multiplication and division of signed numbers

1. Multiplication and Division of Signed Numbers

2. Rule for Multiplication of Signed Numbers
Multiplication and Division of Signed Numbers

3. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
Multiplication and Division of Signed Numbers

4. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
Multiplication and Division of Signed Numbers

5. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Multiplication and Division of Signed Numbers

6. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Multiplication and Division of Signed Numbers
Two numbers with the same sign multiplied yield positive
products.

7. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Multiplication and Division of Signed Numbers
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.

8. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4)
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.

9. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.

10. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4)
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.

11. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.

12. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
In algebra, multiplication operations are indicated in many ways.

13. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
In algebra, multiplication operations are indicated in many ways.
We use the following rules to identify multiplication operations.
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.

14. Multiplication and Division of Signed Numbers
● If there is no operation indicated between two quantities,
the operation between them is multiplication.

15. Multiplication and Division of Signed Numbers
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

16. Multiplication and Division of Signed Numbers
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

17. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

18. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15,
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

19. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

20. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25,
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

21. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

22. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
To multiply many signed numbers together,
we always determine the sign of the product first,
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.

23. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
To multiply many signed numbers together,
we always determine the sign of the product first:
the sign of the product is determined by the Even–Odd Rules,
then multiply just the (absolute values of the) numbers.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)

24. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
Multiplication and Division of Signed Numbers

25. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Multiplication and Division of Signed Numbers

26. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1)
Multiplication and Division of Signed Numbers

27. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1)
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative

28. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
4 came from 1*2*2*1 (just the numbers)

29. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative

30. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2)
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative

31. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2)
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
four negative numbers, so the product is positive

32. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2) = 16
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
four negative numbers, so the product is positive

33. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2) = 16
Fact: A quantity raised to an even power is always positive
i.e. xeven is always positive (except 0).
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
four negative numbers, so the product is positive

34. Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b

35. Rule for the Sign of a Quotient
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b

36. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b

37. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–

38. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–

39. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–

40. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4

41. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5

42. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4)
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5

43. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5

44. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
c.
(–6)2
=
–4

45. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
c.
(–6)2
=
36
–4–4

46. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
c.
(–6)2
=
36
–4 =
–4 –9

47. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers

48. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12

49. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12

50. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
five negative signs
so the product is negative

51. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
4(6)(3)
2(5)(12)
five negative signs
so the product is negative
simplify just the numbers

52. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
4(6)(3)
2(5)(12)
= –
3
5
five negative signs
so the product is negative
simplify just the numbers

53. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
five negative signs
so the product is negative
simplify just the numbers
4(6)(3)
2(5)(12)
= –
3
5
Various form of the Even–Odd Rule extend to algebra and
geometry. It’s the basis of many decisions and conclusions in
mathematics problems.

54. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
five negative signs
so the product is negative
simplify just the numbers
4(6)(3)
2(5)(12)
= –
3
5
Various form of the Even–Odd Rule extend to algebra and
geometry. It’s the basis of many decisions and conclusions in
mathematics problems.
The following is an example of the two types of graphs there
are due to this Even–Odd Rule. (Don’t worry about how they
are produced.)

55. The Even Power Graphs vs. Odd Power Graphs of y = xN
Multiplication and Division of Signed Numbers

56. Make sure that you interpret the operations correctly.
Exercise A. Calculate the following expressions.
1. 3 – 3 2. 3(–3) 3. (3) – 3 4. (–3) – 3
5. –3(–3) 6. –(–3)(–3) 7. (–3) – (–3) 8. –(–3) – (–3)
B.Multiply. Determine the sign first.
9. 2(–3) 10. (–2)(–3) 11. (–1)(–2)(–3)
12. 2(–2)(–3) 13. (–2)(–2)(–2) 14. (–2)(–2)(–2)(–2)
15. (–1)(–2)(–2)(–2)(–2) 16. 2(–1)(3)(–1)(–2)
17. 12
–3
18. –12
–3
19. –24
–8
21. (2)(–6)
–8
C. Simplify. Determine the sign and cancel first.
20. 24
–12
22. (–18)(–6)
–9
23. (–9)(6)
(12)(–3)
24. (15)(–4)
(–8)(–10)
25. (–12)(–9)
(– 27)(15)
26. (–2)(–6)(–1)
(2)(–3)(–2)
27. 3(–5)(–4)
(–2)(–1)(–2)
28. (–2)(3)(–4)5(–6)
(–3)(4)(–5)6(–7)
Multiplication and Division of Signed Numbers