Multiplication and Division of Real Numbers

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Multiplication and Division of Real Numbers. Section 1.8 (60). Objectives (60). Multiply numbers Divide numbers Remove negative signs from denominators Evaluate divisions involving zeroes. 1.8.1 Multiply Numbers.

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### Multiplication and Division of Real Numbers

Section 1.8

(60)

Objectives(60)

Multiply numbers

Divide numbers

Remove negative signs from denominators

Evaluate divisions involving zeroes

1.8.1 Multiply Numbers

The first rule of multiplication is that if you multiply two numbers with the same sign, the product is positive. If you multiply two numbers with different signs, the product is negative.

If there are more than two numbers, this rule can be expanded to, if there is an even number of negative values, the product is positive. If there is an odd number of negative values, the product is negative. It doesn not depend on the number of positive numbers.

Of course, if one of the numbers is zero, the product is zero.

Examples

( -2 ) ( + 3 )

-6

( -4 ) ( - 3 )

+12

( + 5 ) ( + 3 )

+15

( +1) ( +1 ) ( -2 ) ( -1 ) ( +1 ) ( -1 ) ( +1 )

3 (an odd number) of negatives

-2

( +3 ) ( -4 ) ( -8 ) ( 0 ) ( -15 )

0

Hint(61)

Multiplication is indicated by ( ),•, or ×.

( -2 ) - 4 is not a product.

( -2 ) + ( -4 )

-6

-2 ( -4 ) , ( -2 ) ( -4 ) or ( -2 ) • ( -4 ) are all products.

8

Multiple Multiplications

From the book of OOOs, there is a rule about multiplications (or divisions).

If you are multiplying (dividing) more than two numbers with no parentheses except around the number, you do the operations from LEFT to RIGHT.

Example: ( 2 ) ( 4 ) ( 2 )

8 ( 2 )

16

Multiplying Fractions

For now, we will only go as far as if we multiply two fractions, we multiply the numerators and write the product on top and we multiply the denominators and write the product on the bottom.

Example:

( 2/3 ) ( 5/7 )

[( 2 ) ( 5 )] / [( 3 ) ( 7 )]

10/21

Later we will discuss reducing fractions.

Examples

( -3 ) ( 4 ) ( -2 ) ( -3 )

3 negatives, so the product is negative

- ( 3 ) ( 4 ) ( 2 ) ( 3 )

-12 ( 2 ) ( 3 )

-24 ( 3 )

-72

( -2 ) ( 4 ) ( -1 ) ( -3 ) ( -5 )

4 negatives, so the product is positive

+ ( 8 ) ( 1) ( 3 ) ( 5 )

+8 ( 3 ) ( 5 )

+24 ( 5 ) 24

5

120

+120

Examples

( -2/3) ( -5/7 )

[ ( -2 ) ( -5) ] / [ ( 3 ) ( 7 ) ]

+10 / 21

( 3/5 ) ( -1/4 )

[ ( 3 ) ( -1) ] / [ ( 5 ) ( 4 ) ]

-3/20

1.8.2 Divide Numbers(62)

Signs of division

If the numerator and denominator are both the same sign, the quotient is positive

If the numerator and denominator have different signs, the quotient is negative

If a and b are numbers:

- ( a/b ) ≡ ( -a/b ) ≡ a/(-b)

Therefore, - ( 1/3 ) ≡ ( -1/3 ) ≡ 1/(-3)

Examples

-36/ 4

-9/1, better just -9

-45/( -9 )

+5

35/( -7 )

-5

Dividing with Fractions

Remember multiplying by fractions, you multiply numerators, then multiply denominators.

If we divide by a fraction, we invert the divisor, then multiply.

Example:

( -2/7 ) ÷ ( -1/2 )

( -2/7 ) ( -2/1 )

+4/7

Example

Remember if we divide two numbers, the decimal quotient may not terminate.

Sometimes, you may be told to round the value to a particular number of decimals.

Example, if you are rounded to hundredths, you have to compute to the thousandth, then round to hundredth. 0-4 drop, 5-9 add 1 to hundredth.

Examples:

4.393 rounded to hundredth = 4.39

0.785 rounded to tenth = 0.79

5.99 rounded to tenth = 6.00

Example

Find 3.106 ÷ 0.35 to the nearest hundredth

. 8 . 874

0.35. ) 3.10 .600

2 80

30 6

28 0

2 60

2 45

150

140

10

The solution is 8.87

1.8.3 Remove Negative Sign from the Denominator(63)

If you have a negative value in the denominator, just multiply the expression by ( -1 )/( - 1).

1/( -3 ) is really not wrong. It is just that tradition always has the denominator as a positive value.

The book consolidates the signs for addition, subtraction, multiplication, and division in a table on page 64.

Simplification???

If it is addition, check the signs, if the same, just add the absolute values and use the common sign. If they are different, subtract absolute value of larger minus the absolute value of the smaller and use the sign of the larger.

If it is subtraction, convert to addition.

If it is multiplication, same signs the product is positive, if different signs, the product is negative.

If it is division, same rules as multiplication.

1.8.4 Evaluate Divisions Involving Zero(64)

Remember if c = a/b, then c • b = a

Three cases have a zero in a division.

zero divided by a non zero value ( 0/3 )

0

zero divided by zero ( 0/0 )

indefinite.

non zero value divided by zero ( -3/0 )

undefined

To put it in the vernacular, if you divide by zero you are in deep kimche.

Objectives(60)

Multiply numbers

Divide numbers

Remove negative signs from denominators

Evaluate divisions involving zeroes

Section 1.8

(60)