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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 6: PROPERTIES OF REAL NUMBERS

 


SECTION 6.1: COMMUTATIVE PROPERTY OF ADDITION AND MULTIPLICATION

Winifred MerrillWinifred Edgerton Merrill
9/24/1862 - 9/6/1951

Winifred Edgerton, the first American woman to receive a Ph.D. in mathematics, was born in Ripon, Wisconsin. She was a direct descendent of Elder William Brewster of Plymouth Colony. She received her early education from private tutors before earning her B.A. degree from Wellesley College in 1883. After some work at Harvard she was allowed to study mathematics and astronomy at Columbia University. At the end of her second year she petitioned to receive a Ph.D. degree, having fulfilled the required credits and written an original thesis titled "Multiple Integrals" that dealt with geometric interpretations of multiple integrals and translations and relations of various systems of coordinates. Her work in mathematical astronomy included computation of the orbit of the comet of 1883. Despite the support of President Barnard, a campaigner for women's education, the board of trustees refused her application. Barnard suggested that Edgerton personally talk to each trustee. This effort proved successful and at the next meeting the board unanimously voted to award her the Ph.D. in mathematics, which she received in 1886 with highest honors.

Photo Credit: Photograph is used courtesy of Wellesley College Archive

The following two features of real numbers,1 + 5 = 5 + 1 and (7)(2) = (2)(7) are so important that we will give them their own word descriptions. The property displayed in 1 + 5 = 5 + 1 is called the "commutative property of addition." The property displayed in (7)(2) = (2)(7) is called the "commutative property of multiplication."
Both of these properties in effect say that you can switch the order of two numbers and still get the same result.


SECTION 6.2: ASSOCIATIVE PROPERTY OF ADDITION AND MULTIPLICATION

The following two features of real numbers, (1 + 5) + 7 = 1 + (5 + 7) and (7×3)×4 = 7×(3×4) are also so important that we will give them their own word descriptions. The property displayed in (1 + 5) + 7 = 1 + (5 + 7) is called the "associative property of addition." The property displayed in (7×3)×4=7×(3×4) is called the "associative property of multiplication." Both of these properties in effect say that you can switch the position of parentheses from the first pair of numbers to the second pair of numbers and still get the same result. Notice that the order of the numbers DOES NOT change. For example the (1 + 5) + 7 to the left of the equal sign in (1 + 5) + 7 = 1 + (5 + 7) stays in the same positions as 1 + (5 + 7) to the right of the equal sign in (1 + 5) + 7 = 1 + (5 + 7).


SECTION 6.3: DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION

Consider 3(5 + 4). The way that we would evaluate this is via order of operations.

step 1) 3(9) Because 5 and 4 are in parentheses, I would perform the addition of 5 and 4 first to get 9.

step 2) 27 Then I would multiply 3 by 9 to get 27.

Therefore 3(5 + 4) = 27.

Now I'll get the result of 27 in a different way when evaluating 3(5 + 4).

step 1) (3)(5) + (3)(4) :
I took the 3 on the outside of the parentheses in 3(5 + 4), and multiplied it by each term, 5 and 4, which are in the parentheses of 3(5 + 4).
Then I performed the operation that is inside the parentheses, + or addition, on the resulting terms (3)(5) and (3)(4).

step 2)
15 + 12
I multiplied each pair of factors(3)(5) and (3)(4).
step 3) 27 I added the terms.

This longer technique employed above to multiply 3(5 + 4) is called the "distributive property of addition over multiplication."

Here's are a few more examples of this technique:


ex. 1) 7(8 + 12)
QUICK WAY: 7(8 + 12 )→ 7(20) → 140
DISTRIBUTIVE PROPERTY: 7(8 + 12) → 7(8) + 7(12) → 56 + 84 → 140
THIS PROPERTY WILL WORK WITH SUBTRACTION AS WELL!

ex. 2) 7(8 - 12)
QUICK WAY: 7(8 - 12) → 7(-4) → -28
DISTRIBUTIVE PROPERTY: 7(8 - 12) → 7(8) - 7(12) → 56 - 84 → -28


SECTION 6.4: THE IDENTITY PROPERTY OF ADDITION AND MULTIPLICATION

The following characteristic displayed by these three examples, 5 + 0 = 5, -8 + 0 = -8, and 2/3 + 0 = 2/3 is called the "identity property of addition." If you add 0 to any real number then the result will be that real number.

The following characteristic displayed by these three examples, 5(1) = 5, -8(1) = -8 and (2/3)(1) = (2/3) is called the "identity property of multiplication" If you multiply any real number by 1 the result will be that real number.


SECTION 6.5: THE DENSITY PROPERTY OF REAL NUMBERS

Between the numbers 2 and 3, there is another number 2.1. In between the two numbers .00000000001 and .00000000002, there is another number .000000000017.

The "density property of real numbers" states that between any two real numbers there is another number. This property ultimately leads us to the following statement: between any two real numbers there are an infinite number of numbers!

 

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THE SUMMER STUDY GUIDE
BY CHAPTERS

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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: X-INTERCEPT(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