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Physics/Conceptual Physics
Arithmetic Overview:
Numbers and Operations
Number Operations
• Real numbers: all numbers on the number
line. All numbers that appear on this test are
real.
• Integers: all numbers with no fractional or
decimal parts, including negative whole
numbers and zero; multiples of 1. See the
number line below.
Number Operations
• Sum: the result of addition
• Difference: the result of subtraction
• Product: the result of multiplication
• Quotient: the result of division
Number Operations
• Reciprocal: the result of switching the
numerator and denominator of a fraction.
• The reciprocal of 3/5 is 5/3.
• The reciprocal of 2 is 1/2 because 2 can be
considered to be the fraction 2/1
Numbers and the Number Line
• A number line is a straight line, extending
infinitely in either direction, on which real
numbers are represented as points.
• Decimals and fractions can also be depicted
on a number line, as can numbers such as √2..
Numbers and the Number Line
Numbers and the Number Line
• The values of numbers get larger as you move
to the right along the number line.
• Numbers to the right of zero are positive;
• Numbers to the left of zero are negative.
• Zero is neither positive nor negative.
Numbers and the Number Line
• Any positive number is larger than any
negative number. For example, -300 is less
than 4.
Assessment Question 1
All of the following statements are true
EXCEPT:
A. All numbers on the number line are real
numbers.
B. Integers are all numbers with no fractional or
decimal parts. They are whole numbers
C. The reciprocal of 3/7 is 7/3.
D. Zero is neither positive nor negative.
E. Division is used to determine the sum of two
numbers.
Number Operations
• Operations: a process that is performed on
one or more numbers.
• The four basic arithmetic operations are
addition, subtraction, multiplication, and
division.
Laws of Operation: Commutative Law
• Commutative laws of addition and
multiplication
• a+b = b+a
• axb = bxa
Laws of Operation: Commutative Law
• Addition and multiplication are both
commutative.
• Switching the order of any two numbers being
added or multiplied together does not effect
the result.
• For example, 5 + 8 = 8 + 5; both sums equal
13.
• Similarly 2 x 6 = 6 x 2; both products equal 12.
Laws of Operation: Commutative Law
• Subtraction and division are not commutative;
switching the order of the numbers changes
the result.
• For instance, 3-2 ≠ 2-3; the left side yields a
result of 1, whereas the right side yields a
result of -1.
• Similarly, 6÷2 ≠ 2÷6; the left side gives us 3,
whereas the right side gives us ⅓
Laws of Operation: Associative Law
• Associative laws of addition and
multiplication:
• (a + b) + c = a + (b + c)
• (a x b) x c = a x (b x c)
Laws of Operation: Associative Law
• Addition and multiplication are both
associative.
• (3+5) + 8 = 3 + (5+8)
• (8) +8 = 3 + (13)
• 16 = 16
Laws of Operation: Associative Law
• Addition and multiplication are both
associative.
• (4 x 5) x 6 = 4 x (5 x 6)
• (20) x 6 = 4 x (30)
• 120 = 120
Laws of Operation: Associative Law
• Because addition and multiplication are both
commutative and associative, numbers can be
added or multiplied in any order.
• Subtraction and division are not associative.
Laws of Operation: Associative Law
• Example:
• 7 - (10 - 4) = 7 – (6) = 1
• (7 - 10) - 4 = (-3) - 4 = -7
• So 7 - (10 - 4) ≠ (7 - 10) - 4.
Laws of Operation: Associative Law
• Example:
• 24 ÷ (12 ÷ 3) = 24 ÷ (4) = 6
• (24 ÷ 12) ÷ 3 = (2) ÷ 3 = ⅔
• So 24 ÷ (12 ÷ 3) ≠ (24 ÷ 12) ÷ 3
Assessment Question 2
All of the following statements are true
EXCEPT:
A. The four basic arithmetic operations are addition,
subtraction, multiplication, and division.
B. Numbers can be added, multiplied, subtracted or
divided in any order to get the same result.
C. 5 + 8 = 8 + 5 by Commutative law
D. Subtraction and division are not commutative or
associative. switching the order of the numbers
changes the result.
E. (3 x 5) x 7 = 3 x (5 x 7) by Associative law
Laws of Operation: Distributive Law
• In general, the distributive law is:
• a(b + c) = ab + ac
Laws of Operation: Distributive Law
• The distributive law of multiplication allows
you to "distribute" a factor over a group of
numbers that is being added or subtracted, by
multiplying that factor by each number in the
group.
Laws of Operation: Distributive Law
Example:
• 4(3 + 7) =
• (4 x 3) + (4 x 7)=
Laws of Operation: Distributive Law
• Example:
• 4(3 + 7) = (4 x 3) + (4 x 7)
• 4 x (10) = (12) + (28)
• 40 = 40
Laws of Operation: Distributive Law
• Division can be distributed in a similar way,
because dividing by a number is equivalent to
multiplying by that number's reciprocal.
Laws of Operation: Distributive Law
• Example:
•
=
(3 + 5) =
•
(8)
== =
=
(3) +
(5)
Laws of Operation: Distributive Law
• Don't get carried away, though.
• When the sum or difference is in the
denominator-that is, when you're dividing by
a sum or difference-no distribution is possible.
Laws of Operation: Distributive Law
Laws of Operation: Distributive Law
So their sum can not
equal 1
Assessment Question 3
All of the following statements are true
EXCEPT:
A.
A.
B.
C.
D.
In general, the distributive law is:
a(b + c) = ab + ac
Division can be distributed
5 x (8 x 7) = (5 x 8) + (5 x 7)
8/(3+2) = 8/3 + 8/2
Dividing by a number is equivalent to
multiplying by that number's reciprocal.
Operations with Signed Numbers
• Numbers can be treated as though they had
two parts:
• A positive (+) or negative (-) sign
• A number part
Operations with Signed Numbers
• For example: -3
• The sign of the number -3 is negative
• The number part is 3.
• Numbers without any sign are understood to
be positive.
Operations with Signed Numbers
• To add two numbers that have the same sign,
add the number parts and keep the sign.
• Example: What is the sum of -6 and -3 ?
Operations with Signed Numbers
•
•
•
•
•
Example:
What is the sum of -6 and -3 ?
To find (-6) + (-3)
Add 6 and 3
Then attach the negative sign from the
original numbers to the sum.
• (-6) + (-3) = -9
Operations with Signed Numbers
•
•
•
•
•
Example:
What is the sum of -6 and -3 ?
To find (-6) + (-3)
Add 6 and 3
Then attach the negative sign from the
original numbers to the sum.
• (-6) + (-3) = -9
Assessment Question 4
What is the sum of -7 and -2 ?
A.
B.
C.
D.
E.
-5
-9
14
27
-72
Operations with Signed Numbers
• To add two numbers that have different signs,
find the difference between the number parts,
and keep the sign of the number whose
number part is larger.
• Example: What is the sum of -7 and +4?
Operations with Signed Numbers
• Example: What is the sum of -7 and +4?
• To find (-7) + (+4), subtract 4 from 7 to get 3
• 7>4
• The number part of-7 is greater than the
number part of +4 so the final sum will be
negative.
• (-7) + (+4) = -3
Operations with Signed Numbers
• Subtraction is the opposite of addition.
• You can rephrase any subtraction problem as
an addition problem by changing the
operation sign from a minus to a plus and
switching the sign on the second number.
• For instance:
8 - 5 = 8 + (-5).
Operations with Signed Numbers
• There's no real advantage to rephrasing if you
are subtracting a smaller positive number
from a larger positive number.
• But the concept comes in very handy when
you are subtracting a negative number from
any other number, a positive number from a
negative number, or a larger positive number
from a smaller positive number.
Operations with Signed Numbers
• To subtract a negative number:
• Rephrase as an addition problem and follow
the rules for addition of signed numbers.
• For instance:
9 - (-10) =9 + 10 =19
Operations with Signed Numbers
• Here's another example:
• (-5) - (-2) = (-5) + 2;
• The difference between 5 and 2 is 3
• The number with the larger number part is -5
• So the answer is -3.
Assessment Question 5
(-7) – (-2) =
A.
B.
C.
D.
E.
-5
-9
14
27
-72
Operations with Signed Numbers
• To subtract a positive number from a negative
number or from a smaller positive number:
• Change the sign of the number that you are
subtracting from positive to negative
• Then follow the rules for addition of signed
numbers.
• For example, (-4) - 1 = (-4) + (-1) = -5.
Operations with Signed Numbers
• Example: Subtract 8 from 2.
• 2 - 8 = 2 + (-8)
• The difference between the number parts is 6,
and the -8 has the larger number part.
• So the answer is -6.
Operations with Signed Numbers
• Multiplication and division of signed numbers:
• Multiplying or dividing two numbers with the
same sign gives a positive result.
• (-4) x (-7) = +28
• (-50) ÷ (-5) = +10
Operations with Signed Numbers
• Multiplying or dividing two numbers with
different signs gives a negative result:
• (-2) x (+3) =-6
• 8 ÷ (-4) =-2
Assessment Question 6
All of the following statements are true
EXCEPT:
A.
B.
C.
D.
E.
-8 - 9 = -17
8 x -8 = -64
(-8 x -7) = (8 x 7)
-8/-2 = -4
2 - 7 = -5
Properties of Zero
• Adding zero to or subtracting zero from a
number does not change the number.
• x+0=x
• 0+x=x
• x–0=x
Properties of Zero
• Example:
• 5+0=5
• 0+(-3)=-3
• 4-0=4
Properties of Zero
• Notice, however, that subtracting a number
from zero switches the number's sign.
• It's easy to see why if you rephrase the problem
as an addition problem.
Properties of Zero
• Example: Subtract 5 from 0.
• 0-5 = -5
• That's because 0 - 5 = 0 + (-5)
• According to the properties of zero:
• 0 + (-5) = -5.
Properties of Zero
• The product of zero and any number is zero:
• 0xz=0
• z x 0 =0
• Example:
• 0 x 12 = 0
Properties of Zero
• Division by zero is undefined.
• For practical purposes, that translates as "it
can't be done."
• Because fractions are essentially division (that is
¼ means 1 ÷ 4), any fraction with zero in the
denominator is also undefined.
Assessment Question 7
All of the following statements are true
EXCEPT:
A.
B.
C.
D.
E.
0 - 9 = -9
8x0=0
(-8 x 0) = (0 x 7)
-8/0 = -8
2+0 =2
Properties of 1 and -1
• Multiplying or dividing a number by 1 does not
change the number:
• ax1=a
• 1xa=a
• a÷1=a
Properties of 1 and -1
• Example:
• 4x1=4
• 1 x (-5) =-5
• (-7) ÷ 1 =-7
Properties of 1 and -1
• Multiplying or dividing a nonzero number by -1
changes the sign of the number:
• a x (-1) = -a
• (-1) x a = -a
• a + (-1) = -a
Properties of 1 and -1
• Example:
• 6 x (-1) = -6
• (-3) x (-1) = 3
• (-8) ÷ (-1) =8
Assessment Question 8
All of the following statements are true
EXCEPT:
A.
B.
C.
D.
E.
1x9 =9
-1(4) = -4
(-8 x -1) = (8 x 1)
-8/-1 = 8
1+1 =1x1
Order of Operations
• Whenever you have a string of operations, be
careful to perform them in the proper order.
• Otherwise, you will probably get the wrong
answer.
Order of Operations: PEMDAS
• PEMDAS
• The acronym PEMDAS stands for the correct
order of operations:
Order of Operations: PEMDAS
 Parentheses
 Exponents
 Multiplication
 Division
 Addition
 Subtraction
}Multiplication & Division
} in order from left to
right
}Addition & Subtraction
} in order from left to
right
Order of Operations
• If you have trouble remembering PEMDAS
• Think of the mnemonic phrase:
• Please Excuse My Dear Aunt Sally.
Order of Operations
• Example:
• 66 x (3 -2) ÷ 11
Order of Operations
Example:
66 x (3 -2) ÷ 11
• If you were to perform all the operations
sequentially from left to right
• Without regard to the rules for the order of
operations
• You would arrive at the answer 196/11 .
Order of Operations
Example: 66 x (3 -2) ÷ 11
• To do this correctly, do the operation inside the
parentheses first: (3 – 2) = 1
• Now we have:
• 66 x (1) ÷ 11 =
• 66 ÷ 11 = 6
Assessment Question 9
90 - 4( 3+2) x 4 =
A.
B.
C.
D.
E.
-9
10
145
280
2064
Order of Operations
• Example:
• 30 - 5 x 4 + (7 -3)2 ÷ 8 =
Order of Operations
• Example:
30 - 5 x 4 + (7 -3)2 ÷ 8 =
• First perform any operations within
parentheses. (If the expression has
parentheses within parentheses, work from
the innermost out.)
• 30 - 5 x 4 + (4)2 ÷ 8 =
Order of Operations
• Example:
30 - 5 x 4 + (7 -3)2 ÷ 8 =
• Next, do the exponent:
• 30 - 5 x 4 + (16) ÷ 8 =
Order of Operations
• Example:
30 - 5 x 4 + (7 -3)2 ÷ 8 =
• Then, do all multiplication and division in
order from left to right.
• 30 - 20 + 2 =
Order of Operations
• Example:
30 - 5 x 4 + (7 -3)2 ÷ 8 =
• Last, do all addition and subtraction in order
from left to right.
• 10 +2 =
• The answer is 12.
Assessment Question 10
25 ÷ ( 3+2)2 x 4 =
A.
B.
C.
D.
E.
4
16
350
500
2500