SECTION 2.1: OPERATIONS AND PEMDAS
Consider the following: 2 x 3^{2}. What is the answer? Well, there are two possibilities. Watch!
First possible approach 
2 x 3^{2 } 


6^{2 } 

Replace 2 x 3 by 6 
36 

Replace 6^{2} by 36 

OR 
Second possible approach 
2 x 3^{2 } 


2 x 9^{ } 

Replace 3^{2} by 9 
18 

Replace 2 x 9 by 18 

SO WHICH ANSWER IS CORRECT? 36 OR 18?
EDITH CLARKE
February 10, 1883  October 29, 1959
Edith Clarke's achievements were in the applications of mathematics to engineering. She was born in Ellicott City, Maryland in 1883, one of nine children. Both her parents died by the time she was twelve and with the money she inherited when she turned 18, she decided to study mathematics and astronomy at Vassar College. She graduated Phi Beta Kappa in 1908, then taught mathematics and physics, first at a private girls's school in San Francisco and then at Marshall College in West Virginia. Wanting to be an engineer, however, Clarke enrolled in the civil engineering program at the University of Wisconsin in 1911. A summer job as a mathematical computing assistant at AT&T changed those plans, though, as she decided to remain fulltime at AT&T.
Clarke left AT&T in 1918 to study electrical engineering at MIT in Boston. She earned her master's degree in 1919, the first electrical engineering degree ever awarded to a woman at that institution. Her thesis, supervised by A. E. Kennelly, was entitled "Behavior of a lumpy artificial transmission line as the frequency is indefinitely increased." Upon graduation she accepted a job at General Electric where she worked for the next 26 years, except for two years leave as a professor of physics at the Constantinople Women's College in Turkey
Clarke retired from GE in 1945. Two years later she was appointed a full professor at the University of Texas, becoming the first female professor of electrical engineering in the country. She taught there until 1956.
In 1948 Edith Clarke became the first woman elected as a Fellow of the American Institute of Electrical Engineers (now known as the Institute for Electrical and Electronics Engineers, or IEEE). In 1954 she received the Society of Women Engineers Achievement Award "in recognition of her many original contributions to stability theory and circuit analysis."
There are 4 basic operations in mathematics: addition(+),
subtraction(), multiplication(x), and division(÷) and what I call the two "complex
operations": parentheses and exponents. As a doctor operates on a
patient and changes the patient in some way, the six mathematical operations
operate on numbers and change those numbers in some way.
To prevent the type of issue that we saw above, where we got two different answers, mathematicians have agreed there will be an order in which one will compute a "mathematical
expression" such as
"2 x 3^{2}." This order is called PEMDAS and is explained below.
PEMDAS
 P: First, working from left to right, perform all operations inside Parentheses or Brackets.
 E: Second, working from left to right, perform all Exponents or Radicals.
 MD: Third, working from left to right, perform all Multiplication and Division
 AS: Fourth, working from left to right, perform all Addition and Subtraction
IMPORTANT COMMENT: Remember when two parentheses are next to each other as follows, (5)(7), the operation between the parentheses is understood to be multiplication. Hence, (5)(7) means 5 x 7 or 35.
Here are a few examples of PEMDAS:
example 1: 2 x 5 ÷ 4  (3 + 2)^{2} + 9
2 x 5 ÷ 4  (5)^{2} + 9 

Parentheses: (3 + 2) is replaced by 5 
2 x 5 ÷ 4  25 + 9 

Exponents: 5^{2 }is replaced by 25 
10 ÷ 4  25 + 9 

Multiplication:2 x 5 is replaced by 10 
2.5  25 + 9 

Division: 10 ÷ 4 is replaced by 2.5 
22.5 + 9 

Subtraction: 2.5  25 is replaced by 22.5 
13.5 

Addition: 22.5 + 9 is replaced by 13.5 
example 2: 2 x 5(4 ÷ 2  8) + 4 x 7 ÷ 8  3^{ 2}(2 x 4  4^{ 2})
2 x 5(4 ÷ 2  8) + 4 x 7 ÷ 8  3^{ 2}(2 x 4  4^{ 2}) 

2 sets of Parentheses  perform work inside the parentheses (4 ÷ 2  8) which is furthest left. 
2 x 5(2  8) + 4 x 7 ÷ 8  3^{ 2}(2 x 4  4^{ 2}) 

Replaced 4 ÷ 2 by 2 since division before subtraction 
2 x 5(6) + 4 x 7 ÷ 8  3^{ 2}(2 x 4  4^{ 2}) 

Replaced 2  8 with 6 
2 x 5(6) + 4 x 7 ÷ 8  3^{ 2}(2 x 4  16^{}) 

Started working inside (2 x 4  4^{ 2}): exponents firsts; replace 4^{2} with 16 
2 x 5(6) + 4 x 7 ÷ 8  3^{ 2}(8  16^{}) 

Work inside (2 x 4  16^{}). Multiplication before addition therefore replace 2 x 4 with 8 
2 x 5(6) + 4 x 7 ÷ 8  3^{ 2}(8^{}) 

Work inside (8  16^{}). Replace 8 16 with 8 
2 x 5(6) + 4 x 7 ÷ 8  9^{}(8^{}) 

Exponents: Replaced 3^{2} with 9 
10(6) + 4 x 7 ÷ 8  9^{}(8^{}) 

Multiplication and division from left to right. Replace 2 x 5 by 10. 
60 + 4 x 7 ÷ 8  9^{}(8) 

Multiplication and division from left to right. Replace
10(6) by 60 
60 + 28 ÷ 8  9^{}(8) 

Multiplication and division from left to right. Replace
4 x 7 by 28 
60 + 3.5  9^{}(8) 

Multiplication and division from left to right. Replace
28 ÷ 8 by 3.5 
60 + 3.5  72 

Multiplication and division from left to right. Replace
9^{}(8) by 72 
56.5  72 

Addition and subtraction from left to right. Replace 60 + 3.5 by 56.5

56.5 + 72 

Subtraction followed by a negative sign replaced by addition. 
15.5 

Addition: Replaced 55.5 + 72 by 15.5 
TRY THESE  HAVE FUN!
1) (3 × 4 + 2²) ÷ 2 + 6²(3  10 × 2²) ans: 1324
2) 18 ÷ 3(2  7(4  6)²) + (5 + 2²(60  24  7)  11)² ans: 11944



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
