23 multiplication and division of signed numbers

Multiplication and Division of Signed Numbers

Back to Algebra–Ready Review Content.

http://www.lahc.edu/math/algebra/alg-ready-placement-review.html

Rule for Multiplication of Signed NumbersMultiplication and Division of Signed Numbers

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


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


Rule for Multiplication of Signed NumbersTo 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.




Example A.

a. 5 * (4) = –5 * (–4)




Example A.

a. 5 * (4) = –5 * (–4) = 20




Example A.

a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4)




Example A.

a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4) = –20




Example A.

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




Example A.

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

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

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

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




● 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)







However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine.





However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15,





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,





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,





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.


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.



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.


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).




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

b

Rule for the Sign of a Quotient


b

Rule for the Sign of a QuotientDivision of signed numbers follows the same sign-rules for multiplications.


b



b

++ = –

– = + ++ = – –=–


Two numbers with the same sign divided yield a positive quotient.


b

++ = –

– = + ++ = – –=–


Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient.


b

++ = –

– = + ++ = – –=–




Example C.

a.

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

++ = –

– = + ++ = – –=–

204 = –20

–4




Example C.

a.


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5




Example C.

a.

b . –20 / 4 = 20 / (–4)


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5

c. (–6)2

=–4




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5

c. (–6)2

=36–4–4




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5

c. (–6)2

=36–4 =–4 –9

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




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




five negative numbersso the product is negative




= –





= –


simplify just the numbers 4(6)(3)2(5)(12)




= –



= – 35




= –



= – 35Various 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

Education

Transcript of 23 multiplication and division of signed numbers