Download section 1.1: operations with real numbers

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
10  5  15
10  (5)
or
or
10  5  5
10  (5)
10  5  15
10  (5)  5
5  10  5
10  (5)  15
5  10  15
5  (10)
5  (10)
or
or
5  10  5
5  10  15
5  (10)  5
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
Fractions with Signs (+, -) addition to the sign rules on
the previous page.
Using a factor tree to find the LCD
7 13

36 24
Page 3 of 10
Review of Operations with Mixed Numbers When adding and subtracting, it is
usually a good idea to work the whole numbers and fractions separately.
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.
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.
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/exponentsradicals/scientific-notation/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
16.
Section 1.1
1. 16
2. A = 12.672 in
C = 7.134 in
3. S = 1
B = 3.934 in
D = 2.67 in
Width between centers = 58
5. Width = 76
Height = 31
18.
C = 11
8. Total Rise = 68
Total Run = 104
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
430 x 10-
7.5 x 10
4.2 x 10-8
6.2 x 107
7.5x 103
42 x 10-9
62 x 106
maximum
17.351 in
18.27 cm
26 in
minimum
17.283 in
17.97 cm
26 in
320.06 cm
319.94 cm
7.5 kW
42 nF
62MH
Difference
- in
b) Yes there is to spare, although this is
not much when divided among 9 spaces.
7. A = 3
B=5
4.3 x 10-4
480
kHz
430 µV
6
17.
6. a)
480 x 103
3
H=
4. R = 20
4.8 x 105
- .05 mm
+ .041 in
19. 27 Rows, 41 Columns, this design will
produce 1107 parts.
20. 32 Rows, 35 Columns, this design will
produce 1120 parts.
21.
Design
Length
Tolerance
Fraction
Max
Min
3.7 in
±
in
3
in
3
in
3
in
36.62 in
±
in
36
in
36
in
36
in
80.5 in
±
in
80
in
80
in
80
in
100.29 in
±
in
100
in
100
in
100
in
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