Sir Isaac Newton((16421727)
Sir Isaac Newton was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is considered by many scholars and members of the general public to be one of the most influential people in human history. His 1687 publication of the Philosophiæ Naturalis Principia Mathematica (usually called the Principia) is considered to be among the most influential books in the history of science, laying the groundwork for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the scientific revolution.
SECTION 4.1: SIGNED NUMBERS
A "signed number" is a digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
or combination of digits with a positive(+) or negative() sign in from of it.
A signed number with "+" symbol in front of it is called a "positive
number". In general, mathematicians do not place a "+" symbol in
front of a number to designate a positive number. If there is no "sign symbol"
in front of a number, the number is "understood" to be positive.
A signed number with "" symbol in front of it is called a "negative
number".
ex. 1) +7 is called positive 7, but most books will write 7 for +7
ex. 2) 12 is called negative 12
SECTION 4.2: OPPOSITE NUMBERS
The numbers 3 and 3, 5 and 5, 2/3 and 2/3, etc. are called "opposite numbers". Opposite numbers are made up of the same
digits in the same order but have opposite signs.
SECTION 4.3: ABSOLUTE VALUES
The following symbolism,  7 , is asking you a question. It's asking, "How
many units is 7 from 0 on the number line?" Your answer would be 7 units from
0.
 5  is asking, "How many units is 5 from 0 on the number line?" The answer
is 5 units from 0.
The symbol,  , is called the "absolute value symbol". The
number inside the absolute value symbol is called the "argument," and the result you get is called the "absolute value".
MEANING OF THE ABSOLUTE VALUE
The absolute value represents a distance from 0.
Distance is a positive number by agreement. 
ex. 1)  6  = 6; and 6 is called the absolute value and 6 is 6 units from 0
ex. 2)  6  = 6; and 6 is called the absolute value and 6 is 6 units from 0
The distance from 0 to 6 and 0 to 6 is exactly the same. The distance is 6
units.
ex. 3)  25  = 25; and 25 is called the absolute value and 25 is 25 units
from 0
SECTION 4.4: ADDING POSITIVE NUMBERS
At this point in your math career you know you can add positive numbers, add
positive and negative numbers, and add negative numbers. We'll review each one
of these cases.
RULE FOR ADDING POSITIVE NUMBERS
Use traditional addition. The result is understood to be positive. 
ex. 1) 5 + 10 = +15 ex. 2) 3/5 + 7/5 = +10/5 ex. 3) 2/3 + 3/4 = +17/12
Comment: In general the positive symbol "+" is dropped in an answer. So +15 would be written as 15.
SECTION 4.5: ADDING NEGATIVE NUMBERS
RULE FOR ADDING NEGATIVE NUMBERS
Step 1) Find the absolute value of each number
Step 2) Add the absolute values
Step 3) The result of step 2 takes a negative sign. 
ex. 1) 3 + 5
Step 1)  3  = 3 and  5  = 5
Step 2) 3 + 5 = 8
Step 3) ans: 8 

ex. 2) 7 + 11
Step 1)  7  = 7 and  11  = 11
Step 2) 7 + 11 = 18
Step 3) ans: 18 
SECTION 4.6: ADDING A POSITIVE
AND A NEGATIVE NUMBER
RULE FOR ADDING A POSITIVE AND A NEGATIVE NUMBER
Step 1) Find the absolute value of each number
Step 2) Subtract the smallest absolute value from the largest absolute value.
Step 3) The result in step 2 is given the sign of the argument with the
largest absolute value. 
ex. 1) 11 + 7
step 1)  11  = 11 and  7  = 7
step 2) 11  7 = 4
step 3) Of 11 and 7 in step 1, 11 is larger.
Take the sign of the argument of
 11 .
The argument is 11: the sign is negative.
The negative sign is given
to the result in step 2.
ans: 4 

ex. 2) 6 + 24
step 1)  6  = 6 and  24  = 24
step 2) 24  6 = 18
step 3) Of 6 and 24 in step 1 , 24 is larger.
Take the sign of the argument of
the  24 .
The argument is +24: the sign is positive.
The positive sign is
given to the result in step 2.
ans: +18 or 18 
SECTION 4.7: SUBTRACTING
SIGNED NUMBERS
RULE FOR SUBTRACTING SIGNED NUMBERS
Please note that a "signed number" is a number with
with either a positive or negative sign.
Step 1) Change the sign of the number to the right of subtraction to the
opposite sign.
Step 2) Change the subtraction sign to addition.
Step 3) Now apply the rule of adding signed numbers.
Teacher Comment: I know this rule may seem weird but think about 7  2.
7  2 can be rewritten as 7 + (2) 
ex. 1) 12  11
step 1) 12  11
I changed the sign of the number to the right of subtraction,
11, to the
opposite sign to get 11.
step 2) 12 + 11
I changed the subtraction sign to addition.
step 3) I will use the rules of adding signed numbers to determine 12 + 11.
12 + 11 = 1.
Therefore, 12  11 = 1
ex. 2) 17  21
step 1) 17  21
I changed the sign of the number to the right of subtraction, 21, to the
opposite sign to get 21.
step 2) 17 + 21
I changed the subtraction sign to addition.
step 3) I will use the rules of adding signed numbers to determine 17 + 21.
17 + 21= 4.
Therefore, 17  21 = 4
 AN ASIDE 
DEFINITIONS OF SUM AND
DIFFERENCE
The "sum of two numbers x and y" is the result of adding
the numbers x and y.
ex. ) 5 + 7 = 12 and 12 is the sum
The "difference of two numbers x and y" is the result of
subtracting the number y from x.
ex. ) 12  7 = 5 and 5 is the difference 
SECTION 4.8: RULES FOR
MULTIPLYING AND DIVIDING SIGNED NUMBERS
What is a "product"? It is the result of multiplying numbers.
So, in 4 x 7 = 28,
28 is the product.
MULTIPLICATION RULE FOR MULTIPLYING TWO SIGNED NUMBERS
When multiplying 2 SIGNED numbers:
Step 1) Multiply the absolute values of the numbers
Step 2) If the two ORIGINAL numbers have signs that are
a) positive: the product in step 1) takes a positive sign.
b) negative: the product in step 1) takes a positive sign.
c) opposites: the product in step 1) takes a negative sign 
ex. 1) 5 x 7 
s1) 5 x 7= 35 
s2a) +
35 
ex. 2) 5 x 7 
s1) 5 x 7= 35 
s2b) + 35 
ex. 3) 5 x 7 
s1) 5 x 7= 35 
s2c) 
35 
What is a "quotient"? It is the result of dividing numbers.
So, in 40/5 = 8,
8 is the quotient.
DIVISION RULE FOR DIVIDING TWO SIGNED NUMBERS
When dividing 2 SIGNED numbers:
Step 1) Divide the absolute values of the numbers
Step 2) If the two ORIGINAL numbers have signs that are
a) positive: the quotient in step 1) takes a positive sign.
b) negative: the quotient in step 1) takes a positive sign.
c) opposite: the quotient in step 1) takes a negative sign 
ex. 1) 5/7 
s1) 5 / 7 = 5/7 
s2a) + 5/7 
ex. 2) 5/7 
s1) 5 / 7 = 5/7 
s2b) + 5/7 
ex. 3) 5/7 
s1) 5 / 7 = 5/7 
s2c)  5/7 
SECTION 4.9: INEQUALITY SYMBOLS
Instead of using the word phrases "less than," "greater than," "not equal to,"
"less than or equal to" or "greater than or equal to," mathematicians have
some symbols to take the place of these word phrases. These symbols are called
"inequality symbols."
The inequality symbols are as follows:



THE SUMMER STUDY GUIDE
BY CHAPTERS

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
 CHAPTER 1: THE NUMBER SYSTEM
 CHAPTER 2: ORDER OF OPERATIONS
 CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
 CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
 CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
 CHAPTER 6: PROPERTIES OF NUMBERS
 CHAPTER 7: THE WORLD OF FRACTIONS
 CHAPTER 8: EXPONENTS
 CHAPTER 9: ROOTS
 CHAPTER 10: ALGEBRAIC EXPRESSIONS
 CHAPTER 11: CARTESIAN COORDINATE SYSTEM
 CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
 CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
 CHAPTER 14: XINTERCEPT(ZERO) AND Y INTERCEPT(B)
 CHAPTER 15: LINES
 CHAPTER 16: FUNCTIONS
 CHAPTER 17: MULTIPLYING POLYNOMIALS
 CHAPTER 18: FACTORING
 CHAPTER 19: RATIONAL EXPRESSIONS
 CHAPTER 20: SOLVING EQUATIONS
 CHAPTER 21:SOLVING INEQUALITIES
 CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
 CHAPTER 23: QUADRATICS
 CHAPTER 24: CIRCLES
 CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
 CHAPTER 26: VOLUMES
