1. Signed Numbers, Powers, & Roots
Chapter 2 Sections
Chapter 1C Section 1.15

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2.1
Addition of Signed Numbers
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3. Addition of Signed Numbers
Technicians use negative numbers in many ways. In an experiment using low temperatures, for example, you would record 10 below zero as –10. Or consider sea level as zero altitude. If a submarine dives 75 m, you could consider its depth as –75 m (75 m below sea level). See Figure 2.1.
(a)
(b)
Figure 2.1

4. Addition of Signed Numbers
These measurements indicate a need for numbers other than positive integers, which are the only numbers that we have used up to now.
To illustrate the graphical relationship of these numbers, we draw a number line as in Figure 2.2 with a point representing zero and with evenly spaced points that represent the positive integers (1, 2, 3, . . .) to the right as shown.
The real number line
Figure 2.2

5. Addition of Signed Numbers
Then we mark off similarly evenly spaced points to the left of zero.
These points correspond to the negative integers (–1, –2, –3, . . .) as shown.
The negative integers are preceded by a negative (–) sign; –3 is read “negative 3,” and –5 is read “negative 5.”
Each positive integer corresponds to a negative integer.
For example, 3 and –3 are corresponding integers. Note that the distances from 0 to 3 and from 0 to –3 are equal.

6. Addition of Signed Numbers
The rational numbers are defined as those numbers that can be written as the ratio of two integers; that is, a/b, where b ≠ 0. The irrational numbers are those numbers that cannot be written as the ratio of two integers, such as , , or the square root of any nonperfect square;  ; and several other kinds of numbers.

7. Addition of Signed Numbers
The real numbers consist of the rational and irrational numbers and are represented on the real number line as shown in Figure 2.2.
The real number line is dense or full with real numbers; that is, each point on the number line represents a distinct real number, and each real number is represented by a distinct point on the number line.
The real number line
Figure 2.2

8. Addition of Signed Numbers
Examples of real numbers as illustrated in
Your text on page 109

9. Addition of Signed Numbers
The absolute value of a number is its distance from zero on the number line.
Because distance is always considered positive, the absolute value of a number is never negative.
We write the absolute value of a number x as | x |; it is read “the absolute value of x.” Thus, | x |  0. (“” means “is greater than or equal to.”) For example, | +6 | = 6, | 4 | = 4, and | 0 | = 0.

10. Addition of Signed Numbers
However, if a number is less than 0 (negative), its absolute value is the corresponding positive number.
For example, | –6 | = 6 and | –7 | = 7. Remember:

11. Example 1
Find the absolute value of each number: a. +3, b. –5, c. 0, d. –10, e. 15.
a. |+3| = 3
b. | –5 | = 5
c. | 0 | = 0
d. | –10 | = 10
e. | 15 | = | +15 | = 15
The distance between 0 and +3 on the number line is 3 units.
The distance between 0 and –5 on the number line is 5 units.
The distance is 0 units.
The distance between 0 and –10 on the number line is 10 units.
The distance between 0 and +15 on the number line is 15 units.

12. Addition of Signed Numbers
One number is larger than another number if the first number is to the right of the second on the number line in Figure Thus, 5 is larger than 1, 0 is larger than –3, and 2 is larger than –4. Similarly, one number is smaller than another if the first number is to the left of the second on the number line in Figure 2.2. Thus, 0 is smaller than 3, –1 is smaller than 4, and –5 is smaller than –2.
The real number line
Figure 2.2

13. Addition of Signed Numbers
The use of signed numbers (positive and negative numbers) is one of the most important operations that we will study. Adding Two Numbers with Like Signs (the Same Signs) 1. To add two positive numbers, add their absolute values. The result is positive. A positive sign may or may not be used before the result. It is usually omitted.
2. To add two negative numbers, add their absolute values and place a negative sign before the result.

14. Example 2 Add: a. (+2) + (+3) = +5 b. (–4) + (–6) = –10
c. (+4) + (+5) = +9
d. (–8) + (–3) = –11

15. Addition of Signed Numbers
Adding Two Numbers with Unlike Signs
To add a negative number and a positive number, find the difference of their absolute values.
The sign of the number having the larger absolute value is placed before the result.

16. Example 3 Add: a. (+4) + (–7) = –3 b. (–3) + (+8) = +5
c. (+6) + (–1) = +5
d. (–8) + (+6) = –2
e. (–2) + (+5) = +3
f. (+3) + (–11) = –8

17. Addition of Signed Numbers
Adding Three or More Signed Numbers
Step 1: Add the positive numbers.
Step 2: Add the negative numbers.
Step 3: Add the sums from Steps 1 and 2 according to the rules for addition of two signed numbers.

18. Example 4 Add (–8) + (+12) + (–7) + (–10) + (+3).
Step 1: (+12) + (+3) = +15
Step 2: (–8) + (–7) + (–10) = –25
Step 3: (+15) + (–25) = –10
Therefore, (–8) + (+12) + (–7) + (–10) + (+3) = –10.
Add the positive numbers.
Add the negative numbers.
Add the sums from Steps 1 and 2.

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2.2
Subtraction of Signed Numbers
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20. Subtraction of Signed Numbers
Subtracting Two Signed Numbers
To subtract two signed numbers, change (or reverse) the sign of the number being subtracted (second number) and add according to the rules for addition of signed numbers. This is because subtraction reverses the direction we travel on the number line.

21. Example 1 Subtract: a. (+2) – (+5) = (+2) + (–5) = –3
= –1
c. (+6) – (–4) = (+6) + (+4)
= +10
To subtract, change the sign of the number being subtracted, +5, and add.
To subtract, change the sign of the number being subtracted, –6, and add.
To subtract, change the sign of the number being subtracted, –4, and add.

22. Example 1 d. (+1) – (+6) = (+1) + (–6) = –5
cont’d
d. (+1) – (+6) = (+1) + (–6)
= –5
e. (–8) – (–10) = (–8) + (+10)
= +2
f. (+9) – (–6) = (+9) + (+6)
= +15
g. (–4) – (+7) = (–4) + (–7)
= –11
To subtract, change the sign of the number being subtracted, +6, and add.

23. Example 2 Subtract: (–4) – (–6) – (+2) – (–5) – (+7)
= (–4) + (+6) + (–2) + (+5) + (–7)
Step 1: (+6) + (+5) = +11
Step 2: (–4) + (–2) + (–7) = –13
Step 3: (+11) + (–13) = –2
Therefore, (–4) – (–6) – (+2) – (–5) – (+7) = –2.
Change the sign of each number being subtracted and add the resulting signed numbers.

24. Subtraction of Signed Numbers
Adding and Subtracting Combinations of Signed Numbers
When combinations of additions and subtractions of signed numbers occur in the same problem, change only the sign of each number being subtracted. Then add the resulting signed numbers.

25. Example 3 Perform the indicated operations:
(+4) – (–5) + (–6) – (+8) – (–2) + (+5) – (+1)
= (+4) + (+5) + (–6) + (–8) + (+2) + (+5) + (–1)
Step 1: (+4) + (+5) + (+2) + (+5) = +16
Step 2: (–6) + (–8) + (–1) = –15
Change only the sign of each number being subtracted and add the resulting signed numbers.

26. Example 3 Step 3: (+16) + (–15) = +1 Therefore,
cont’d
Step 3: (+16) + (–15) = +1
Therefore,
(+4) – (–5) + (–6) – (+8) – (–2) + (+5) – (+1) = +1.

27. 2.3 Multiplication and Division of Signed Numbers
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28. Multiplication and Division of Signed Numbers
Multiplying/Dividing Two Signed Numbers
1. If the two numbers have the same sign, multiply or divide their absolute values. This product is always positive.
2. If the two numbers have different signs, multiply or divide their absolute values and place a negative sign before the product.

29. Example 1 Multiply: a. (+2)(+3) = +6 b. (–4)(–7) = +28
c. (–2)(+4) = –8
d. (–6)(+5) = –30
Multiply the absolute values of the signed numbers; the product is positive because
the two numbers have the same sign.
Multiply the absolute values of the signed numbers; the product is positive because
the two numbers have the same sign.
Multiply the absolute values of the signed numbers; the product is negative because the two numbers have different signs.
Multiply the absolute values of the signed numbers; the product is negative because the two numbers have different signs.

30. Example 1 e. (+3)(+4) = +12 f. (–6)(–9) = +54 g. (–5)(+7) = –35
cont’d
e. (+3)(+4) = +12
f. (–6)(–9) = +54
g. (–5)(+7) = –35
h. (+4)(–9) = –36

31. Multiplication and Division of Signed Numbers
Multiplying More Than Two Signed Numbers
1. If the number of negative factors is even (divisible by ) multiply the absolute values of the numbers. This product is positive.
2. If the number of negative factors is odd, multiply the absolute values of the numbers and place a negative sign before the product.

32. Example 2 Multiply: (–11)(+3)(–6) = +198
The number of negative factors is 2, which is even; therefore, the product is positive.

33. Example 4
Divide:
Divide the absolute values of the signed numbers; the quotient is positive because the two numbers have the same sign.
Divide the absolute values of the signed numbers; the quotient is positive because the two numbers have the same sign.
Divide the absolute values of the signed numbers; the quotient is negative because the two numbers have different signs.
Divide the absolute values of the signed numbers; the quotient is negative because the two numbers have different signs.

34. Example 4 e. (+30)  (+5) = +6 f. (–42)  (–2) = +21
cont’d
e. (+30)  (+5) = +6
f. (–42)  (–2) = +21
g. (+16)  (–4) = –4
h. (–45)  (+9) = –5

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2.4
Signed Fractions
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36. Example 1
Add:
The LCD is 16.
Combine the numerators.

37. Signed Fractions Equivalent Signed Fractions
That is, a negative fraction may be written in three different but equivalent forms.
However, the form is the customary form.
For example,
Note
using the rules for dividing signed numbers.

38. Example 12 Add: Change to customary form. The LCD is 12.
Combine the numerators.

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1.15
Powers and Roots
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40. Powers and Roots
The square of a number is the product of that number times itself. The square of 3 is 3  3 or 32 or 9. The square of a number may be found with a calculator as follows.

41. Powers and Roots
The square root of a number is that positive number which, when multiplied by itself, gives the original number. The square root of 25 is 5 and is written as
The symbol is called a radical.

42. Example 3 Find the square roots of a. 16, b. 64, c. 100, and d. 144.
a = 4 because 4  4 = 16
b = 8 because 8  8 = 64
c = 10 because 10  10 = 100
d = 12 because 12  12 = 144
Numbers whose square roots are whole numbers are called perfect squares. For example,1, 4, 9, 16, 25, 36, 49, and 64 are perfect squares.

43. Powers and Roots
The cube of a number is the product of that number times itself three times. The cube of 5 is 5  5  5 or 53 or 125.

44. Example 6 Find the cubes of a. 2, b. 3, c. 4, and d. 10.
= 8
= 27
= 64
= 1000

45. Powers and Roots
The cube root of a number is that number which, when multiplied by itself three times, gives the original number. The cube root of 8 is 2 and is written as
(Note: 2  2  2 = 8. The small 3 in the radical is called the index.)

46. Example 9 Find the cube roots of a. 8, b. 27, and c. 125.
a = 2 because 2  2  2 = 8
b = 3 because 3  3  3 = 27
c = 5 because 5  5  5 = 125

47. Powers and Roots
Numbers whose cube roots are whole numbers are called perfect cubes. For example, 1, 8, 27, 64, 125, and 216 are perfect cubes.
In general, in a power of a number, the exponent indicates the number of times the base is used as a factor. For example, the 4th power of 3 is written 34, which means that 3 is used as a factor 4 times (34 = 3  3  3  3 = 81).

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2.5
Powers of 10
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49. Powers of 10 Multiplying Powers of 10
To multiply two powers of 10, add the exponents as follows:
10a  10b = 10a + b

50. Example 1 Multiply: (102)(103) Method 1: (102)(103) =
(10  10)(10  10  10)
= 105
Method 2: (102)(103) =
Add the exponents.

51. Powers of 10 Dividing Powers of 10
To divide two powers of 10, subtract the exponents as follows:
10a  10b = 10a – b

52. Example 4
Divide:
Method 1:
Method 2:
Subtract the exponents.

53. Powers of 10 Raising a Power of 10 to a Power
To raise a power of 10 to a power, multiply the exponents as follows:
(10a)b = 10ab

54. Example 7 Find the power (102)3. Method 1: (102)3 = 102  102  102=
= 106
Method 2: (102)3 = 10(2)(3)
Use the product of powers rule.
Multiply the exponents.

55. Powers of 10 For example, and
In a similar manner, we can also show that

56. Powers of 10 For example, and
Combinations of multiplications and divisions of powers of 10 can also be done easily using the rules of exponents.

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2.6
Scientific Notation
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58. Scientific Notation
Scientific Notation Scientific notation is a method that is especially useful for writing very large or very small numbers. To write a number in scientific notation, write it as a product of a number between 1 and 10 and a power of 10.

59. Example 1 Write 226 in scientific notation. 226 = 2.26  102
Remember that 102 is a short way of writing 10  10 = 100.
Note that multiplying 2.26 by 100 gives 226.

60. Scientific Notation Writing a Decimal Number in Scientific Notation
To write a decimal number in scientific notation,
1. Reading from left to right, place a decimal point after the first nonzero digit.
2. Place a caret (^) at the position of the original decimal point.
3. If the decimal point is to the left of the caret, the exponent of the power of 10 is the same as the number of decimal places from the caret to the decimal point.

61. Scientific Notation
4. If the decimal point is to the right of the caret, the exponent of the power of 10 is the same as the negative of the number of places from the caret to the decimal point.
5. If the decimal point is already after the first nonzero digit, the exponent of 10 is zero.
2.15 = 2.15  100

62. Example 3
Write 2738 in scientific notation.

63. Scientific Notation
Writing a Number in Scientific Notation in Decimal Form
To change a number in scientific notation to decimal form,
1. Multiply the decimal part by the given positive power of by moving the decimal point to the right the same number of decimal places as indicated by the exponent of 10. Supply zeros when needed.
2. Multiply the decimal part by the given negative power of by moving the decimal point to the left the same number of decimal places as indicated by the exponent of 10. Supply zeros when needed.

64. Example 5 Write 2.67  102 as a decimal. 2.67  102 = 267
Move the decimal point two places to the right, since the exponent of 10 is +2.

65. Scientific Notation
You may find it useful to note that a number in scientific notation with
a. a positive exponent greater than 1 is greater than 10, and b. a negative exponent is between 0 and 1. That is, a number in scientific notation with a positive exponent represents a relatively large number. A number in scientific notation with a negative exponent represents a relatively small number.

66. Scientific Notation
Scientific notation may be used to compare two positive numbers expressed as decimals. First, write both numbers in scientific notation.
The number having the greater power of 10 is the larger. If the powers of 10 are equal, compare the parts of the numbers that are between 1 and 10. Scientific notation is especially helpful for multiplying and dividing very large and very small numbers.

67. Scientific Notation
To perform these operations, you must first know some rules for exponents.
Multiplying Numbers in Scientific Notation
To multiply numbers in scientific notation, multiply the decimals between 1 and 10.
Then add the exponents of the powers of 10.

68. Example 10
Multiply (4.5  108)(5.2  10–14). Write the result in scientific notation.
(4.5  108)(5.2  10–14) = (4.5)(5.2)  (108)(10–14)
= 23.4  10–6
= (2.34  101)  10–6
= 2.34  10–5
Note that 23.4  10–6 is not in scientific notation, because 23.4 is not between 1 and 10.

69. Scientific Notation Dividing Numbers in Scientific Notation
To divide numbers in scientific notation, divide the decimals between 1 and 10.
Then subtract the exponents of the powers of 10.

70. Scientific Notation Powers of Numbers in Scientific Notation
To find the power of a number in scientific notation, find the power of the decimal between 1 and 10. Then multiply the exponent of the power of 10 by this same power.

71. Example 13
Find the power (4.5  106)2. Write the result in scientific notation.
(4.5  106)2 = (4.5)2  (106)2
=  1012
= (2.025  101)  1012
=  1013
Note that is not between 1 and 10.

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2.7
Engineering Notation
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73. Engineering Notation
Numbers may also be written in engineering notation, similar to scientific notation, as follows:
Engineering Notation
Engineering notation is used to write a number with its decimal part between 1 and 1000 and a power of 10 whose exponent is divisible by 3.

74. Engineering Notation Writing a Decimal Number in Engineering Notation
To write a decimal number in engineering notation,
1. Move the decimal point in groups of three digits until the decimal point indicates a number between 1 and If the decimal point has been moved to the left, the exponent of the power of 10 in engineering notation is the same as the number of places the decimal point was moved.

75. Engineering Notation
3. If the decimal point has been moved to the right, the exponent of the power of 10 in engineering notation is the same as the negative of the number of places the decimal point was moved. In any case, the exponent will be divisible by 3.

76. Example 1 Write 48,500 in engineering notation. 48,500 = 48.5  103
Check The exponent of the power of 10 must be divisible by 3.
Move the decimal point in groups of three decimal places until the decimal part is between 1 and 1000.

77. Engineering Notation
Writing a number in engineering notation in decimal form is similar to writing a number in scientific notation in decimal form. Operations with numbers in engineering notation using a calculator are very similar to operations with numbers in scientific notation. If your calculator has an engineering notation mode, set it in this mode. If not, use scientific notation and convert the result to engineering notation.

78. Engineering Notation
For comparison purposes, the following table shows six numbers written in both scientific notation and engineering notation: