we start at zero, take two steps to the right,
and then take three more steps to the right,
which causes us to land on positive 5. -5 -4 -3
-2 -1 0 1 2 3 4 5 Chapter 1: The Numbers of
Arithmetic 9 ADDITION OF NEGATIVE
NUMBERS What does it mean to add nega
REDUCING, AND DIVIDING FRACTIONS
EQUIVALENT FRACTIONS Equivalent fractions
are fractions that have the same value, for
example 8 4 6 3 4 2 2 1 = = = etc. Although all
these fractions are written differently, they all
represent the same quantity. You can m
MP = P D where D = 0 0 0 0 1 0 0 0 3 .
(e) Yes, the direction given by the eigenvector
1 1 1 because its eigenvalue is zero.
This is probably a bad design for a bridge
because it can be displaced in this direction
with no force! 12. (a) If we call M = a
reciprocal gives you back what you started
with. This allows us to define division as
multiplication by the reciprocal: a b = a (1/b)
This is usually the most convenient way to
think of division when you are doing algebra.
NOTATION FOR DIVISION Instead of
are 1. The associative and commutative laws
for addition and multiplication 2. The
existence of the additive and multiplicative
identities (0 and 1) Chapter 1: The Numbers of
Arithmetic 14 3. The existence of the additive
inverse (opposites, or negatives)
the nearest tenth. The digit to the right of the
tenths place is a 6, so we have to round up.
But when we round up the 9 it becomes a
10, forcing the one to be added to the left.
Unfortunately, we find another 9 there and
the process is repeated for each
3 first, and then add 5 to the result, given the
incorrect answer of 6.3333. To make it perform
the addition first, use parentheses: 4 (3 + 5) =
0.5 In our example problem above, the 2 + 3
in the exponent is an implied grouping, and
you would need to use
Numbers of Arithmetic 4 Fractions can be
numbers smaller than 1, like 1/2 or 3/4 (called
proper fractions), or they can be numbers
bigger than 1 (called improper fractions), like
two-and-ahalf, which we could also write as
5/2 All integers can also be tho
number 4 could be factored into 2 2). 2x 3 4 2
- x + 1 Factor SIMPLIFYING ALGEBRAIC
EXPRESSIONS By simplifying an algebraic
expression, we mean writing it in the most
compact or efficient manner, without changing
the value of the expression. This mainly
i
problem that way, you should be able to see
what you need to do in order to solve it.
Converting between percents and their
decimal equivalents is so simple that it is
usually best to express all percents in decimal
form when you are working percent probl
In this last example, the first fraction has a
decimal in it, which is not a proper way to
represent a fraction. To clear the decimal, just
multiply both the numerator and the
denominator by 10 to produce an equivalent
fraction written with whole numbers.
of them (2x + 4x). The only thing left is the
constants 2 + 7 = 9. Putting this all together we
get x 2 + 2x + 3x 2 + 2 + 4x + 7 = 4x 2 + 6x + 9
PARENTHESES Parentheses must be
multiplied out before collecting like terms You
cannot combine things in paren
decimal places: 8.602 7480 1122 2.3 3.74 +
Division You can divide decimal numbers using
the familiar (?) technique of long division. This
can be awkward, though, because it is hard to
guess at products of decimals (long division,
you may recall, is basi
Divide the percentage by 100 (or move the
decimal point two places to the left). Chapter
1: The Numbers of Arithmetic 33 Since 100 % x
x = , the decimal equivalent is just the
percentage divided by 100. But dividing by 100
just causes the decimal point t
have zero as an eigenvalue. iii. Since MX = 0
has no non-zero solutions, the matrix M is
invertible. (b) Now we suppose that M is an
invertible matrix. i. Since M is invertible, the
system MX = 0 has no non-zero solutions. But
LX is the same as MX, so the
is extremely important, and it is impossible to
understand algebra without being thoroughly
familiar with this law. Example: 2(3 + 4)
According to the order of operations rules, we
should evaluate this expression by first doing
the addition inside the par
denominator (but not necessarily the Least
Common Denominator). LEAST COMMON
DENOMINATOR (LCD) By Inspection The
smallest number that is evenly divisible by all
the denominators In General The LCD is the
product of all the prime factors of all the
denomin
the third line was just algebra and the fourth
used the definition of (L1 + L2) again. (h)
Yes. The easiest way to see this is the
identification above of these maps with bitvalued column vectors. In that notation, a
basis is n 1 0 0 , 0 1 0 , 0 0 1 o . 3
of that something. In fact, we have one whole
plus one more quarter (if you have 5 quarters
in change, you have a dollar and a quarter).
Chapter 1: The Numbers of Arithmetic 20
MIXED NUMBER NOTATION One way of
expressing the improper fraction 5/4 is as th
works for the repeating fraction part of a
number. If you have a number like 2.33333.,
you should just work with the decimal part
and rejoin it with the whole part after you
have converted it to a fraction. o Irrational
numbers like p or 2 have non-repeat
there exists its opposite b, and we can define
subtraction as adding the opposite: a b = a +
(-b) In algebra it usually best to always think
of subtraction as adding the opposite
DISTINCTION BETWEEN SUBTRACTION AND
NEGATION The symbol means two
different
exponents: (42 ) 3 = (42 )(42 )(42 ) = (4 4) (4 4)
(4 4) = 4 4 4 4 4 4 = 4 6 There are more
rules for combining numbers with exponents,
but this is enough for now. ORDER OF
OPERATIONS When we encounter an
expression such as 3 + 15 3 + 5 2 2+3, it
makes qu
minus 2 x 2 difference between the
difference between a number and 8 x 8
from 2 from a number n 2 less a number
less 3 n 3 less than 3 less than a number y
3 fewer than 2 fewer than a number y 2
decreased by a number decreased by 2 x 2
Subtraction take a
division. 4. Addition and Subtraction, left to
right The left-to-right order does not matter if
only addition is involved, but it matters for
subtraction. Example: Going back to our
original example, 3 + 15 3 + 5 2 2+3 Given: 3
+ 15 3 + 5 2 2+3 The expone
to add these fractions by finding a common
denominator. In this example, the common
denominator is 1000, and we get 1000 2345
1000 5 1000 40 1000 300 1000 2000 1000 5 100
4 10 3 1 2 2.345 = = + + + = + + + This suggests a
general rule for converting a dec
we can say that 1000x = 345 + x Now solve for
x: 333 115 999 345 999 345 1000 345 1000 345
= = = - = = + x x x x x x CONVERTING FRACTIONS
TO DECIMALS We know the decimal
equivalents for some common fractions
without having to think about it: 1/2 = 0.5, 3/
would have to add 3/2 Chapter 2: Introduction
to Algebra 51 to both sides of the equation to
isolate the x. It is usually more convenient,
though, to use the addition principle first:
Given: 2x 3 = 5 Add 3 to both sides: 2 8 3 3 2
3 5 = = - = x x At this
we finally have the complete set of real
numbers. Any number that represents an
amount of something, such as a weight, a
volume, or the distance between two points,
will always be a real number. The following
diagram illustrates the relationships of the s
and 5/8 inches, or maybe 2 and 23/32 inches.
It seems that however close I look it is going to
be some fraction. However, this is not always
the case. Imagine a line segment exactly one
unit long: Now draw another line one unit
long, perpendicular to the
are decreasing by 2 each time, so if we let the
values for a continue into the negative
numbers we should keep decreasing the
product by 2: a b a b 3 2 6 2 2 4 1 2 2 0 2 0 1
2 2 2 2 4 3 2 6 We can make a bigger
multiplication table that shows many differe