3.3 Arithmetic Operations with Signed Numbers

In the previous chapters, we learned that positive real numbers can be represented by points on a number line from zero to the right of the zero. That is, whole numbers and positive rational and irrational numbers can be represented on a number line from zero to the right of the zero.

Every positive number has a negative number known as its opposite, which lies to the left of zero on the number line. We use the negative sign '−' to represent negative numbers, and the positive sign '+' to represent positive numbers. Zero, '0', is neither positive nor negative.

For example, plotting the positive numbers [latex] \displaystyle{\frac{3}{4}} [/latex], 4, and 6.5 and their opposites on the number line:

The arrowhead on either end shows that the number line continues indefinitely in both the positive and negative directions.

Positive and negative numbers are collectively referred to as signed numbers. Since numbers are naturally positive, when we read or write positive numbers, we usually omit the word ‘positive’ or the positive sign (+). However, when the number is negative, we must read or write it as ‘negative’ or include the negative sign (−). For example, ‘+7’ is read as ‘seven’ and written as ‘7’. However, ‘−7’ is read as ‘negative seven’ and written with the negative sign as ‘−7’.

Any positive number and its negative (opposite) will be at an equal distance from zero (origin) on the number line.

Numbers that lie to the left of a number on the number line are less than that number, and numbers that lie to the right of a number on a number line are greater than that number.

For example,

• 3 is greater than −2,   i.e., 3 > –2
• −3 is greater than −5,   i.e., –3 > –5
• −5 is less than −4,   i.e., –5 < –4
• −1 is less than 2,   i.e., –1 < 2

Absolute Value

The absolute value of a number is its distance from the origin ‘0’ on the number line. Since it is a distance, it is always positive and the direction does not matter.

For example, −5 and +5 are both 5 units from the origin ‘0’.

Therefore, the absolute value of both −5 and +5 is 5.

The absolute value of a number ‘a’ is denoted by |a|. The vertical bars used in the representation of the absolute value differ from how brackets are used.

For example, | −4 | = 4, whereas (−4) = −4

Example 3.3-a: Simplifying Arithmetic Expressions Involving Absolute Values

Simplify the following expressions:

1. [latex] \displaystyle{-\left|\frac{-4}{3}\right|} [/latex]

   
   

2. [latex] -|8(-2)| [/latex]

   
   

Solution

1. [latex] \displaystyle{-\left|\frac{-4}{3}\right| = -\left(\frac{4}{3}\right) = -\frac{4}{3}} [/latex]

   Therefore, [latex] \displaystyle{-\left|\frac{-4}{3}\right| = -\frac{4}{3}} [/latex].

   
   

2. [latex] -|8(-2)| = -|-16| = -(16) = -16 [/latex]

   Therefore, [latex] -|8(-2)| = -16 [/latex].

   
   

Example 3.3-b: Adding and Subtracting Arithmetic Expressions Involving Absolute Values

Simplify the following expressions:

1. [latex] 10 - |8 - 15| [/latex]

   
   

2. [latex] -5 + |-10 + 8| [/latex]

   
   

Solution

1. [latex] 10 - |8 - 15| = 10 - |-7| = 10 - 7 = 3 [/latex]

   Therefore, [latex] 10 - |8 - 15| = 3 [/latex].

   
   

2. [latex] -5 + |-10 + 8| = -5 + |-2| = -5 + 2 = -3 [/latex]

   Therefore, [latex] -5 + |-10 + 8| = -3 [/latex].

   
   

Note: In the previous two examples, we performed arithmetic with signed numbers. The rules we follow when adding, subtracting, multiplying, and dividing signed numbers are explained in detail below.

Addition and Subtraction of Signed Numbers

• When adding two positive numbers, the answer is always positive (+).

  For example,

  Adding +5 and +3:

  (+5) + (+3) = 5 + 3 = 8

  This is the same as +8.

  
  

• When adding two negative numbers, the answer is always negative (−).

  For example,

  Adding −4 and −3:

  (−4) + (−3) = −4 − 3 = −7

  
  

• When adding numbers that have different signs, subtract the smaller absolute value from the larger absolute value, and keep the sign of the number with the larger absolute value.

  For example,

  1. Adding +8 and −12 (or −12 and +8):

     (+8) + (−12) = 8 − 12 = −4

     
     

  2. Adding −5 and +8 (or +8 and −5):

     (−5) + (+8) = −5 + 8 = 3 (or +3)

     
     

• When subtracting negative numbers, first change all the subtraction problems to addition problems by adding the opposite, and then follow the rules for addition of signed numbers.

  For example,

  1. Subtracting 12 from 18:

     18 – 12 = 18 + (–12) = 6

     
     

  2. Subtracting –12 from 18:

     18 – (–12) = 18 + 12 = 30

     
     

  3. Subtracting 12 from –18:

     –18 – 12 = –18 + (–12) = –30

     
     

  4. Subtracting –12 from –18:

     –18 – (–12) = –18 + 12 = –6

     
     

Multiplication and Division of Signed Numbers

When multiplying two signed numbers:

• The product of two numbers with the same sign is positive.

  For example,

  1. (+5)(+4) = +20

     
     

  2. (–5)(–4) = +20

     
     

• The product of two numbers with different signs is negative.

  For example,

  1. (+5)(−4) = −20

     
     

  2. (−5)(+4) = −20

     
     

When dividing two signed numbers:

• The quotient of two numbers with the same sign is positive.

  For example,

  1. [latex] \displaystyle{\frac{+12}{+8} = \frac{3}{2}} [/latex]

     
     

  2. [latex] \displaystyle{\frac{-12}{-8} = \frac{3}{2}} [/latex]

     
     

• The quotient of two numbers with different signs is negative.

  For example,

  1. [latex] \displaystyle{\frac{-12}{+8} = -\frac{3}{2}} [/latex]

     
     

  2. [latex] \displaystyle{\frac{+12}{-8} = -\frac{3}{2}} [/latex]

     
     

When multiplying or dividing more than two signed numbers, group them into pairs to determine the sign using the rules for multiplication and division of signed numbers.

For example,

1. [latex] (-3)(-2)(+4)(-1)(-5) = (6)(-4)(-5) [/latex]

   
   

   [latex] = (-24)(-5) = 120 [/latex]

   
   

2. [latex] \displaystyle{\frac{(-15)(+8)(-50)}{(-25)(14)} = \frac{-(15 \times 8)(-50)}{-(25 \times 14)} = \frac{+(15 \times 8 \times 50)}{-(25 \times 14)}} [/latex]

   
   

   [latex] \displaystyle{= \frac{+(15 \times 4 \times 2)}{-(1 \times 7)} = -\frac{15 \times 4 \times 2}{7} = -\frac{120}{7}} [/latex]

   
   

Powers with Negative Bases

When a power has a negative base, there are four possible scenarios, as outlined in the following table:

Table 3.3-a: Powers with Negative Bases

From the above scenarios you will note:

• A negative base with an even exponent produces a positive result (because pairs of negatives become positive).
• A negative base with an odd exponent produces a negative result (because after negatives are paired, one negative will be left over).

A negative base of a power expressed within a bracket, as in [latex] (-a)^n [/latex], results in a different answer than a negative base expressed without a bracket, as in [latex] -a^n [/latex].

In [latex] (-a)^n [/latex], the exponent applies to both the negative sign and [latex] a [/latex].

In [latex] -a^n [/latex], the exponent applies only to [latex] a [/latex] and the negative sign is applied to the answer.

For example,

1. In [latex] (-5)^4 [/latex], [latex] (-5) [/latex] is multiplied [latex] 4 [/latex] times; i.e., [latex] (-5)^4 = (-5)(-5)(-5)(-5) = 625 [/latex]

   
   

2. In [latex] (-5)^3 [/latex], [latex] (-5) [/latex] is multiplied [latex] 3 [/latex] times; i.e., [latex] (-5)^3 = (-5)(-5)(-5) = -125 [/latex]

   
   

3. In [latex] -5^4 [/latex], only [latex] 5 [/latex] is multiplied [latex] 4 [/latex] times and the answer is negative; i.e., [latex] -5^4 = -[5 \times 5 \times 5 \times 5] = -625 [/latex]

   
   

4. In [latex] -5^3 [/latex], only [latex] 5 [/latex] is multiplied [latex] 3 [/latex] times and the answer is negative; i.e., [latex] -5^3 = -[5 \times 5 \times 5] = -125 [/latex]

   
   

Example 3.3-c: Evaluating Expressions with Negative Bases using the Product Rule

Evaluate the following expressions:

1. [latex] (-5)^4 \times (-5)^{-1} [/latex]

   
   

2. [latex] (-2)^5 \times (-2)^2 \times (-2)^0 \times 2 [/latex]

   
   

Solution

1. [latex] (-5)^4 \times (-5)^{-1} = (-5)^{(4 - 1)} = (-5)^3 = -125 [/latex]

   
   

2. [latex] (-2)^5 \times (-2)^2 \times (-2)^0 \times 2 = (-2)^{(5 + 2 + 0)} \times 2 [/latex]

   [latex] = (-2)^7 \times 2 = -128 \times 2 = -256 [/latex]

   
   

Example 3.3-d: Evaluating Expressions with Negative Bases using the Quotient Rule

Evaluate the following expressions:

1. [latex] (-3)^7 \div (-3)^2 [/latex]

   
   

2. [latex] (-5)^3 \div (-5)^0 [/latex]

   
   

Solution

1. [latex] (-3)^7 \div (-3)^2 = (-3)^{(7 - 2)} = (-3)^5 = -243 [/latex]

   
   

2. [latex] (-5)^3 \div (-5)^0 = (-5)^{(3 - 0)} = (-5)^3 = -125 [/latex]

   
   

Example 3.3-e: Evaluating Expressions with Negative Bases using the Power of a Product Rule

Evaluate the following expressions:

1. [latex] (-5 \times 2)^3 [/latex]

   
   

2. [latex](-3 \times 2)^{-2} [/latex]

   
   

Solution

1. [latex] (-5 \times 2)^3 = (-5)^3 \times 2^3 = -125 \times 8 = -1,000 [/latex]

   or

   [latex] (-5 \times 2)^3 = (-10)^3 = -1,000 [/latex]

   
   

2. [latex] \displaystyle{(-3 \times 2)^{-2} = (-3)^{-2} \times 2^{-2} = \frac{1}{(-3)^2} \times \frac{1}{2^2} = \frac{1}{9} \times \frac{1}{4} = \frac{1}{36}} [/latex]

   or

   [latex] \displaystyle{(-3 \times 2)^{-2} = (-6)^{-2} = \frac{1}{(-6)^2} = \frac{1}{36}} [/latex]

   
   

Example 3.3-f: Evaluating Expressions with Negative Bases using the Power of a Quotient Rule

Evaluate the following expressions:

1. [latex] (-2 \div 3)^{-2} [/latex]

   
   

2. [latex] (3 \div (-2))^{-3} [/latex]

   
   

Solution

1. [latex] \displaystyle{(-2 \div 3)^{-2} = \left(\frac{-2}{3}\right)^{-2} = \left(\frac{3}{-2}\right)^2 = \frac{3^2}{(-2)^2} = \frac{9}{4}} [/latex]

   
   

2. [latex] \displaystyle{(3 \div (-2))^{-3} = \left(\frac{3}{-2}\right)^{-3} = \left(\frac{-2}{3}\right)^3 = \frac{(-2)^3}{3^3} = \frac{-8}{27} = -\frac{8}{27}} [/latex]

   
   

Example 3.3-g: Evaluating Expressions with Negative Bases using the Power of a Power Rule

Evaluate the following expressions:

1. [latex] (-2^3)^3 [/latex]

   
   

2. [latex] (-3^3)^2 [/latex]

   
   

Solution

1. [latex] (-2^3)^3 = (-2)^{3 \times 3} = (-2)^9 = -512 [/latex]

   
   

2. [latex] (-3^3)^2 = (-3)^{3 \times 2} = (-3)^6 = 729 [/latex]

   
   

Principal Roots

Roots of Positive Numbers

• When the index of the root is even, any positive number will have two real number solutions, with one being the negative of the other. The positive solution is known as its principal root.

  • For example, [latex] \sqrt{9} [/latex] has two roots: +3 and –3, because (3) × (3) = 9 and (−3) × (−3) = 9 This is usually written as ±3 and read as “plus or minus 3”, and the principal root is 3.
  • [latex] \sqrt[4]{16} [/latex] has two roots: +2 and –2, because (2) × (2) × (2) × (2) = 16 and (–2) × (–2) × (–2) × (–2) = 16 i.e., the roots are ±2 and the principal root is 2.

• When the index of the root is odd, there is only one real number solution and it is positive. This positive solution is the principal root.

  • For example, [latex] \sqrt[3]{27} = 3 [/latex] because (3) × (3) × (3) = 27; i.e., the principal root is 3.
  • Similary, [latex] \sqrt[5]{32} = 2 [/latex] because (2) × (2) × (2) × (2) × (2) = 32; i.e., the principal root is 2.

Roots of Negative Numbers

• When the index of the root is even, there is no real number solution to any negative number.

  • For example, [latex] \sqrt[4]{-81} [/latex] has no real roots.

• When the index of the root is odd, there is only one real number solution to any negative number and it is negative. This negative solution is the principal root.

  • For example, [latex] \sqrt[5]{-32} = -2 [/latex] because (−2) (−2) (−2) (−2) (−2) = −32; i.e., the principal root is –2.

Real Numbers

Real numbers include all positive numbers and negative numbers. A summary of the types of real numbers, which we have learned about in the last three chapters, is provided below:

Table 3.3-b: Types of Real Numbers

Type	Description	Examples
Natural Numbers	Counting numbers (numbers starting from 1).
Whole Numbers	Natural numbers, including zero.
Integers	Natural numbers (positive integers), their negatives (negative integers), and zero.
Rational Numbers	Numbers that can be expressed as one integer divided by another non-zero integer; i.e., numbers that can be written as a quotient of integers with non-zero divisors.	[latex] \displaystyle{-\frac{5}{2}, 0.75, \frac{3}{2}} [/latex]
Irrational Numbers	Numbers that cannot be expressed as a rational number.	[latex] sqrt{2}, \pi, 2.718281... [/latex]

Note: Terminating decimals (decimals that end) and repeating decimals (decimals that do not end but show a repeating pattern) are also rational numbers because they can be expressed as a quotient of integers.

For example,

• [latex] 0.375 [/latex] can be expressed as [latex] \displaystyle{\frac{3}{8}} [/latex].

  
  

• [latex] 0.185185... [/latex] is usually written as [latex] 0.\overline{185} [/latex] and can be expressed as [latex] \displaystyle{\frac{5}{27}} [/latex].

  
  

3.3 Exercises

For Problems 1 to 6, place the correct sign ‘<’ or ‘>’ in the space between the following pairs of numbers.

1. a. –5 ▢ 0   b. –2 ▢ +6

   
   

2. a. 0 ▢ –3   b. –5 ▢ +3

   
   

3. a. +8 ▢ –3   b. +1 ▢ –2

   
   

4. a. –5 ▢+4   b. +3 ▢ –7

   
   

5. a. –6 ▢ –8   b. –5 ▢ –2

   
   

6. a. –7 ▢–9   b. –8 ▢ –4

   
   

For Problems 7 to 10, arrange the numbers in order from least to greatest.

7. a. 5, –6, 8, –8, –5, 2

   b. –8, 4, –6, 3, –9, 7

   
   

8. a. –2, –3, 5, 2, –1, 4

   b. 15, –14, 17, 4, –5, –7

   
   

9. a. 9, –5, –8, 3, 7, 10

   b. 12, –13, 15, 2, –8, –3

   
   

10. a. –3, 6, 1, –7, –1, 7

    b. –12, 0, 12, –16, 15, –5]

    
    

Evaluate Problems 11 to 20.

11. a. |–16|   b. –|3|

    
    

12. a. |–8|   b. –|12|

    
    

13. a. –|–5|   b. –[–|–9|]

    
    

14. a. –|–7|   b. –[–|–3|]

    
    

15. a. |–4| – |–7|   b. –|–8| + |–3|

    
    

16. a. –|–15| – |–3|   b. |–5| + |–4|

    
    

17. a. |–10| × |–5|   b. |–15| ÷ |–3|

    
    

18. a. |–10| × |–2|   b. |–12| ÷ |–4|

    
    

19. a. –|6| × |–3|   b. –|–20| ÷ |–5|

    
    

20. a. |–8| × |4|   b. –|–24| ÷ |–6|

    
    

Evaluate Problems 21 to 44.

21. a. –8 + (–5 – 7)

    b. 2 – (–3) + 1

    
    

22. a. –9 + (–3 – 8)

    b. 5 – (–7) + 8

    
    

23. a. –3 – (–7) + 8

    b. (–4 + 9) – (–3 – 6)

    
    

24. a. –7 – (–9) – 1

    b. (–5 + 3) + (–4 – 9)

    
    

25. a. 4 + (–3) – [5 + (–11)]

    b. –6 + (–4) – [–(15 – 8)]

    
    

26. a. 5 + (–4) – [7 + (–9)]

    b. –8 + (–15) – [–(6 – 7)]

    
    

27. a. 2(–3)(–5)

    b. –4(–3)(–2)

    
    

28. a. 6(–2)(–4)

    b. –5(–3)(–2)

    
    

29. a. –64 ÷ (–8)

    b. 45 ÷ (–5)

    
    

30. a. –48 ÷ (–6)

    b. 36 ÷ (–4)

    
    

31. a. –5 + (–2)(–5) – (6 – 3)

    b. 7(2 – 3) – 4(–7 + 1)

    
    

32. a. –8(5 – 6) – 3(–6 + 2)

    b. –7 + (–3)(–4) – (–8 – 3)

    
    

33. a. [latex] (5 + 7)^2 - 5^2 - 7^2 [/latex]

    b. [latex] 2^2 - 2^4 - 20 \times 3 [/latex]

    
    

34. a. [latex] (9 - 12)^2 - 9^2 - 12^2 [/latex]

    b. [latex] 7 \times 8 - 3^2 - 2^3 [/latex]

    
    

35. a. [latex] (20 \times 4 - 8^2)^2 + 9 [/latex]

    b. [latex] (3 \times 9 - 3^2) + 12 [/latex]

    
    

36. a. [latex] -12 ÷ 4 - 3 \times 2^4 [/latex]

    b. [latex] -5 \times 3^3 \div 9 + 5^2 [/latex]

    
    

37. [latex] -11^2 - 4 \times 54 \div (5 - 2)^3 - 3 [/latex]

    
    

38. [latex] [(1 + 12)(1 - 5)]^2 \div (5 - 3 \times 2^2 - 4) [/latex]

    
    

39. [latex] \displaystyle{-31 - [(15 \div 3) \times 32] \div \left(2^2 \times \frac{23}{46}\right)} [/latex]

    
    

40. [latex] 3(7^2 + 2 \times 15 \div 3) - (1 - 3 \times 4)^2 [/latex]

    
    

41. [latex] [(-8) + 4]3 \times (-2) \div (6 + |-5| \times 2) [/latex]

    
    

42. [latex] |-7 + 3|3 \times (-3) \div (4 + |-2| \times 4) [/latex]

    
    

43. [latex] |-3 - 1|2 \div (-2 + 2 \times |-3|) - (-3 - 2) [/latex]

    
    

44. [latex] |-2 - 1|3 \div |3 - 6| \times (-4 + |-5| \times 3) [/latex]

    
    

Express Problems 45 to 50 as a single power and then evaluate.

45. a. [latex] (-6)^5 \times (-6)^3 [/latex]

    b. [latex] 8^{-6} \times 8^9 [/latex]

    
    

46. a. [latex] (-2)^5 \times (-2)^6 [/latex]

    b. [latex] 4^{-7} \times 4^8 [/latex]

    
    

47. a. [latex] (-4)^5 \div (-4)^3 [/latex]

    b. [latex] -5^4 \div (-5)^2 [/latex]

    
    

48. a. [latex] (-2)^7 \div (-2)^4 [/latex]

    b. [latex] -4^6 \div (-4)^4 [/latex]

    
    

49. a. [latex] (-2^3)^2 \div (-2)^5 [/latex]

    b. [latex] -(3^2) \div (-3)^6 [/latex]

    
    

50. a. [latex] (-3^2)^3 \div (-3)^7 [/latex]

    b. [latex] -(2^2)^2 \div (-2)^3 [/latex]

    
    

Evaluate Problems 51 to 54.

51. a. [latex] (-2)^2 + (-3)^3 [/latex]

    b. [latex] (-3)^2 - (-2)^3 [/latex]

    
    

52. a. [latex] (-4)^2 + (-5)^3 [/latex]

    b. [latex] (-5)^2 - (-4)^2 [/latex]

    
    

53. a. [latex] (-2)^3 - (-3)^2 - 1^5 [/latex]

    b. [latex] (-7)^1 + (5)^0 - (-3)^2 [/latex]

    
    

54. a. [latex] (-3)^0 + (-4)^1 - (-2)^3 [/latex]

    b. [latex] (-3)^2 - (-2)^3 - 2^2 [/latex]