23 multiplication and division of signed numbers

Multiplication and Division of Signed Numbers
Back to Algebra–Ready Review Content.

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 positive
products.

+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.

+ * + = – * – = + ;
+ * – = – * + = – ;
product.
product.
Example A.
a. 5 * (4) = –5 * (–4)

+ * + = – * – = + ;
+ * – = – * + = – ;
product.
product.
Example A.
a. 5 * (4) = –5 * (–4) = 20

+ * + = – * – = + ;
+ * – = – * + = – ;
product.
product.
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4)

+ * + = – * – = + ;
+ * – = – * + = – ;
product.
product.
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20

+ * + = – * – = + ;
+ * – = – * + = – ;
product.
product.
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
In algebra, multiplication operation are not always written
down explicitly.

+ * + = – * – = + ;
+ * – = – * + = – ;
product.
product.
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
In algebra, multiplication operation are not always written
down explicitly. Instead we use the following rules to identify
multiplication operations.

● If there is no operation indicated between two quantities, the
operation between them is multiplication.

operation between them is multiplication. Hence xy means x * y.

● 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 sets of ( )’s,
the operation between them is multiplication.

Hence (x + y)(a + b) = (x + y) * (a + b)

Hence (x + y)(a + b) = (x + y) * (a + b)
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.

Hence (x + y)(a + b) = (x + y) * (a + b)
Hence 3(+5) = (+5)3 =15,

Hence (x + y)(a + b) = (x + y) * (a + b)
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,

Hence (x + y)(a + b) = (x + y) * (a + b)
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25,

Hence (x + y)(a + b) = (x + y) * (a + b)
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.

Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
Hence (x + y)(a + b) = (x + y) * (a + b)
To multiply many signed numbers together, we always
determine the sign of the product first, then multiply just the
numbers themselves. The sign of the product is determined by
the following Even–Odd Rules.

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)

Example B.
a. –1(–2 ) 2 (–1)
three negative numbers, so the product is negative

Example B.
a. –1(–2 ) 2 (–1) = – 4
4 came from 1*2*2*1 (just the numbers)

Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4

Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2)

Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2)
four negative numbers, so the product is positive

Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2) = 16

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).

In algebra, a ÷ b is written as a/b or .
a
b

Rule for the Sign of a Quotient
a
b

Division of signed numbers follows the same sign-rules for
multiplications.
a
b

multiplications.
a
b
+
+
=
–
– = +
+
+
=
–
–=
–

multiplications.
Two numbers with the same sign divided yield a positive
quotient.
a
b
+
+
=
–
– = +
+
+
=
–
–=
–

multiplications.
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
a
b
+
+
=
–
– = +
+
+
=
–
–=
–

multiplications.
quotient.
quotient.
Example C.
a.
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
=
–20
–4

multiplications.
quotient.
quotient.
Example C.
a.
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
=
–20
–4
= 5

multiplications.
quotient.
quotient.
Example C.
a.
b . –20 / 4 = 20 / (–4)
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
=
–20
–4
= 5

multiplications.
quotient.
quotient.
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
=
–20
–4
= 5

multiplications.
quotient.
quotient.
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
=
–20
–4
= 5
c.
(–6)2
=
–4

multiplications.
quotient.
quotient.
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
=
–20
–4
= 5
c.
(–6)2
=
36
–4–4

multiplications.
quotient.
quotient.
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
=
–20
–4
= 5
c.
(–6)2
=
36
–4
=
–4
–9

The Even–Odd Rule applies to more length * and / operations
problems.

problems.
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12

problems.
(– 4)6(–1)(–3)
(–2)(–5)12
five negative numbers
so the product is negative

problems.
(– 4)6(–1)(–3)
(–2)(–5)12
= –

problems.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
simplify just the numbers4(6)(3)
2(5)(12)

problems.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
2(5)(12)
= –
3
5

problems.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
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.)

The Even Power Graphs vs. Odd Power Graphs of y = xN

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)

23 multiplication and division of signed numbers

More Related Content

What's hot (20)

Similar to 23 multiplication and division of signed numbers (20)

More from alg-ready-review (20)

Recently uploaded (20)

23 multiplication and division of signed numbers