37. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units. • Next it will turn around because it sees the subtraction symbol – . • Click to see animation.

  38. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units. • Next it will turn around because it sees the subtraction symbol – .

  39. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units. • Next it will turn around because it sees the subtraction symbol – .

  40. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units. • Next it will turn around because it sees the subtraction symbol – . • Our car still starts at 0 facing right. • It then moves forward 5 units. • Next it will turn around because it sees the subtraction symbol – . • Finally it will move forward (to the left) by 3 units. • click to see animation.

  41. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units. • Next it will turn around because it sees the subtraction symbol – . • Finally it will move forward (to the left) by 3 units.

  42. Example 3: 5 – 3 • Our car still starts at 0 facing right. • It then moves forward 5 units. • Next it will turn around because it sees the subtraction symbol – . • Finally it will move forward (to the left) by 3 units. Since the car stops at 2, the answer to 5 – 3 must be 2. Click to see the next example.

  43. Example 4: (-4) – 2 • Our car still starts at 0 facing right. • It then backs up 4 units because it sees the negative symbol - in front of 4.(Click to see animation.)

  44. Example 4: (-4) – 2 • Our car still starts at 0 facing right. • It then backs up 4 units because it sees the negative symbol - in front of 4.

  45. Example 4: (-4) – 2 • Our car still starts at 0 facing right. • It then backs up 4 units because it sees the negative symbol - in front of 4.

  46. Example 4: (-4) – 2 • Our car still starts at 0 facing right. • It then backs up 4 units because it sees the negative symbol - in front of 4.

  47. Example 4: (-4) – 2 • Our car still starts at 0 facing right. • It then backs up 4 units because it sees the negative symbol - in front of 4. • Now it has turn around because of the subtraction symbol –. (click to go on)

  48. Example 4: (-4) – 2 • Our car still starts at 0 facing right. • It then backs up 4 units because it sees the negative symbol - in front of 4. • Now it has turn around because of the subtraction symbol –.

  49. Example 4: (-4) – 2 • Our car still starts at 0 facing right. • It then backs up 4 units because it sees the negative symbol - in front of 4. • Now it has turn around because of the subtraction symbol –.

  50. Example 4: (-4) –2 • Finally it moves forward by 2 units. (click to go on)