1.1 Numbers, Data, and Problem Solving
The Real Number System
start with
1, 2, 3, . .
natural
numbers
and do the following:
add
to get the which are:
larger
class of
negative whole
integers
. . . -3, -2, -1, 0, 1, 2, 3, . . .
numbers and 0
fractions:
rational
all the above +
e,g. 3/2, -2/3,
numbers
3/2, -2/3, etc.
irrational numbers: real
all the above +
(1) roots that can’t numbers
2 , 3 7 , , etc.
be simplified to
expressions without
roots
(2) numbers like 
 3/2 is a rational number and a real number (but is not an
integer or natural number)
 any number in a class also belongs to all larger classes
 6/3 is an integer, because 6/3 = 2
 but 4/3 is not (not expressible as a whole number)
 non-terminating, repeating, decimals, e.g. 1/3 = .333…,
are rational numbers
 non-terminating, but not repeating, decimals are
irrational numbers
1.1-1
% change:
 if the price of a dress goes from $100 to $110, the %
change is obviously +10%
 mathematically, it is computed by:
change in price
o
= 10/100 = .10 = 10%
base price
 so if a $52 pair if shoes goes up to $63
o % change is 11/52 = .212 = 21.2%
 if the shoes go down to $47, the change is negative, so
o % change is -5/52 = -.096 = -9.6%
Scientific notation:
e.g. 431,000 is expressed as 4.31 x 105
 scientific notation is a number with one digit before the
decimal point, times
 the power of ten that makes it correct
 you have to move the decimal point for 4.31 5 places
to the right to re-capture the original number
 thus the “x 105”
 similarly, .000431 = 4.31 x 10-4
 have to move decimal point for 4.31 4 places to the
left to recapture the original number
 “x 10-4” does this for you
1.1-2
Computing with scientific notation:
431,000 x 5,200,000 = 4.31 x 105 x 5.2 x 106
= 4.31 x 5.2 x 105 x 106 = 22.412 x 1011 = 2.2412 x 1012
Note the last step:
 you have to re-express the answer into scientific
notation
 move the decimal to the left one place
 but this makes it smaller
 so compensate
 by increasing the exponent by one . . .
 makes it bigger
 and restores the original value
1.1-3