Download LectureSection1.1OperationsWithRealNumbers

SECTION 1.1: OPERATIONS WITH REAL NUMBERS
Skill Review and Development
Review of Operations With Signed Numbers
Adding Signed Numbers:
positive + positive
Add the absolute values. The sign of the answer is positive.
8 + 6 = 14
negative + negative
Add the absolute values. The sign of the answer is negative.
-8 + (-6) = -14
positive + negative or negative + positive
Subtract the absolute values (smallest from largest). The sign of the answer is the sign of
the largest numeral.
8 + (-6) = 2
-8 + 6 = -2
Subtracting Signed Numbers: Some problems can be easily done directly. Others are easier to
do if you do a little reformatting first. When you reformat, change the subtraction to the sum of
the opposite, and then follow the rules for addition.
10  5  5
or
10  5  15
or
10  ( 5)  5
5  10  5
or
10  (5)  15
5  10  15
or
5  (10)  5
5  (10)  15
10  (5)
10  5  5
5  (10)
5  10  5
10  (5)
10  5  15
5  (10)
5  10  15
TAKE THE SHORTCUT When adding and subtracting integers
(positive and negative numbers) there is a much easier approach.
1. Get rid of all the double signs.
+ + becomes +
- - becomes +
+ - becomes –
- + becomes –
2. Now look at each integer separately and combine them as if they
were numbers tossed on a table. You will still need to follow the rules
for addition, but this is a much easier approach.
Page 1 of 10
Multiplying and Dividing Signed Numbers:
Multiply/Divide the absolute values. If the signs of the values are:
 Two Positives - The answer is positive.
 Two Negatives - The answer is positive.
 One Positive and One Negative - The answer is negative.
18  2  36
18  2  36
18  (2)  36
18  (2)  36
18  2  9
18  2  9
18  (2)  9
18  (2)  9
TAKE THE SHORTCUT
1. If the problem is multiplication, just multiply.
If the problem is division, just divide.
2. If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.
Page 2 of 10
Review of Operations With Fractions
Operation
Rule
Addition/Subtraction
Same Denominators
Add/Subtract the numerators.
Denominator stays the same.
Simplify if possible.
Addition/Subtraction
Different Denominators
Convert the fractions to LCD,
then Add/Subtract the
numerators. Denominator stays
the same. Simplify if possible.
Multiplication
Multiply the numerators.
Multiply the denominators.
Simplify if possible.
Division
Change to multiplication of the
inverse. Multiply the
numerators. Multiply the
denominators. Simplify if
possible.
Example
4 3 7
2
  or 1
5 5 5
5
4 2

5 3
12 10 2
 
15 15 15
4 3 12 3
 

5 8 40 10
Reduce before or after
multiplying.
3 3

7 10
3 1 10 10
3
 
or 1
7 31 7
7
Use the rules above in addition
Fractions with Signs (+, -) to the sign rules on the previous
page.
Using a factor tree to find the LCD
7 13

36 24
Review of Operations with Mixed Numbers When adding and subtracting, it is usually a
good idea to work the whole numbers and fractions separately.
Page 3 of 10
Example 1: Find the space between each fence slat if the spaces are all equal.
Page 4 of 10
Review of Operations with Decimals
Operation
Rule
Write the problem vertically
Addition/Subtraction lining up the decimals points.
Add/Subtract the values and bring
down the decimal.
Example
14.68 – 3.901
14.68
- 3.901
10.779
14.68 x 3.901
Multiplication
Right justify the numbers.
Multiply the numbers, insert the
decimal the total number of
places moving from right to left.
Division
Move the decimal in the divisor
to the far right. Move the decimal
in the dividend the same number
of places. Bring the decimal up
in the quotient. Divide the
numbers. Directions for rounding
would probably apply.
14.68 2 places
x 3.901 3 places
1468
132120
4404___
57.26668 5 places
___10 .1_
2.43 ) 24.600
243
300
243
57
Example 2:
Find the new dimensions for the part below if it is scaled up by a multiple of 6.
Find the new dimensions for the part below if it is scaled down by a factor of 4.
Page 5 of 10
Example 3:
Calculate the overall width and height of plate
steel necessary to lay out 3 rows and 5 columns
of the part below if they are spaces ¾ in apart
to allow for the cutting tool. Assume there is
no allowance between the part and the edge of
the plate (the part goes all the way to the
edge). All dimensions are in inches. Round
your answer to the nearest 16th of an inch.
Hint: Drawing a picture will probably help.
Example 4:
Calculate dimensions X
& Y so that all 8 gaps
(between the circles and
between the circle and
edge of the plate) are
the same size. Give your
answer to the nearest
16th of an inch.
Page 6 of 10
Page 7 of 10
From - http://www.ehow.com/how_8630346_convert-decimals-engineeringnotation.html#ixzz2xVGpkmyW
Instructions
Move the decimal to the right or the left three places at a time while keeping track of the total number
of decimal places moved. If the number is very large, move it to the left; stop before the number
would become less than one. If the number is very small, move it to the right until the number is
greater than one and less than one thousand. Range for first value is 1-999.
Rewrite the number with the decimal place in the new location. If the number is not greater than one
and less than one thousand, Step 1 has been performed incorrectly.
Add to the end of the number "x 10" with the proper exponent. The exponent will be equivalent to the
the number of places the decimal was moved. It will be negative for small numbers and positive for
large numbers. The exponent should always be a multiple of 3 or -3.
Page 8 of 10
Homework: You are assigned problems 1-14, 17-22.
Show all of your work in an organized manner. Label your work so it can be easily
followed. If you draw pictures, do so with care. Be careful with your units and label
what you are doing. You will be graded on your presentation.
We will do problems 15 and 16 together in class; they are in your lecture notes. You are
expected to turn these in (completed) attached to the homework from section 1.1.
Make sure you check your answers on the homework; see the last page of the lecture
handout.
Here are some links you may find helpful:
Adding and Subtracting Positive and Negative Numbers https://www.khanacademy.org/math/arithmetic/addition-subtraction
Scientific Notation - https://www.khanacademy.org/math/arithmetic/exponents-radicals/scientificnotation/v/scientific-notation
Decimals - https://www.khanacademy.org/search?page_search_query=decimals
Fractions - https://www.khanacademy.org/search?page_search_query=fractions
Page 9 of 10
Math 63 Solutions:
Section 1.1
16.
4.8 x 105
4.3 x 10-4
7.5 x 103
4.2 x 10-8
6.2 x 107
17.
maximum
17.351 in
18.27 cm
1. 16
2. A = 12.672 in
B = 3.934 in
C = 7.134 in
D = 2.67 in
26
3. S = 1
480 x 103
430 x 10-6
7.5x 103
42 x 10-9
62 x 106
minimum
17.283 in
17.97 cm
in
26
320.06 cm
H=
18.
4. R = 20
480 kHz
430 µV
7.5 kW
42 nF
62MH
in
319.94 cm
Difference
-
Width between centers = 58
in
- .05 mm
+ .041 in
5. Width = 76
Height = 31
6. a)
b) Yes there is
to spare, although this is not
much when divided among 9 spaces.
19. 27 Rows, 41 Columns, this design will produce
1107 parts.
20. 32 Rows, 35 Columns, this design will produce
1120 parts.
21.
7. A = 3
B=5
Design
Length
C = 11
3.7 in
±
in
3
in
3
in
3
in
16.62 in
±
in
16
in
16
in
16
in
80.5 in
±
in
80
in
80
in
80
in
100.29 in
±
in
100
in
100
in
100
in
8. Total Rise = 68
Tolerance
Total Run = 104
Fraction
Max
Min
9. A = 109
B = 122
, this design will produce 84 parts.
10. A = 102.506 in
B = 75.39 in, this design will produce 96 parts.
11. Area = 89
square inches
Perimeter = 39
12. Width = 2
Height = 1
13. A = 2
B=3
,
14. 32
15. a) 6,320,000,000 V
b) .0000000098 A
c) .00000047 F
d) 14,200,000 W
22.
Design
Dimension
in
Tolerance
± in
Maximum
Dimension
in
Minimum
Dimension
in
in
± in
in
in
in
± in
in
in
in
±
in
in
in
in
± in
in
in
in
± in
in
in
Page 10 of 10