CLASS NOTES
Section 1.1 Standard Notation
I reminded everyone how important the notation is by asking
How many 5’s in 55? There are
2 answers, 11 • 5 = 55 and,
There are 2, a 5 and a 5.
Section 1.2 Addition
The notation for this is: addend + addend = sum
One I did not cover in class is what is:
Carry
1
1 1
5 1 6 3 4 7
+ 8 4 3 1 7 4
1 3 5 9 5 2 1
the above 1’s are
being carried to the next digit because the
previous digit’s
addition was greater than or equal to 10
The last part of section 1.2 we covered is
finding the perimeter
To determine the parameter of the above
square, add all of the
sides together
keeping the units of measurement to obtain
4 in. + 6 in. + 4 in. + 6
in. = 20 in.
Section 1.3 Subtraction
The notation for this is:
Again one we did not cover in class is:
Replace 2
4
- 3
1 1 6 4
1 3 1 5 3
The dashes between the 3
indicate that it is being replaced by a 2
The order of addition and
subtraction is to go from right to left or
add or subtract the
lowest value of the integer first, then keep
adding or subtracting
the next left integer until finished.
Section 1.4 Rounding and
Estimating; Order
Some
examples are:
2346 rounded to the nearest tens is 2350.
1.449 to the nearest hundredth is 1.45.
If we round 71.9 it = 72. Likewise if we round 45.4 it = 45.
If we take the sum of 72 and 45, we get 117
which is a ballpark
Figure of 71.9 + 45.4 =
117.3.
Read page 30 Rounding Whole
Numbers
By rounding whole numbers, you can get an
estimated answer.
An example of this is:
Say you are traveling 460 miles before your
first stop, and the
total
trip is 679 miles, how much further do you have to
go? If we round to the nearest ten,
680 miles – 460 miles = 220 miles in our estimation. The actual
miles would be, 679 miles – 460 miles = 219 miles.
Here
we are off by 1 mile
Order of whole numbers on
page 34
inequality symbols
<
is less than 4 < 6
>
is greater than 12 > 8
and ≠ is not equal to 22 ≠ 12
2 other ones I spoke about
are
≤ is less than or equal to
3 ≤ 4 and 4 ≤ 4
≥ is greater than or equal to
18 ≥ 3 and 10 ≥ 10
Section 1.5 Multiplication
The notation we use for this operation is:
factor ∙ factor = product
I prefer the ∙ for multiplication,
but X or * can be used.
8 X 4 = 8 ∙ 4 = ( 8 ) ( 4 ) = 8 ( 4 ) = 32
notice that (8)(4) = 8(4) = 32 but 84 ≠ 32, the numbers must be
separated by parenthesis if no operation symbol is to be used.
Looking at tab 11,
First carry 1 1 2 2
Second carry 1 2 2 2
2 3 4 4
X 6
0 0 5
1 1 7 2 0
+ 1 4
0 6 4 0 0 0 → the reason for the 3 zero’s is we
are multiplying by 6000
1 4 0 7 5 7 2 0
The order to multiply is from
right to left.
The multiplicative identity
is 1 because 1 ∙ n = n for any n in the whole numbers.
The next part was finding area.
Using the figure from above
(as seen on next page)
to find the area, multiply length X width
here 4 in ∙ 6 in = 24 square inches = 24 in2
do not forget to get the correct units of measurement. For further information, see page 43 in the
book.
Section 1.6 Division
The notation for division is:
Dividend ¸ Divisor = Quotient = Dividend / Divisor
For the symbol to imply the
division operation I usually use / but ¸ can be used as well.
An example of how to do
division well is:
328/84
3
8 4) 3 2 8
- 2 5 2
7 6
because 76 < 84 the remainder is 72 therefore the answer is
3 R 76.
The
check for division is:
Dividend = (Quotient ∙ Divisor) + Remainder
Using that rule, 3 ∙ 84
= 252 + 76 = 328
Again 1 is the division
identity because n / 1 = n for any n that is part of the real numbers.
For any number a, a/a =1 (page 51)
For the next 2 formula’s let a be any of the whole numbers.
0/a = 0 because 0 ∙ a = 0 for any a.
a/0 is undefined because 0 ∙
n ≠ a for any n that is a whole
number
(see page 51).
0/0 is undefined because if 0/0=n for any n that is a whole
number
the test works for every number 0 ∙ n = 0,
there is
more than 1 answer, and therefore it is undefined (see page 51 of Basic
Mathematics 9th Edition by Marvin L. Bittinger).
See page 52 of Basic
Mathematics 9th Edition by Marvin L. Bittinger
for an explanation of remainder shown earlier.
Another example would be tab
15, page 54 of Basic Mathematics 9th Edition by Marvin L. Bittinger
8 0 7
6 ) 4 8 4 6
- 4 8 0 0
4 6
- 4 2
4
Therefore the answer is 807 R
4
the properties of the
operations.
Adding is: Addend + Addend = Sum
0 is the additive identity
A property the additive
identity is:
the number + the identity = the number
Let n be any number. n + 0 = n.
Let a, b, c be any numbers.
Two properties of addition is
the associative law:
a + (b + c) = (a + b) + c
an example of this is let a = 6, b = 11, c = 19.
a + (b + c) = 6 + (11 + 19) =
6 + (30) = 36
(a + b) + c = (6 + 11) +19 =
(17) + 19 = 36
then 6 + (11 +19) = (6 + 11) + 19
and the communitive law:
a + b = b + a
For en example, using the above
values
a + b = 6 + 11 = 17
b + a = 11 + 6 = 17
so a + b = b + a.
Subtraction is: Minuend – Subtrahend = Difference
Let a, b, c be any numbers
such that a – b = c. Then a = c + b
For an example, a = 5, b = 2,
c = 3 then
a – b = c implies 5 – 2 = 3. To check this, c + b = 2 + 3 = 5 = a
To find missing numbers lets
look at 50 - □ = 32
We want to get □ all by
itself.
50 - □ = 32
-32 + □ = -32 + □
18 = □
0 is the subtractive identity
because
6 – 0 = 6 implies 6 = 0 + 6
which is true.
The multiplication term is:
Factor ∙ Factor = Product
Properties of multiplication
are:
Let a, b, c be any integers.
Distributive law.
a ∙(b + c) = (a
∙ b) + (a ∙ c)
for an example of this would be:
let a = 9, b = 21, c = 11. Then
9 ∙ (21 + 11) = 9
∙ (32) = 288
(9 ∙ 21) + (9 ∙
11) = (189) + (99) = 288
Associative law of multiplication.
a ∙ (b ∙ c) =
(a ∙b) ∙c
Using the same values for a,
b, and c given above; an example of the above property is:
9 ∙ (21 ∙ 11) = 9
∙ (231) = 2079
(9 ∙ 21) ∙ 11 =
(189) ∙ 11 = 2079
Commutative Law of Multiplication.
a ∙ b = b ∙ a
For the example, let a = 5, b
= 11. Then
a ∙ b = 5
∙ 11 = 55
b ∙ a = 11
∙ 5 = 55
Therefore a ∙ b = b ∙ a
Remember that language is
important in Mathematics. An ambiguous
question would be how many 5’s in 55?
One answer is 11 because 11 ∙ 5 = 55, another answer would be 2, a
5 and then another 5, go figure. Another
ambiguous question would be what is the product of 3 + 5
since product is the answer in a multiplication problem. A proper phrasing would be what is the sum of
3 + 5? The answer is 8.
Some math shorthand symbols,
you all will not be tested on, but may help when you take notes are:
∑ is the sum of
∏ is the product of
∆ is the change in
$ is
for all
' is
such that
\ is
therefore
Q is because
Section 1.7 Solving equations
Solution: What makes the equation true.
Solved: When all variable value(s) are found
Reminder what you do to one
side of the equation, you must do to the other side to keep the equation true.
Example:
10 + x = 23
-10 =-10
x = 13
Section 1.8 Applications and Problem Solving
Five steps for problem
solving
1) Familiarize yourself with what is being asked
a) Very carefully
read the problem until you know what is
being asked.
b) Draw diagrams
or review formulas that will aid in the
situation.
c) Assign a
letter to the variable or unknown
2) Translate the problem to en equation using the variable.
3) Solve the equation
4) Check the answer with the problem
5) State the answer clearly with appropriate units.
Section 1.9
Exponential Notation & Order
of Operation
The little number in the superscript, indicates the number of times the big number is
multiplied by itself. This is called exponential notation
52 = 5 ∙ 5 =
25 ; 52 is also called 5 squared
33 = 3 ∙ 3
∙ 3 = 27 ; 33 is also called 3 cubed
8 ∙ 8 ∙ 8 ∙
8 ∙ 8 = 85
Order of the Operation
1: Do the
operations in the parenthesis (), the brackets [], or braces {} starting with
the inner most and working to the outer most.
Example:
3 + ( 3 + ( 2 + 2 ) ) = 3
+ ( 3 + 4) = 3 + 7 = 10
2: Evaluate all exponential equations.
The above equation could be as:
3 ( 3 + 22 ) =
3 + ( 3 + 4 ) = 3 + 7 = 10
3: Multiplication and Division from left to right.
4: Addition and Subtraction from left to right.
Section 2.1 Factorizations
Remember section 1.5 of Basic
Mathematics 9th Edition by Marvin L. Bittinger
multiplication where we had two factors = a product. To find the factorization we start with a
product and find the factors. An example
would be:
72 = 8 ∙ 9
Where 8 ∙ 9 is the
answer.
If we divide a number n by a
divisor d and get a remainder of 0, we then say that d is a factor of n. Be aware that every integer is a factor of 0 [ 0 = 0 ∙ any number ] and 1 is a factor of every
number.
A number n is divisible by d
if there exists a c such that n = d ∙ c for
instance:
39 is divisible by 3 because 39 = 13 ∙ 3 so n = 39, d
= 3, and
c = 13.
3 is also a factor of 39.
A prime number is a number that
only has 1 and itself as factors.
See page 104 for a list of
prime numbers from 2 to 157
1 is not prime! The reasoning is due to the fact that 12
= 1
Likewise 0 is not prime 02
= 0.
If a number n has more than 1
and n as a factor, the number is then a composite number. an example is:
24 factors into 1, 24, 2, 12, 3, 8, 4, 6
it is then a composite number.
Section 2.2 Divisibility
A number is considered even if the far right digit is either a
2, 4, 6, 8, 0, otherwise it
is considered odd. If it is even, it is divisible by 2.
A number is divisible 3 if
all digits, added together down to
1 digit = 3, 6, 9. an example would be:
69 and adding digits together is 12, again adding digits
together
is 3 so 69 is divisible by 3
If it is divisible by 3 and
is even, it is divisible by 6.
A number is divisible 9 if
all digits, added together down to
1 digit = 9.
A number is divisible 10 if
the last number is a 0.
A number is divisible 5 if
the last number is a 0 or 5.
A number is divisible by 4 if
the number formed by the last 2 digits is.
For instance; 124 is because 24 is.
124/4 = 31.
A number is divisible by 8 if
the number formed by the last 3 digits is.
Section 2.3 Fractions and Fraction Notation
Let n be any real number,
then n/n = 1.
Example:
1265/1265 = 1 because 1265 = 1
∙ 1265.
Let n be any real number,
then 0/n = 0.
Example:
0/31 = 0 because 0 = 31 ∙ 0
Let n be any real number, then
n/0 is undefined.
Example:
4556/0 is undefined because 4556 ≠ 0 ∙
any number
Likewise n/0
is undefined. Because any number ∙
0 = 0 there are more than 1 answer which is a contradiction.
Let n be any real number,
then n/1 = n.
Example:
125/1 = 125 because 125 = 125 ∙
1.
Section 2.4 Multiplication and Applications
When multiplying fractions,
always remember to multiply the numerators (tops) together, and the
denominators (bottoms) together like: 5 ∙ 4/23 = 5/1
∙ 4/23 = 20/23
Check out example 10 on page
129 of Basic Mathematics 9th Edition by Marvin L. Bittinger to show how to do story problems.
Section 2.5 Simplifying
For help of other fractions
out there, see the blue tab on page 133 of Basic Mathematics 9th
Edition by Marvin L. Bittinger
The quickest way is to break
down prime factors the
numerator (top) and denominator (bottom), cancel like terms,
and multiply out what is left.
Let k, l, m,
n be real numbers. Say you simplified k/l
= m/n then a way that answer can be checked is l ∙
m = k ∙ n. Example:
Does 4/5 = 16/20 ? 5 ∙ 16 = 80, 4 ∙ 20 = 80 and
because
5 ∙ 16 = 4 ∙ 20 then it is true.
Section 2.6 Multiplication,
Simplification, and Applications
See blue tab on page 140 of
Basic Mathematics 9th Edition by Marvin L. Bittinger
on how To Multiply and Simplify
Another way to do section c
is to a prime decomposition on the
factors,
cancel out, and multiply out.
Section 2.7 Division and Applications
The reciprocal is the
denominator over the numerator. An
example is:
The reciprocal of 5/9 is 9/5
any number multiplied by its
reciprocal = 1
Let n be any real number.
0/n has no reciprocal because n/0
is
Undefined
To divide, multiply the
numerator by the reciprocal of the denominator